On Wireless Link Scheduling and Flow Control

On Wireless Link Sc heduling and Flo w Con trol A thesis submitted in parti al fulﬁll men t of the requir emen ts for th e deg ree of Do ctor of Phil osoph y b y Ash utosh Deepak Gore (Roll num b er: 0240 7 007) Advisor : Prof. Abha y Karandi k ar Department of El ect rical Eng i neering, Indian Inst itute of T ec hno logy Bo m bay , P o w ai, Mum bai, 4 0007 6 . Decem b er 20 08 Indian Institute of T ec hnology B o m ba y Certiﬁcate of Course W ork This is to certify that Ash utosh Deepak Gore w as admitted to the candidacy of the Ph.D. d egree in Jan uary 2003 after s uccessfully completing all the cours es req uired for the Ph.D. degree programme. The details of the course w ork done are give n b elow: Sr. No. Course code Course name Credits 1 EE708 Information Theory and Co ding 6.00 2 MA402 Algebra 8.00 3 EES801 Seminar 4.00 4 EE659 A First Course in Optimization 6.00 5 EE621 Mark ov Chains and Queueing Syste ms 6.00 6 MA403 Real Analysis 8.00 7 HS699 Comm unication Skills 4.00 I IT Bom bay Date:........ Deput y Registrar (Academic) iii Ac kno wledgmen ts I joined the Ph.D . programme at my alma mater with the in ten t of honing m y kno wledge in net working and wireless comm unicatio ns. In retrosp ect, I feel that I ha v e gained kno wledge in many other domains as w ell. This is primarily due to close in teraction with intellec tuals (both facult y and students ) at I IT Bomba y . A do ctoral thesis can nev er b e pro duced by the thoughts and actions of a single p erson. Rep eated tec hnical discussions, mathematical work outs and sim ulations ar e the ma jor factors that contribute to the “ev olutio n” o f a thesis. In this space, I wish to explicitly thank v arious individuals who ha ve help ed me during m y do ctoral adv en ture. I would lik e to thank m y exub eran t advisor Prof. Abha y Ka r a ndik ar , who has taugh t me engineering in t he tr ue sense of the w ord. His keen insight into the nitti-gritty of ev ery problem and his p erfectionism in tec hnical do cumen tation hav e signiﬁcan tly moulded my g rey matter. I will alw a ys r emember his w ords “A Ph.D. thesis is a piece of sc holarly w ork. It is not a sequence of pap ers stapled tog ether!” I hav e also sharp ened m y kno wledge and pedagogy as a teaching assistan t in v arious courses t augh t b y Prof. Karandik ar. I would lik e to express m y gra t itude to my researc h pro gress committee mem b ers, namely , Prof. H. Nar ay anan, Prof. Harish Pillai and Prof. V arsha Apte. They hav e pro vided v aluable tips a nd guidance throughout m y researc h career. I would esp ecially lik e t o thank Prof. Na ra yanan for encouraging me to pursue a Ph.D. at I IT Bo m bay . I also wish t o t ha nk m y Ph.D. thesis review ers for their insightful commen ts whic h helped to improv e the qualit y of the ﬁna l t hesis. I hav e closely in teracted with man y brigh t p eople a t Info rmation Net works La b ora- tory , which has b een m y second ho me for the past six y ears. In particular, I would like to thank my p eers, Nitin Sa lo dk ar, Heman t Ra th and Punit R a tho d, a nd m y juniors, v Mukul Agarwal and S. Sundhar Ram, for man y a discussion, b oth tec hnical and non- tec hnical. I also wish to thank Srik an th Jagabathula and N. Praneeth Kumar, who ha ve b een my collab or a tors in some of my w ork. I wish to sincerely thank my wife Chaitali for her constan t lov e and supp ort. Our w onderful bab y gir l, b orn on 9 th Decem b er 2008, has infused a lot of energy in me o ver the past few w eeks! My brother Hrishik esh and cousin siste r Namrata ha v e en th used me at v arious stages of m y do ctoral j o urney . My father, Deepak Keshav Gore, and m y mother, Jayshree Deepak Gore, had rec- ognized my pro clivit y for mathematics righ t from my c hildho o d. They did not ﬂinc h a bit when I decided to tread t he oﬀ- b eaten trac k tow ards a Ph.D. Their unconditiona l lo ve, inspiration and ethics hav e b een the pillars of m y motiv ation all a lo ng. This thesis is dedicated to them. Ash utosh D eepak Gore 26 th Decem b er 2008 Abstract This thesis fo cuses on link sche duling in wireles s mesh net works b y taking into accoun t ph ysical la yer characteristics . The assumption made throughout is that a pack et is receiv ed succes sfully only if the Signal to In terference a nd Noise R atio (SINR) a t the receiv er exceeds a certain t hreshold, termed a s communic ation threshold. The thesis also discusse s the complemen t a ry problem of ﬂow con t r ol. First, we consider v arious problems o n cen tralized link sche duling in Spatial Time Division Multiple Access (STDMA) wireless mesh net works . W e motiv ate the use of spatial reuse as p erfor ma nce metric and pro vide an explicit characterization of spatial reuse. W e prop ose link sc heduling algorithms base d on certain graph mo dels (comm uni- cation graph, SINR g r a ph) of the net w ork. Our algorit hms achie v e highe r spatial reuse than that of existing algorit hms, with only a slight incre ase in computat io nal complexit y . Next, w e in v estigate a related scenario in v olving link sc heduling, namely random access algorithms in wireless netw orks. W e assume that the receiv er is capable o f p o wer- based capture and pro p ose a splitting a lgorithm that v aries transmission p ow ers of us ers on the basis of quaternary channel feedbac k. W e mo del the algorithm dynamics b y a Dis- crete Time Marko v Chain and c onsequen tly sho w t ha t its maxim um stable thro ughput is 0.5518. Our alg orithm ac hiev es higher maxim um stable throughput and signiﬁcan tly lo wer dela y than the F ir st Come First Serve (F CFS) splitting algor ithm with uniform transmission p o w er. Finally , w e consider the complemen tary problem of ﬂow control in pac k et net w orks from an information-theoretic p ersp ectiv e. W e deriv e the maximum en trop y of a ﬂow whic h confo rms to traﬃc constraints imp osed by a generalized token buc ket regulato r, b y taking in t o account the co v ert information presen t in the randomness of pac k et lengths. Our results demonstrate that the optimal generalized tok en buc k et regulator has a near uniform buc ke t depth sequence and a decreasing tok en incremen t s equence. vii viii Con ten ts List of Acronym s xiii List of Sym b ols xvii List of T ables xxi List of Figures xxvi 1 In t ro duction 1 1.1 Link Sc heduling in Wireless Net works . . . . . . . . . . . . . . . . . . . . 1 1.2 Motiv ation for the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Ov erview and Contributions of the Thesis . . . . . . . . . . . . . . . . . 7 2 A F ramew ork for Link Sched uling Algorithms for STDMA Wir eless Net w orks 11 2.1 Sys tem Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Link Sc heduling b ased on Proto col In terference Mo del . . . . . . . . . . . 17 2.2.1 Equiv alence of Link Sc heduling and Graph Edge C oloring . . . . . 17 2.2.2 Review of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Limitations of Algorithms based on Prot o col Inte rference Mo del . . . . . 24 2.4 Link Sc heduling based on Comm unication Graph Mo del and SINR Con- ditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Link Sc heduling b ased on SINR G raph Mo del . . . . . . . . . . . . . . . 33 2.6 Spatial Reuse as P erformance M etric . . . . . . . . . . . . . . . . . . . . 35 3 P oin t to P oin t Link Sc heduling based on Comm unication Graph Mo del 39 3.1 ArboricalLinkSch edule Algor it hm Revisited . . . . . . . . . . . . . . . . 40 ix 3.1.1 P erformance Results . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.2 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 A High Spatial Reuse Link Sc heduling Algorithm . . . . . . . . . . . . . 48 3.2.1 Problem F orm ulation . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.2 Motiv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.3 ConﬂictF reeLinkSc hedule Algorithm . . . . . . . . . . . . . . . . 50 3.2.4 P erformance Results . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.5 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 P oin t t o Poin t Link Sche duling based on SINR Graph Mo del 63 4.1 Motiv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 SINRGraphLinkSc hedule Algorithm . . . . . . . . . . . . . . . . . . . . . 65 4.2.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Pe rformance Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.5 D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 P oin t t o Multip oint Link Sc heduling: A Hybrid Approac h 83 5.1 System Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Equiv alence of Link Sc heduling a nd G r aph V ertex Coloring . . . . . . . . 87 5.3 Limitatio ns of Algorithms based on Proto col In terference M o del . . . . . 89 5.4 Problem F ormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5 MaxAv erageSINRSc hedule Algorithm . . . . . . . . . . . . . . . . . . . . 94 5.6 Pe rformance Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.7 Analytical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.8 D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6 A Review of Random A ccess Algorithms for Wireless Netw orks 99 6.1 T raditional Random Access Algorithms . . . . . . . . . . . . . . . . . . . 101 6.2 Signal Pro cessing in Random Access . . . . . . . . . . . . . . . . . . . . 1 08 x 6.3 Channel-Aw are ALOHA Algorithms . . . . . . . . . . . . . . . . . . . . . 109 6.4 Splitting Algo rithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.5 T ow ards P ow er Con trolled Random Access . . . . . . . . . . . . . . . . . 113 7 P o wer Con trolled F CFS Splitting Algor ithm for Wireless Net w or ks 117 7.1 System Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7 7.2 Motiv ation and Problem F orm ula t io n . . . . . . . . . . . . . . . . . . . . 119 7.3 PCFC FS In terv al Splitting Algorithm . . . . . . . . . . . . . . . . . . . . 1 21 7.3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.4 Throughput Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8 Flo w Con tr ol: An Information Theory Viewp oint 147 8.1 System Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 8 8.2 Generalized T ok en Buc k et Regulator . . . . . . . . . . . . . . . . . . . . 151 8.3 Notio n of Information Utility . . . . . . . . . . . . . . . . . . . . . . . . 153 8.4 Problem F ormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8.5.1 Analytical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 58 8.5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.6 Informat io n-Theoretic Inte rpretation . . . . . . . . . . . . . . . . . . . . 164 8.7 D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9 Conclusions 167 A P ro ofs of Limiting T ransition Probabilities 173 A.1 Pro of of (7.54) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 A.2 Pro of of (7.55) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 A.3 Pro of of (7.56) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 A.4 Pro of of (7.57) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5 xi xii List of Acron yms 3GPP L TE 3 r d Generation P a r t nership Pro ject Long T erm Ev olut io n 3GPP2 3 r d Generation P a r t nership Pro ject 2 A CK Ackno wledgmen t ALS Arb oricalLinkSc hedule A W GN Additiv e White Gaussian Noise BS Base Station BS BroadcastSc hedule BT A Basic T ree Algorithm CAA Channel Access Algorithm CDMA Co de D ivision M ultiple Access CFLS ConﬂictF r eeLinkSc hedule CRP Collision Resolution P erio d CSI Channel State Information CSMA/CA Carrier Sense Multiple Access with Collision Avoidance CSMA/CD Carrier Sense Multiple Access with Collision Detection CTS Clear T o Send DTMC D iscrete Time Marko v Chain F CFC FirstConﬂictF reeColor F CFS F irst Come First Serv ed FDMA F requency Division Multiple Access FEC F orward Error Correction GP GreedyPh ysical GTBR Generalized T ok en Buck et Regulato r IETF In ternet Engineering T ask F orce xiii i.i.d. independen t a nd identic ally distributed ISP In ternet S ervice Prov ider LAN Lo cal Area Net w ork LMMSE Linear Minim um Mean Square Erro r OFDM Orthogonal F requency Division Multiplexing PCF CFS P ow er Controlled First Come First Serv ed MA C Medium Access Control MANET Mobile Ad Ho c Netw ork MASC MaxAv erageSINRC olor MASS MaxAv erageSINRSche dule MIMO M ultiple Input Multiple Output MPR MultiP a ck et Rece ption MT A Mo diﬁed T ree Algorit hm NDMA Net work-As sisted Div ersit y Multiple Access NP Non-deterministic P olynomial time p df probabilit y densit y function pmf probabilit y mass f unction QoS Qualit y o f Service R TS Request T o Send SGLS SINR G raphLinkSc hedule SINR Signal to Interferenc e a nd Noise Ratio SLA Service Lev el Agree men t SS Subscriber Station STBR Standard T ok en Bu c ket Regulator STDMA Spatial Time Division Multiple Access TBR T oken Buc k et Regulator TCP T ransmission Control Proto col TDMA Time D ivision Multiple Access TGSA T runcated Graph Based Sc heduling Algorithm VBR V a riable Bit R ate WiMAX W o rldwide Interoperability fo r Micro w av e Acc ess xiv WLAN Wireless Lo cal Area Netw ork WMAN Wireless Metrop olitan Area Net work WMN Wireless Mesh Net w ork xv xvi List of Sy m b ols N n umber o f no des in STDMA wireless net work ( X j , Y j ) Cartesian co ordinates of j th no de ( R j , Θ j ) p olar co ordinates of j th no de P p o w er with whic h a no de transmits its pac k et N 0 thermal noise p o w er sp ectral density β path loss exp onent D ( j, k ) Euclidean distance b etw een no des j and k C n umber o f slots (colors) in STDMA link sc hedule γ c comm unication thres hold γ i in terference thres hold R c comm unication range R i in terference range Φ( · ) STDMA wireless net w ork V set of vertic es E set of directed edges E c set of communication edges E i set of interference edges G c ( V , E c ) comm unication graph represen tation of STDMA net work G ( V , E c ∪ E i ) t wo-tier gra ph represen ta tion of STDMA net w ork Ψ( · ) p oin t to p o int link sc hedule for STDMA net work σ spatial r euse o f p oint to p oin t link sc hedule v n umber o f v ertices in comm unicatio n graph e num b er of edges in comm unicatio n graph θ thickn ess of comm unication g raph xvii ρ maxim um degree of an y v ertex t i,j index of j th transmitter in i th slot r i,j index of j th receiv er in i th slot S i set of t r ansmissions in i th slot of p o int to p oint link sc hedule M i n umber o f concurren t transmitters in i th slot SINR r i,j SINR a t receiv er r i,j SNR r i,j SNR at receiv er r i,j G c ( · ) undirected equiv alen t o f comm unication graph I ( · ) indicator function v i i th v ertex in comm unication or t w o-t ier g r a ph T i i th orien ted graph C ( x ) colour assigned to edge x L ( u ) lab el assigned to vertex u ω maxim um num b er of neigh b ors with low er lab els τ [ k 1 , k 2 ] n um b er of success ful links from slot k 1 to slot k 2 η [ k 1 , k 2 ] n umber o f successful links p er time slot from slot k 1 to slot k 2 G r ( · ) residual subgraph of communication gr aph C set of e xisting color s C c set of conﬂicting color s C 1 set of colo rs with primary edge conﬂict C 2 set of colo rs with secondary edge conﬂict C cf set of conﬂict- f ree colors C nc set of no n-conﬂicting colo r s R radius o f circular deploym en t region V ( · ) fading c ha nnel gain W ( · ) s hado wing c hannel gain measured in b els f X ( x ) probability densit y function of random v ariable X t j j th transmitter in a give n time slot r j j th receiv er in a given time slot M n umber o f concurren t transmiss ions in a given time slot V ′ set of vertice s of SINR graph xviii E ′ set of directed edges of SINR graph G ′ ( V ′ , E ′ ) SINR g raph represen tatio n of STDMA net work w ij in terference w eigh t func tion for edges i, j ∈ E c w ′ ij co-sc hedulability weigh t function for edges i, j ∈ E c N ( v ′ ) normalized noise p ow er for v ertex v ′ ∈ V ′ V ′ uc set of uncolor ed ve rtices of V ′ V ′ c p set of vertice s of V ′ colored with color p C ( v ′ ) color assigned to vertex v ′ ∈ V ′ E ′ t set of directed edges of truncated SINR graph G ′ t ( V ′ , E ′ t ) t r uncated SINR graph V ′ cc set of co- colored v ertices of V ′ Ω( · ) p oint to m ult ip oin t link sc hedule for STDMA netw ork r i,j,k index of k th receiv er of j th transmission in time slot i ς spatial reuse of p o in t to m ultip oin t link sc hedule B i set of t r ansmissions in i th slot of p o int to m ultip oin t link sc hedule SINR r i,j,k SINR a t receiv er r i,j,k η ( j ) n um b er of neigh b ors of no de j C p set of colo rs with primary v ertex conﬂict C s set of colo rs with secondary v ertex conﬂict λ P oisson pac ket arriv al rate D av erage pac k et de la y T throughput T ( k ) left endp oin t o f allo catio n interv al for slot k φ ( k ) length o f allo catio n in terv al for slot k of PCF CFS algorithm φ 0 maxim um size of allo cation inte rv al of PCF CFS alg orithm a i arriv al time o f i th pac ke t d i departure time of i th pac ke t P i ( k ) transmission p o w er of i th pac ke t in slot k P 1 nominal transmission p o w er P 2 higher t ransmission p ow er L left tag xix R righ t tag σ ( k ) tag of allo catio n in terv al in slot k L left allo cation in terv al R righ t allocation in terv al α 0 maxim um size of allo cation inte rv al of FC FS algorithm G i exp ected n um b er of pac kets in an in terv al split i times P A i ,B j transition probabilit y from state ( A, i ) to ( B , j ) x Z n umber o f pac k ets in allo catio n in terv al Z Q X i probabilit y of hitting state ( X , i ) in a CRP K random v ariable denoting num b er of slots in a CRP F random v ariable denoting f r a ction of original allo cation interv al returned to waiting interv al U X i probabilit y that state ( X, i ) has a collision or a capture D exp ected c hange in time ba c klog τ n umber o f slots for whic h algorithm op erates n suc n umber o f successful pack ets in [0 , τ ) r tok en incremen t rate of STBR B buc ke t de pth (maxim um bu rst size) of STBR S n um b er of slots of op eration of TBR r k tok en inc remen t of GTBR in slot k B k buc k et depth of GTBR in slot k ℓ k length o f pack et transmitted b y GTBR in slot k u k n umber o f residual tokens of GTBR at s tart o f slot k r token incremen t se quence of GTBR B buc k et depth sequence of GTBR R s ( · ) standard tok en buc k et regulato r R g ( · ) generalized tok en buc k et regulato r p ℓ k ( u k ) probabilit y of transmitting pac ket of length ℓ k bits with u k residual tokens H k ( u k ) ﬂow en tropy of GTBR in slot k with u k residual tok ens H ∗ k ( u k ) optimal ﬂow en t r op y o f GTBR in slot k with u k residual tokens µ i maxim um num b er of tok ens p ossible in slot i xx List of T ables 2.1 System parameters for STDMA net works sho wn in Figures 2.1(a ), 2.5 and 2.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1 System parameters for the STDMA net work shown in Figure 4.1. . . . . 68 4.2 Interfere nce and co-sc hedulabilit y w eigh t f unctions f or edges of SINR graph shown in Figure 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Normalized noise p ow ers at v ertices of SINR gr aph sho wn in Figure 4 .3 . . 71 4.4 Output of SGLS algorithm for S TDMA net work described b y Figur e 4.1 and T able 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1 System parameters for STDMA net w orks shown in Figures 5.1(a) and 5.4. 87 6.1 T ransmitting a nd w aiting sets for basic tree alg orithm sho wn in Figure 6.1. 104 6.2 T ransmitting and w aiting sets for mo diﬁed tree algorithm sho wn in Figure 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.1 System para meters for p erforma nce ev aluation of PCFC FS and FC FS algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 42 8.1 Entrop y-maximizing G TBR for giv en data transmission time, tok en rate and buc k et depth of a comparable STBR. . . . . . . . . . . . . . . . . . 161 9.1 Link sc heduling algo rithms inv estigated in Chapters 3, 4 a nd 5. . . . . . 169 xxi xxii List of Fi gures 1.1 Wireless mesh net w ork, adapted from [1]. . . . . . . . . . . . . . . . . . . 2 1.2 Poten tial applications of link sc heduling and ﬂow con trol in wireless net- w orks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Example of STDMA net work and p o in t to p oint link sc hedule. . . . . . . 14 2.2 Commu nication g r aph mo del of STDMA netw ork describ ed b y Figure 2.1(a) and T able 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Tw o-tier g r aph mo del of STD MA net w ork des crib ed by Figure 2.1(a) and T able 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Edge coloring of communication graph sho wn in Fig ure 2.2 corresp onding to the link sc hedule sho wn in Figure 2.1(b). . . . . . . . . . . . . . . . . 20 2.5 An STDMA wireless net work with six no des. . . . . . . . . . . . . . . . . 25 2.6 Tw o-tier graph mo del of the STDMA wireless netw ork describ ed by Fig- ure 2 .5 and T able 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Subgraph of tw o-tier graph sho wn in Fig ure 2.6. . . . . . . . . . . . . . . 25 2.8 Coloring of subgraph show n in Figure 2.7. . . . . . . . . . . . . . . . . . 26 2.9 Poin t to p oin t link sc heduling algorithms based on prot o col in terference mo del can lead to high in terf erence. . . . . . . . . . . . . . . . . . . . . . 26 2.10 An STDMA wireless net work with four no des. . . . . . . . . . . . . . . . 27 2.11 Two-tier gr a ph mo del of STDMA wireless netw ork described b y Figure 2.10 a nd T able 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.12 Subgr a ph of tw o-tier graph shown in Figure 2.1 1. . . . . . . . . . . . . . 28 2.13 Colo r ing of subgraph show n in Figure 2.12. . . . . . . . . . . . . . . . . . 28 2.14 Poin t to p oin t link sc heduling algorithms based on p roto col in t erference mo del can lead to higher num b er of colors. . . . . . . . . . . . . . . . . . 29 xxiii 2.15 Alternat ive coloring of subgraph show n in Figure 2.12. . . . . . . . . . . 29 2.16 A p o int to p oint link sch edule corresponding to Figure 2.15 tha t yields lo wer n um b er of colors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1 Sche dule length vs. n umber of no des. . . . . . . . . . . . . . . . . . . . . 44 3.2 Poten tial conﬂicting edges when coloring edge ( u, v ) . . . . . . . . . . . . 46 3.3 Spatial reuse vs. n um b er of no des f o r Exp erimen t 1. . . . . . . . . . . . . 54 3.4 Spatial reuse vs. n um b er of no des f o r Exp erimen t 2. . . . . . . . . . . . . 55 3.5 Spatial reuse vs. n um b er of no des for Exp erimen t 1 under multipath fading and shadowing c ha nnel conditions. . . . . . . . . . . . . . . . . . . 57 3.6 Spatial reuse vs. n um b er of no des for Exp erimen t 2 under multipath fading and shadowing c ha nnel conditions. . . . . . . . . . . . . . . . . . . 57 3.7 Comparison of thic kness and n umber of edges with num b er of v ertices. . 59 4.1 An STDMA wireless net work with fo ur no des. . . . . . . . . . . . . . . . 68 4.2 Commu nication graph mo del of STDMA net w ork des crib ed by Figure 4.1 and T able 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 SINR gra ph mo del of comm unication graph sho wn in F igure 4 .2. . . . . . 70 4.4 T runcated SINR graph deriv ed from SINR graph show n in F igure 4 .3 and w eight v alues giv en in T a bles 4.2 and 4.3. . . . . . . . . . . . . . . . . . . 70 4.5 Coloring of ve rtices of truncated SINR g raph after ﬁrst iteration of SGLS algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Coloring of v ertices of truncated SINR graph af t er second iteration o f SGLS alg o rithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.7 Coloring of vertice s o f truncated SINR graph after third iteratio n o f SGLS algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.8 Coloring of v ertices of truncated SINR graph after complete execution of SGLS alg o rithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.9 Output of SGLS algorithm for S TDMA net work described b y Figur e 4.1 and T able 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.10 Spat ial reuse vs. n um b er of no des fo r Exp erimen t 1. . . . . . . . . . . . . 80 4.11 Spat ial reuse vs. n um b er of no des fo r Exp erimen t 2. . . . . . . . . . . . . 81 xxiv 5.1 Example of STDMA net work and p o in t to m ultip oin t link sc hedule. . . . 85 5.2 Commu nication g r aph mo del of STDMA netw ork describ ed b y Figure 5.1(a) and T able 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3 V ertex coloring of comm unication graph sho wn in Figure 5.2 corresp ond- ing to the link sc hedule sho wn in Figure 5 .1(b). . . . . . . . . . . . . . . 88 5.4 An STDMA wireless net work with six no des. . . . . . . . . . . . . . . . . 90 5.5 Commu nication graph mo del of STDMA net w ork des crib ed by Figure 5.4 and T able 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.6 Coloring of v ertices v 1 and v 4 of gr a ph show n in Figure 5.4. . . . . . . . . 91 5.7 Poin t to m ultip oin t link sc heduling algorithms ba sed on pro t o col interfer- ence mo del can lead to high in terference. . . . . . . . . . . . . . . . . . . 91 5.8 Ave rage spatial reuse vs. num b er of no des for Exp erimen t 1. . . . . . . . 96 5.9 Ave rage spatial reuse vs. num b er of no des for Exp erimen t 2. . . . . . . . 97 6.1 Basic T ree Algo rithm fo r three no des a , b and c . . . . . . . . . . . . . . . 1 0 4 6.2 Stack represen tation of tra nsmitting and w a iting no des for basic tree al- gorithm shown in Figure 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 Mo diﬁed T ree Algorithm for t hr ee nodes a , b a nd c . . . . . . . . . . . . . 105 6.4 Stack represen tat ion of transmitting and w a it ing no des for mo diﬁed tree algorithm shown in Figure 6.3. . . . . . . . . . . . . . . . . . . . . . . . . 106 7.1 PCFC FS splitting algo r ithm illustrating a collision follow ed b y an idle. . 124 7.2 FCFS splitting algorithm illustrating a collision follow ed b y an idle. . . . 125 7.3 PCFC FS splitting algorithm illustrating a collision follow ed b y another collision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.4 FCFS splitting algorithm illustrating a collision follow ed b y another col- lision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 7 7.5 D iscrete Time Mark ov Chain represen ting a CRP of PCF CFS splitting algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.6 Nota t io n for n umber of pack ets in left and right sub in t erv als of the original allo cation in terv al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.7 Plot of ζ v ersus λφ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.8 Throughput ve rsus arriv al rate for PCF CFS and FCFS algorithms. . . . 143 xxv 7.9 Ave rage dela y v ersus arriv al rate f o r PCFCFS and F CFS algorithms. . . 143 7.10 Average p ow er v ersus arriv al rate fo r PCF CFS and FCFS algorithms. . . 144 8.1 Flow con trol o f a source’s pac k ets o v er a pac k et netw ork. . . . . . . . . . 148 8.2 T ok en buc ket regulatio n o f a source’s pac kets ov er a pac k et netw ork. . . . 150 8.3 Relative time instan ts o f parameters deﬁned in (8.4 ) . . . . . . . . . . . . 1 52 8.4 Informat io n utility of GTBR vs. buc k et de pth of comparable STBR. . . . 162 8.5 Informat io n ut ility of GTBR vs. toke n incremen t rate o f comparable STBR. 162 xxvi Chapter 1 In tro duction 1.1 Link Scheduling in Wirel ess Ne t w o rks Wireless and mobile comm unications hav e rev olutionized t he wa y w e comm unicate ov er the past decade. This impact has b een fe lt both in voic e comm unications and wirele ss In ternet access. The eve r-increasing need f o r applicatio ns lik e video and imag es ha v e driv en the need for technologies like 3 r d Generation Partnership Pro j ect Long T erm Ev olution (3GPP L TE), 3 r d Generation P art nership Pro j ect 2 (3GPP2), IEEE 802.1 6 W o rldwide In terop erability for Micro wa v e Access (WiMAX) netw orks and IEEE 802.1 1 Wireless Lo cal Area Netw orks (WLANs) whic h promise broadband data ra tes to wireless users. This revolution in wireless comm unications has had a great impact in India, where the num b er o f cellular su bscrib ers is 250 million (as of Nov em b er 2008) and is gro wing at a ra te of appro ximately 3% p er mon t h [2 ]. Wireless net works can b e broa dly classiﬁed in to cellular netw orks and ad ho c net- w orks. A wireless ad ho c net w ork is a collection of wireless no des that can dynamically self-organize in to an ar bitr ary top ology to form a net work without necessarily using any pre-existing infrastructure. Based on their application, a d ho c net w orks can b e further classiﬁed in to Mobile Ad Ho c Netw orks ( MANETs ), wireless mesh netw orks and wireless sensor net w orks. A wireless mesh net work can b e considered to b e an infrastructure- based ad ho c netw ork with a mesh backbone carrying most of the traﬃc. Wireless Mesh Net w orks (WMNs) hav e b een recen tly adv o cated to prov ide connectiv- it y and co v erage, esp ecially in sparsely p o pula t ed and rural areas. F or example, sev eral 1 2 Chapter 1. In tr o d uction backbone Wireless mesh Internet Sensor network Mesh router Mesh router with gateway/bridge Wireless link Wired link Access point Base station Base station Sink node Wired client Wireless client WLAN Cellular network WiMAX network Figure 1.1: Wireless mesh net w ork, adapted from [1]. Wireless Comm unity Net w orks (W CNs) are op erational in Europe, Au stralia and USA [3]. P eer to p eer wireless tec hnology is also b eing dev elop ed by companies suc h as [4]. WMNs are dynamically self-organized and self-conﬁgured, with no des in the netw ork au- tomatically e stablishing an a d ho c netw ork and main taining me sh connectivit y [1]. An example of a WMN is sho wn in Figure 1.1. T ypically , a WMN compris es of tw o t yp es of no des: mesh routers and mesh clien ts. A mesh router consists of gatew a y/bridge functions and the capability to supp ort mesh net w o r king. Mesh routers hav e little or no mobilit y and form a wireless backbone for mesh clien ts. The gatewa y/bridge func- tionalities in mesh routers aid in the in tegration of WMN s with heterogeneous netw orks suc h as Ethernet [5], cellular netw orks, WLANs [6], WiMAX net works [7] and sensor net works . WMNs are witnessing commercialization in v arious a pplications lik e bro a d- band home netw orks, en terprise netw orks, comm unit y net w orks and metrop olitan area net works . Moreov er, WM Ns dive rsify the functionalities of ad ho c netw orks, ins tead of just b eing another t yp e of a d ho c netw ork. These additional functionalities neces sitate no vel design principles and eﬃcien t algorithms for the realization of WMNs. Signiﬁcan t researc h eﬀorts are required to realize the full p oten tial of WMNs. Among 1.1. Link Sc h ed uling in Wirele ss Net works 3 the man y c hallenging issues in the design of WMNs, the des ign of the phy sical as w ell as the Medium Access Con trol (MA C) la y ers is important, esp ecially from a p ersp ectiv e of achie ving hig h netw ork throughput. At the ph ysical lay er, t echniq ues lik e adaptiv e mo dulation and co ding, Ortho gonal F requency Division Multiplexing (OFD M) [8], [9] and Multiple Input Multiple Output (MIMO) tec hniques [10] can b e used to increase the capacit y of a wireless channel a nd achie v e high data transmission rates. At the MA C lay er, v ario us solutions lik e directional a ntenna ba sed MA C [11], MA C with p ow er con tro l [1 2] and mu lti-c hannel MA C [13] ha v e b een prop osed in the literature. In this thesis, w e primarily focus on the design o f the MAC lay er for wireless mesh net works . W e abstract o ut essen tial features of the MA C and ph ysical la yers of a WMN and prop ose tec hniques that deliv er hig h net w o rk throughput. W e tak e in to account wireless c hannel eﬀects suc h a s propagation path loss, fa ding a nd shado wing [14]. T o- w ards the end of the t hesis, we pro vide an information-theoretic p ersp ectiv e on ﬂow con tro l. The main b o dy o f this thesis, how ev er, fo cuses o n MA C lay er design for tw o t yp es of netw orks: Spat ia l Time Division Multiple Access (STDMA) net works and ran- dom access net w orks. W e next describe these t w o t yp es of net w orks along with their p oten tia l a pplicatio ns in WMNs. An STDMA netw ork can be thought of as a mesh netw ork in w hic h m ultiple trans- mitter receiv er pairs can comm unicate at the same t ime. More sp eciﬁcally , consider a WM N comprising of store-a nd-forw a r d no des connected by “point to p oin t” wireless comm unication c hannels (links). A link is an ordered pair ( t, r ), where t is a t r a nsmitter and r is a receiv er. Time is d ivided into ﬁxed-length interv als called slots. In STDMA, w e allo w concurren t comm unications bet w een collections of no des tha t a re “reasonably far” from e ac h other, i.e., w e exploit spatial reuse. An STDMA link sche dule des crib es the transmission rights for each t ime slot in suc h a w a y that comm unicating entities assigned to the same slot do not “ collide”. In this thesis , we design cen tralized STDMA link s c heduling algorithms that tak e in to accoun t ph ysical lay er c haracteristics suc h as Signal to Interferenc e a nd Noise Ratio (SINR) at a r eceiv er. STDMA link sc heduling alg orithms can b e implemen ted at the MAC la yer of wireless mesh net w orks, as sho wn in Fig ure 1.2. A mesh net w ork can b e constructed with mesh routers and mesh clien ts functioning as relay no des in addition to their sender and receiv er roles. The link sc hedule can b e computed b y a des ignated mesh router a nd then 4 Chapter 1. In tr o d uction Internet network Wireless mesh links scheduled by Wireless BS SS SS SS SS links scheduled by BS flow control at STDMA algorithm multipoint to point network random access algorithm network ingress Figure 1.2: P oten tial applications of link sc heduling and ﬂow con tro l in wireless net works . disseminated to all other no des. The mesh routers fo rm the mesh ba c kb one to prov ide connectivit y to (p ossibly mobile) mesh clients . In a related pro blem in v olving link sc heduling, we consider a m ultip oin t t o p oin t wire- less netw ork with ra ndom access. When random access alg orithms are directly t r anslated from a wired net work to a wireless netw ork, they yield equal or low er throughput. This is b ecause they do not consider the time v ariation of the w ireless c hannel and in terfer- ence conditions at the receiv er. In this thesis, we design a distributed random access algorithm tha t tak es into accoun t wireless c hannel att r ibutes suc h as propaga t ion pa t h loss and phys ical lay er c hara cteristics suc h a s SINR at the receiv er. Random access algor it hms can be applied to the MA C la y er of wireless net works, as sho wn in Figure 1.2. The BS and SSs are organized in to a cell-lik e structure. Both uplink (from SS to BS) and downlin k (from BS to SS) channels are shared among the SSs. This mo de requires all SSs to b e within the communic ation range and line of sigh t of the BS. A ra ndom access algorithm can b e implemen ted in the SSs to resolv e conte n t io ns on the uplink c hannel. In a complemen tary problem, w e consider a pac k et lev el ﬂo w from a source to a destination o v er a data net w ork. The pack ets transmitted b y the source are regulated 1.2. Motiv ation for the Thesis 5 at the ingress of the netw ork, as sho wn in Fig ure 1.2. In this thesis, we inv estigate the maxim um amount of information that can b e transmitted from the source to the destination b y utilizing the idea of co v ert information channe ls. T o summarize, this thesis deals with the design of MAC la yer algorit hms (equiv- alen tly , link sch eduling algorithms) for mesh net w orks. The prop osed link sc heduling algorithms take in to account ph ysical lay er c haracteristics suc h as SINR at a receiv er. Finally , we also consider the problem o f ﬂow con trol. V a rious solutions to t he link sche duling problem hav e b een prop osed in literature dep ending on the mo deling of the wireless net w o r k and in t erference c onditions. In the next section, w e motiv ate our w ork b y brieﬂy outlining the es sen tial diﬀerences b etw een our approa c h and the metho dology of existing approache s. 1.2 Motiv ati o n for the The sis Consider the problem of de termining a link sc hedule for an STDMA w ireless net w ork. STDMA link sc hedules can b e classiﬁed in to p o in t to p oint and p oint to m ult ip oin t link sc hedules. In a p oin t to p oin t link sc hedule, the transmission righ t in eac h slot is assigned to ce rtain links , while in a point to m ultip oin t link sche dule, the transmission righ t in eac h slot is assigned to certain no des. An STDMA sc heduling algo rithm is a set of rules t hat is used to determine a link sc hedule so as to satisfy certain ob jective s. An STDMA link sc hedule should b e so designed that, in ev ery time slot, all pac kets transmitted by the sche duled transmitters are receiv ed succe ssfully at the corresponding (in tended) r eceiv ers. Tw o mo dels hav e b een prop osed in literature f or sp ecifying t he criteria fo r successful pac ke t reception. According to the pro to col in terference mo del [15], a pack et is receiv ed success fully at a r eceiv er only if its in tended transmitter is within the communication range a nd other unin tended transmitters are out side the in terference range of the re- ceiv er. In essence, the pro t o col in terference mo del mandat es a “silence zone” around ev ery sc heduled receiv er in a time slot. On t he other hand, according to the ph ysical in terference mo del [15], a pac k et is rece iv ed successfully at a receiv er only if the SINR at the receiv er is no less than a certain threshold, called comm unication threshold. Throughout this thesis, w e assume that a pac ke t is r eceiv ed successfully if the SINR at 6 Chapter 1. In tr o d uction the receiv er is great er than or equal to the comm unication threshold, i.e., w e employ the ph ysical in t erference model. Moreov er, w e ass ume that, a s long as the SINR threshold condition is satisﬁed at the receiv er of a link, a constan t rat e of dat a transfer o ccurs alo ng that link. In other w ords, the existence of a c hannel co ding tec hnique that guaran tees a ﬁxed data ra te is assumed, when the SINR threshold conditio n is satisﬁed. T o maximize the ag gregate traﬃc transp orted b y an STDMA wirele ss net w ork, most link sc heduling algorithms emplo y the proto col interference model and see k to minimiz e the sc hedule length. These a lg orithms mo del the netw ork b y a comm unication graph and emplo y no v el t echniq ues to color all the edges of the graph using minim um num b er of colors [16]. Suc h approac hes ha ve three lacunae. First, they transform the link sc heduling problem to an edge coloring problem in a graph, whic h is a simpliﬁcation of the true system mo del. Second, they do not incorp orate wireless c hannel eﬀects lik e propagatio n path loss , fading and shadow ing. Finally , they do no t consider SINR threshold conditions at a r eceiv er. In this thesis, w e seek to address these issues by designing p olynomial time link sc heduling algorithms that emplo y the ph ysical in terference mo del, pro vide a reasonably accurate represen tation of the wireless netw ork and a im to maximize the num b er o f success ful pac k et transmis sions p er time s lot. These alg o rithms take into accoun t wire- less c ha nnel eﬀects like propagation path loss, fading a nd shado wing, as w ell a s SINR conditions at a receiv er. W e design and ev aluate algorithms for b oth p oin t to p oin t and p oin t to multipoint link sch eduling. Our work falls under the realm of j o in t PHY-MA C design of wireless net w orks. In a related scenario in v olving link sc heduling, consider t he problem of designing a random access algorithm for a m ultip oin t to p oin t wireless net w ork. When traditional random access algorithms lik e ALOHA [17] and tree-lik e algorithms [18] are emplo y ed in a wire less netw ork, they yield equal or lo we r throughput compared to the wired case. This is b ecause suc h algorithms are incognizan t of wireless c hannel eﬀects and ph ysical la yer c hara cteristics. Th us, it is imp ortant t o design a random access alg o rithm that incorp orates wireless c hannel eﬀects and exploits ﬂexibilities pro vided by the phys ical la yer. T ow ards this step, we assume a rece iv er that is capable of p ow er-based capture [19]. Also, we assume that users can v ary their transmission p ow ers to increase t he c hances of successful pack et reception under the ph ysical in t erference mo del. Conse- 1.3. Ov er v iew and Contributions of the Th esis 7 quen tly , w e design and analyze a v a riable-p o w er tree-like algorithm for a random acces s wireless net work. In the ﬁnal scenario, w e fo rm ulate the problem of analyzing ﬂo w con tr o l in pack et net works fro m an info rmation-theoretic p ersp ectiv e. W e fo cus on the problem of a na lyz- ing regulated ﬂo ws in a point to p oin t net work. It is we ll-kno wn that info r mation (in the Shannon sense) can b e transmitted from a source to a destination only b y enco ding it in the conten ts, lengths and timings of dat a pack ets from the source to the destination [20], [21]. W e inv estigate the maxim um amoun t of information that can b e tra nsmitted by a source whose ﬂow is linearly bounded. Sp eciﬁcally , w e assume that c o v ert informatio n is con vey ed b y randomness in pac k et lengths and in v estigate prop erties of the regulating mec hanism that leads to maximum informatio n transfer. 1.3 Ov erview and C on trib u tions of the Thesis In the ﬁrst part of the thesis (Chapters 2 to 5 ), we consider v arious problems o n cen tral- ized link sc heduling in STDMA wireless netw orks; eac h problem represen ts a diﬀeren t n uance o f t he ov erall link sc heduling problem. In the second par t of the thesis (Chapters 6 and 7), w e consider a related link sc heduling problem, namely , distributed medium access control in a random access wireless net work. In the third and ﬁnal par t of the thesis (Chapter 8), w e consider ﬂo w con trol in net w o rks from an information-theoretic p ersp ectiv e. Chapter 2 presen ts a generic framework a nd system mo del for link sc heduling in STDMA wireless net works. W e describ e the system parameters of an STDMA wire- less net w ork and explain t wo prev a len t mo dels used to specify the criteria for successful pac ke t reception, namely proto col in terference mo del and ph ysical interference mo del [15]. W e argue t ha t STDMA link sc heduling algorit hms can b e classiﬁed in to three classes: alg o rithms based on mo deling the net w or k b y a tw o-tier o r commu nication graph, “h ybrid” a lgorithms based on mo deling the netw ork b y a commu nication graph and v erifying SINR conditions and algorithms based o n mo deling the netw ork by a n SINR graph. W e review repres en tative researc h pap ers from eac h of these classes. W e explain the relativ e merits and demerits of eac h class o f alg o rithms in terms of com- putational complexit y , p erformance and a ccuracy of the netw ork mo del. W e discuss 8 Chapter 1. In tr o d uction limitations of link sc heduling algorithms based only on the comm unication gra ph mo del b y pro viding illustrative examples. Finally , to compar e the p erfor ma nce of v arious link sc heduling algorithms, w e motiv ate and introduce spatial reuse as a p erformance metric. V a rious “spinoﬀs” of the “paren t ” link sc heduling problem constitute the subproblem s considered in Chapters 3 , 4 and 5. In Chapter 3, we consider STDMA p oin t to p oin t link sc heduling algorithms which utilize a communic ation graph represen tation of the netw ork. Initially , w e examine the Arb oricalLinkSc hedule (ALS) algo rithm [16], whic h represen ts the netw ork by a com- m unication graph, partitions the graph into minim um n um b er of planar subgraphs a nd colors each subgraph in a greedy manner. W e suggest a mo diﬁcation to the ALS al- gorithm based on reusing colors from previously c olored subgraphs to color the current subgraph. W e compare the p erformance of the mo diﬁed algorithm with the ALS algo- rithm and derive its running time complexit y . Subsequen tly , we prop ose the Conﬂict- F reeLinkSc hedule algor ithm, whic h is a h ybrid algorithm based on the comm unication graph and v erifying SINR conditions. Under v arious wireless c hannel conditions, we demonstrate that ConﬂictF reeLinkSc hedule ac hieve s higher spatial reuse than existing link sc heduling algorithms based on the c omm unication graph. Ho w ev er, this impro v e- men t in perf o rmance is achiev ed at a cost of sligh tly higher computational complexit y . In Chapter 4, w e consider the p oin t to p oint link sc heduling pro blem under the ph ysical in terference mo del. The STDMA net work is represen ted b y an SINR g raph, in whic h w eights of edges correspo nd to interferenc es b et w een pairs of nodes and w eigh ts of ve rtices corresp ond to normalized noise p ow ers at receiving no des. W e pro p ose a link sc heduling algorithm base d on the SINR graph represen tatio n of the net w ork. W e pro ve the correctness of the algorithm a nd sho w that it has p olynomial running time complexit y . F ina lly , w e demonstrate t hat the prop osed algorithm achiev es higher spatial reuse than ConﬂictF reeLinkSc hedule. In Chapter 5, w e consider p oin t to multipoint link sc heduling (broadcast sc heduling) under the ph ysical in t erference mo del. The problem addressed herein can b e considered as the “dual” of the problem considered in Chapters 3 and 4. W e generalize the deﬁnition of spatial reuse to the p oint to m ultip o in t link sc heduling problem. W e prop o se a greedy sc heduling algorithm whic h has demonstrably hig her spatial reuse than ex isting algorithms, without an y increase in computational complexity . 1.3. Ov er v iew and Contributions of the Th esis 9 In Chapter 6, w e consider a no ther ﬂa vor of the link sc heduling problem, namely random access algorithms for wireless net w orks. While random access algorithms f or satellite net w or ks, pac k et radio net w orks, multidrop telephone lines a nd m ultita p bus (“traditional random access alg orithms”) is a w ell-researc hed and mature sub ject, the study of random acce ss algorithms for wireless net works that tak e in to account ph ysical la yer c haracteristics suc h as SINR a nd channe l v ariations has y et to gain momen tum. This chapter reviews represen tativ e researc h w o rk whic h in v estigate suc h random access algorithms, most of them b eing generalizations of the ALOHA proto col (b y ada pting the retransmiss ion probability) or the tree algorithm (b y adapting the set of con tending users). W e motiv ate the use of v ariable transmis sion p o wer to increase the throughput in r andom access wireless netw orks. W e consider random access for wireless net w orks under the phy sical in terference mo del in Chapter 7. W e design an algorithm that adapts the set of contending users and their corresp onding tra nsmission p ow ers based on quaternary (2 bit) c hannel f eedback . W e mo del the algo rithm dynamics by a Discrete Time Marko v Chain and subsequen tly deriv e its maximum stable throughput. Finally , w e demonstrate tha t the prop o sed algorithm a c hiev es higher throughput and substan tially lo w er dela y than the w ell-kno wn First Come Fir st Serve splitting algorithm [22]. In Chapter 8, w e form ulate the problem of analyzing ﬂo w con tro l in pac k et netw orks from a p erspective of maximizing m utual information b et w een a source and a destination. W e fo cus on the simpler, ye t insightful, problem of analyzing regulated ﬂo ws in a p oint to p oin t netw ork. More sp eciﬁcally , w e consider a source whose ﬂo w is b ounded b y a “generalized” T ok en Buc ket Regulator ( TBR) and analyze t he maxim um amount of information (in the Shannon sense) that the source can conv ey to it s destination by enco ding information in the ra ndomness of pack et lengths. This c hapter rev eals tw o in teresting r esults. First, under certain “bandwid th” constrain ts on cum ulativ e tok ens and cumu lativ e buck et depth, we demonstrate that a generalized TBR can achie v e higher ﬂo w en trop y than that of a standard TBR. Second, we provide infor mation-theoretic argumen ts for the observ ations that the optimal g eneralized TBR has a decreasing tok en incremen t se quence and a near- uniform buc k et depth sequence. In Chapter 9, we summarize the thesis a nd provide p ossible directions for future w ork. Sp eciﬁcally , w e suggest generalizations of the tw o-lev el p ow er control algorithm 10 Chapter 1. In tr o d uction prop osed in Chapter 7. W e also prov ide p ointers for deriving the appro ximation fa cto r s of the algo rithms prop o sed in Chapters 3 and 4. Chapter 2 A F ramew ork for L i nk Sc heduli ng Algorithms for STDMA Wireles s Net w orks An STDMA wireless ne t work cons ists of a ﬁnite set of no des wherein multiple p airs of no des can comm unicate concurrently , as discusse d in Chapter 1. In t his c hapter, we outline a framew o rk for mo deling STDMA link sc heduling algorithms. W e consider a general represen tatio n of an STDMA wireless netw ork, i.e., this mo del is not sp eciﬁc to an y technology or proto col. This abstraction lends simplicit y to the net w ork mo del and helps us fo cus on the de sign of sc heduling alg orithms fo r the netw ork. Since the problem of determining an optimal link sche dule is NP-hard [16], researc hers ha v e prop osed v arious heuristics to obtain close -to-optima l solutions. In our view, su c h heuristics can b e br o adly classiﬁed in to three categories: algorithms based on mo deling the net work b y a tw o-tier or comm unication graph, “hy brid” algorithms based on modeling the netw ork b y a comm unication graph and v erifying SINR conditions and algorithms based on mo deling the net work b y an SINR graph. W e review represen tat ive researc h pap ers f r o m eac h o f these classes. The relative merits and demerits of eac h class of algor ithms are also elucidated in the chapter. Our observ ations motiv ate us to prop ose a p erformance metric that is prop ortional t o agg r egate net w ork throughput. The r est of this c hapter is structured as fo llo ws. In Section 2.1, we describe the system mo del o f an STDMA wireless netw ork and explain the proto col and phy sical 11 12 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks in terference mo dels. In Section 2.2, we elucidate the equiv alence b et we en a p oint t o p oin t link sche dule for an STDMA net w ork and the colo rs of edges o f the comm unication gr a ph mo del of the netw ork. This is follo w ed b y a review of researc h w ork on po in t to p oin t link sc heduling a lgorithms based on the proto col inte rference mo del. In Section 2.3, w e describe the limitations of algo r it hms based on the p roto col in terference mo del from a p ersp ectiv e of maximizing net work throughput in wireless net w orks. W e review rese arc h w ork on link sc heduling algorithms based on the physic al interferenc e mo del in Sections 2.4 and 2 .5. Sp eciﬁcally , Section 2.4 reviews algorithms ba sed on comm unication gra ph mo del of the net w ork and SINR conditions, while Section 2.5 rev iews algorithms bas ed on an SINR gr a ph mo del of the net work. F inally , in Section 2.6, we prop o se spatial reuse as a p erfo rmance metric and argue that it corresp onds to net work throug hput from a ph ysical lay er viewp oin t. 2.1 System Mo del W e consider a general mo del of an STDMA wireles s netw ork with N static store-a nd- forw a rd no des in a t w o- dimensional plane, where N is a p ositiv e in teger. No des are indexed as 1 , 2 , . . . , N . In a wireless netw ork, a link is an ordered pair of no des ( t, r ), where t is a transmitter and r is a r eceiv er. W e a ssume equal length pac k ets. Time is div ided in to slots of equal duration. Dur ing a time slot, a no de can either transmit, receiv e or remain idle. The slot duration equals the amoun t of time it tak es to transmit one pa c k et ov er the wireless c hannel. W e make the following additional assumptions: • Sync hronized no des: All no des are sync hronized to slot b o undaries. • Homogeneous no des: Ev ery no de has iden t ical receiv er sensitivit y , transmiss ion p o w er and thermal noise characteris tics. • Bac klogged no des: W e a ssume a no de to be con tin uously bac klogged, i.e., a no de alw ays has a pac k et to transmit a nd cannot transmit more than one pac k et in a time slot. 2.1. System Mod el 13 Let: ( x j , y j ) = Cartesian co ordinates of no de j =: r j , P = p ow er with whic h a no de transmits its pack et , N 0 = thermal noise p o w er sp ectral densit y , D ( j, k ) = Euclidean distance bet w een no des j and k . The receiv ed signal p o wer at a distance D from the transmitter is given by P D β , where β is the path lo ss exp onen t 1 . An STDMA link sc hedule is a mapping fr o m the set of links to time slots. W e only consider static link sche dules, i.e., link sch edules that rep eat p erio dically throughout the op eration of the netw ork. Let C denote the n um b er of time slots in a link sc hedule, i.e., the sche dule length. F or a g iven time slot i , j th comm unicating transmitter-receiv er pair is denoted b y t i,j → r i,j , where t i,j denotes the index of the no de whic h transmits a pack et and r i,j denotes t he index of the node whic h receiv es the pac ket. Let M i denote the n umber of concurren t transmitter-receiv er pairs in time slot i . A p oint to p oint link sc hedule for the STDMA netw ork is denoted b y Ψ( S 1 , · · · , S C ), where S i := { t i, 1 → r i, 1 , · · · , t i,M i → r i,M i } = set of transmitter-receiv er pairs whic h c an comm unicate concurren tly in t ime slot i. Note that a link sc hedule rep eats p erio dically thro ug ho ut the op eration o f the net work. More sp eciﬁcally , tra nsmitter-receiv er pairs that communicate concurren tly in time slot i also comm unicate concurren tly in time slots i + C , i + 2 C and so o n. Th us, S i = S i (mo d C ) . Finally , no t e that all transmitters and receiv ers are stationary . Ev ery point to p oin t link sch edule mus t satisfy the follow ing: • Op erational constraint: During a time slot, a no de can transmit to exactly one no de, receiv e from exactly one no de or remain idle, i.e., { t i,j , r i,j } ∩ { t i,k , r i,k } = φ ∀ i = 1 , . . . , C ∀ 1 6 j < k 6 M i . (2.1) 1 W e do not cons ider fading and s hadowing eﬀects. 14 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks Y X 1 ≡ ( − 40 , 5) 2 ≡ (0 , 0) 3 ≡ (95 , 0) 4 ≡ (135 , 0) 5 ≡ ( − 75 , 0) 6 ≡ (0 , − 75) (a) An STDMA wireless netw ork with six nodes. 1 → 2 time 1 2 3 4 5 6 7 8 1 2 3 4 time slo t s 5 → 1 · · · 3 → 4 3 → 2 4 → 3 6 → 2 1 → 5 2 → 5 2 → 1 1 → 6 5 → 2 2 → 3 1 → 2 3 → 4 5 → 1 3 → 2 4 → 3 6 → 2 1 → 5 · · · po i n t to p oint li nk schedule transmitter -receiver pairs 2 → 6 6 → 1 (b) A point to point link sc hedule for the netw o rk shown in Figur e 2.1(a). Figure 2.1: Example of STDMA net work a nd p o in t to p oint link sc hedule. 2.1. System Mod el 15 As an illustration, consider the STDMA wireles s netw ork sho wn in Figure 2.1(a). It consists of six no des whose co o r dina t es (in meters) are 1 ≡ ( − 40 , 5), 2 ≡ (0 , 0), 3 ≡ (95 , 0 ), 4 ≡ (135 , 0 ) , 5 ≡ ( − 75 , 0 ) and 6 ≡ ( 0 , − 75). An example p oint to p o in t link sc hedule for this STDMA netw ork is sho wn in F ig ure 2.1(b). Note that this sc hed- ule is only one o f the sev eral p ossible sc hedules and is give n here o nly for illustrative purp oses. The sche dule length is C = 8 time slots and the sc hedule is deﬁned by Ψ( S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , S 7 , S 8 ), where S 1 = { t 1 , 1 → r 1 , 1 } = { 1 → 2 } , S 2 = { t 2 , 1 → r 2 , 1 , t 2 , 2 → r 2 , 2 , t 2 , 3 → r 2 , 3 } = { 3 → 4 , 5 → 1 , 2 → 6 } , S 3 = { t 3 , 1 → r 3 , 1 } = { 3 → 2 } , S 4 = { t 4 , 1 → r 4 , 1 , t 4 , 2 → r 4 , 2 , t 4 , 3 → r 4 , 3 } = { 4 → 3 , 6 → 2 , 1 → 5 } , S 5 = { t 5 , 1 → r 5 , 1 , t 5 , 2 → r 5 , 2 } = { 2 → 5 , 6 → 1 } , S 6 = { t 6 , 1 → r 6 , 1 } = { 2 → 1 } , S 7 = { t 7 , 1 → r 7 , 1 , t 7 , 2 → r 7 , 2 } = { 1 → 6 , 5 → 2 } , S 8 = { t 8 , 1 → r 8 , 1 } = { 2 → 3 } . After 8 time slots, the sc hedule rep eats p erio dically , as sho wn in F igure 2.1(b). A sc heduling a lgorithm is a set of rules that is used to determine a link sche dule Ψ( · ). Usually , a sc heduling algorithm needs to satisfy certain ob jectiv es. Consider j th receiv er in time slot i , i.e., r eceiv er r i,j . The p o w er receiv ed at r i,j from its in tended transmitter t i,j (signal p o we r) is P D β ( t i,j ,r i,j ) . Similarly , the p ow er receiv ed at r i,j from its u nin tended transmitters (in terference p ow er) is P M i k =1 k 6 = j P D β ( t i,k ,r i,j ) . Th us, the 16 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks Signal to Interferenc e a nd Noise Ratio (SINR) at receiv er r i,j is given by SINR r i,j = P D β ( t i,j ,r i,j ) N 0 + P M i k =1 k 6 = j P D β ( t i,k ,r i,j ) . (2.2) Without cons idering t he inte rference p o wer, the Signa l to Noise Ratio (SNR) at receiv er r i,j is given b y SNR r i,j = P N 0 D β ( t i,j , r i,j ) . (2.3) According to the pr oto c ol interfer enc e mo del [15], tra nsmission t i,j → r i,j is successful if: 1. the SNR at receiv er r i,j is no less than a certain threshold γ c , termed as the c ommunic ation thr eshold . F rom (2.3) , this tra nslates to D ( t i,j , r i,j ) 6  P N 0 γ c  1 β =: R c , (2.4) where R c is termed a s commu nication rang e, and 2. the signa l from an y unin tended transmitter t i,k is rece iv ed at r i,j with an SNR less than a certain threshold γ i , termed as the interfer enc e thr esh o ld . F rom ( 2.3), this translates to D ( t i,k , r i,j ) >  P N 0 γ i  1 β =: R i ∀ k = 1 , . . . , M i , k 6 = j, (2.5) where R i is termed as in terference range. In essence, the transmission on a link is succe ssful if the dis tance betw een the no des is less than or equal to the c omm unic ation r ange and no other no de is transmitting within the in terfer enc e r ange from the receiv er. The STDMA net work is denoted by Φ( N , ( r 1 , . . . , r N ) , P , γ c , γ i , β , N 0 ). No te that 0 < γ i < γ c , th us R i > R c . The relation R i = 2 R c is widely assume d in lit era t ur e [23], [24], [25 ], [2 6]. According to the physic al interfer enc e mo del [1 5], the transmission on a link is suc- cessful if t he SINR at the receiv er is g reater than or equal to the communication threshold 2.2. Link Sc h ed uling based on Pr otocol Interference Model 17 γ c . More sp eciﬁcally , the ph ysical in terference mo del states that transmission t i,j → r i,j is successful if: P D β ( t i,j ,r i,j ) N 0 + P M i k =1 k 6 = j P D β ( t i,k ,r i,j ) > γ c . (2.6) Note that the phy sical in terference mo del is less restrictiv e but more complex. Usually , this represen tation has b een emplo y ed t o mo del mesh netw orks with TDMA lik e access mec hanisms [2 7 ]. W e will dis cuss this asp ect la ter in the thesis. A p o in t to p oin t link sc hedule Ψ( · ) is c onﬂict-fr e e if the SINR at ev ery intende d receiv er does not dro p b elow the comm unication thresh old, i.e., SINR r i,j > γ c ∀ i = 1 , . . . , C , ∀ j = 1 , . . . , M i . (2.7) 2.2 Link Sc h eduling based on Proto col In terferenc e Mo de l 2.2.1 Equiv alence of Link Sc heduling and G r aph Edge Coloring In this se ction, w e describ e the comm unication a nd t w o - tier graph repres en tatio ns of an STDMA wireless netw ork. W e explain the equiv alence b etw een a p o int to p oint link sc hedule for the STDMA netw ork and the colors of e dges of t he comm unication gra ph represen tation o f the net w ork, and illustrate this equiv alence with an example. The STDMA net work Φ( · ) can b e mo deled b y a directed graph G ( V , E ), where V is the set of v ertices and E is the set of edges. Let V = { v 1 , v 2 , . . . , v N } , where vertex v j represen ts no de j in Φ( · ). In the graph represen tation, if no de k is within no de j ’s comm unication range, then there is an edge from v j to v k , denoted b y v j c → v k and termed as communic ation edge. Similarly , if no de k is outside no de j ’s comm unicatio n range but within its in terference range, then there is an edge from v j to v k , denoted by v j i → v k and termed as in terference edge. Th us, E = E c ∪ E i , where E c and E i denote the set of comm unication and inte rference edges resp ectiv ely . The two-tier gr a ph represen tation of the STDMA netw ork Φ( · ) is deﬁned as the graph G ( V , E c ∪ E i ) comprising of all v ertices and b oth comm unicatio n and in terference edges. The c ommunic ation gr aph represen tation of the STDMA net w ork Φ( · ) is deﬁned as the gra ph G c ( V , E c ) comprising 18 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks of all v ertices and comm unication edges only . W e will illustrate these represen tations with an example. P arameter Sym b ol V alue transmission p o w er P 10 mW path loss exp onent β 4 noise p ow er sp ectral densit y N 0 -90 dBm comm unication thres hold γ c 20 dB in terference threshold γ i 10 dB T able 2.1: System parameters for STDMA netw orks sho wn in Figures 2.1(a), 2.5 and 2.10. v 3 v 4 v 2 v 5 v 1 v 6 Figure 2.2: Comm unication g raph mo del of STDMA netw ork describ ed by F ig ure 2.1 ( a ) and T a ble 2.1 . Consider the STDMA wireless netw ork Φ( · ) whose deplo ymen t is show n in Figure 2.1(a). The system par a meters for this net w ork are giv en in T a ble 2.1. F rom (2.4) a nd (2.5), it can b e easily shown that R c = 100 m and R i = 177 . 8 m. The corresp onding comm unication graph represen tation G c ( V , E c ) is sho wn in Figure 2.2. The comm uni- cation graph comprises of 6 vertice s and 14 directed comm unication edges. The v ertex 2.2. Link Sc h ed uling based on Pr otocol Interference Model 19 v 3 v 4 v 2 v 5 v 1 v 6 Figure 2.3: Tw o-tier g raph mo del of STDMA netw ork describ ed b y Figure 2.1(a) and T able 2.1 . and comm unication ed ge sets are given by V = { v 1 , v 2 , v 3 , v 4 , v 5 , v 6 } , (2.8) E c = { v 1 c → v 2 , v 2 c → v 1 , v 1 c → v 5 , v 5 c → v 1 , v 1 c → v 6 , v 6 c → v 1 , v 2 c → v 5 , v 5 c → v 2 , v 2 c → v 6 , v 6 c → v 2 , v 2 c → v 3 , v 3 c → v 2 , v 3 c → v 4 , v 4 c → v 3 } . (2.9) The tw o-tier graph mo del G ( V , E c ∪ E i ) of the STDMA netw ork Φ( · ) is sho wn in Figure 2.3. The t w o-t ier graph comprises of 6 v ertices, 14 directed comm unication edges and 10 directed in terference edges. The verte x and comm unication edge sets are giv en by (2.8) a nd (2.9) resp ectiv ely , while the in terference edge set is give n by E i = { v 1 i → v 4 , v 4 i → v 1 , v 2 i → v 4 , v 4 i → v 2 , v 3 i → v 6 , v 6 i → v 3 , v 4 i → v 6 , v 6 i → v 4 , v 5 i → v 6 , v 6 i → v 5 , } . (2.10) Giv en the ab ov e r epresen tations, a p oin t to p oin t link sc hedule Ψ( · ) for an STDMA wireless netw ork Φ( · ) can b e considered as equiv alen t to assigning a unique color to ev ery edge in the communic ation gra ph, suc h that transmitter-receiv er pairs with the same color transmit sim ultaneously in a par t icular time slot. F or the example net w o r k considered, the link sc hedule sho wn in Figure 2.1(b) corresp onds to the coloring of the edges o f the comm unication graph sho wn in Figure 2.4. Time slots 1, 2 , 3, 4, 5, 6, 7 and 8 in Ψ( · ) corresp ond to colors red, blue, green, magen ta, y ello w, cy an, bro wn and gold in 20 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks v 5 v 1 v 2 v 3 v 4 v 6 Figure 2.4: Edge coloring of comm unication graph sho wn in Figure 2.2 corr esp onding to the link sch edule show n in Figure 2.1(b). E c resp ectiv ely . Note that a coloring algorithm that uses the least nu m b er of colors also minimizes the sc hedule length. This asp ect is further addressed in subseque n t sections. 2.2.2 Review of Algorithms In this section, we pro vide an ov erview of past researc h in the ﬁeld of STDMA p oint to p oin t link sc heduling algorithms based o n the protocol in terference model. The proto col in terference mo del is widely studied in literature because of its simplicit y . It has b een usually employ ed to mo del net works suc h as Carrier Sense Multiple Access with Collision Av oidance ( CSMA/CA) based WLANs 2 [27], [25 ]. Cen tralized a lgorithms [16], [28 ], [29], [30], [25] as w ell as distributed algorithms [31] hav e b een pro p osed for generating link sc hedules base d on the proto col in terference mo del. A link sc heduling algorithm based on the proto col in terference mo del utilizes a com- m unication or t w o- tier graph mo del of the STDMA net w ork to determine a p oin t to p oin t link sc hedule [32], [33]. Algorithms based on t he proto col in terference mo del for assigning links to time slots (equiv alently , colors) require that t wo comm unication edges 2 Consider an IEE E 802.11 based WLAN wherein CSMA with R TS/CTS/ACK is used to protec t unicast transmissions. Due to carrier s ensing, a transmission betw ee n no des j and k may blo ck all transmissions that ar e within a distance of R i from either j (due to sens ing R TS a nd DA T A) or k (due to sensing CTS and ACK). 2.2. Link Sc h ed uling based on Pr otocol Interference Model 21 v i c → v j and v k c → v l can b e colored the same if a nd only if: i. v ertices v i , v j , v k , v l are all m utua lly distinct, i.e., there is no primary e dge c onﬂict, and ii. v i → v l 6∈ G ( · ) and v k → v j 6∈ G ( · ) , i.e, there is no se c ondary e dge c onﬂict . The ﬁrst criterion is based on the op erational constraint (2.1). The second criterion states that a no de cannot receiv e a pa c k et if it lies within the interfere nce range of an y other tra nsmitting no de. A sch eduling alg orithm utilizes v arious graph coloring metho dologies t o o btain a non-conﬂicting link sc hedule, i.e., a link sc hedule dev oid of primary a nd secondary edge conﬂicts. T o maximize the throughput of a n ST DMA net w o rk, algorithms based on the pro- to col in terference mo del 3 seek to minimiz e the total n um b er of colors used to color all the communic ation edges of G ( · ). This will in turn minimize the sc hedule length. It is w ell known that for an arbitr a ry comm unication graph, the problem of determining a minim um length sc hedule (optimal sc hedule) is NP-hard [16], [29]. Hence, the approac h follo w ed in the lit era t ure is to devise algorithms t ha t pro duce close to optimal (sub- optimal) solutions. The eﬃciency of a sub-optimal algorithm is typic ally measured in terms of its computational (run time) complexit y and p erformance guarantee (appro xi- mation factor). The concept o f STDMA for wireless net works w as fo rmalized in [28]. The authors assume a m ultihop pac k et radio net work with ﬁxed no de lo cat io ns and consider the problem of a ssigning an inte gral n um b er of slots to ev ery link in an STDMA cycle (frame). T o solv e this problem, they mo del the net work b y a commun ication graph, determine a s et of maximal clique s and then ass ign a certain num b er o f slots to a ll the links in eac h maximal clique. Finally , the authors dev elop a ﬂuid approximation for the mean system d ela y and v alidate it using sim ula tions. In [29], the authors consider pre-sp eciﬁed link demands in a spread spectrum pac ke t radio netw ork. They formulate the problem a s a linear o ptimization problem and use 3 Link scheduling algorithms base d on the pro to co l interference model ar e sometimes referr e d to as “ graph based algorithms” in literature [3 2], [33]. This t erm is slightly confusing since scheduling algorithms based on the physical in terference model also construct graphs prior to determining a link schedule. 22 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks the ellipsoid algorithm [34] to solv e the problem. They assume tha t the desired link data rates ar e rational n um b ers and dev elop a strongly p olynomial algorithm 4 that computes a minim um length sc hedule. Finally , they consider t he problem of link sc heduling to satisfy pre-sp eciﬁed end-to- end demands in the netw ork. They formu late this problem as a m ulticommo dit y ﬂow problem and describ e a p olynomial time algorithm that computes a minimum length sche dule. As p oin ted out by the autho rs, their a lg orithm is not practical due to its high computational complexit y . A signiﬁcan t w ork in link sc heduling under proto col interference mo del is rep orted in [16], in which the authors show that tree net works can b e sche duled optimally , oriented graphs 5 can b e sch eduled near-o pt ima lly and arbitrary netw orks can b e sc heduled s uc h that the sc hedule is b ounded b y a length pr o p ortional to the graph thic kness 6 times the optim um n um b er of colors. In [16], the autho rs s eem to hav e missed a subtle p oin t that colors from previously colored orien ted graphs c an b e used to color the curren t orien ted graph. Instead, they use a fr esh set of color s to color each success iv e orien t ed graph. Consequen tly , their algorithm leads to a higher n umbers of colors, es p ecially if the n um b er of orien ted gra phs is larg e. The authors emp lo y suc h a heuristic primarily to upp er b ound the n um b er of colors used by the algorithm ([16], Lemma 3 .4 ) and consequen tly obtain b ounds on the running time complexit y and p erfor mance guar an tee of the algorithm ([16], Theorem 3.3). Though the Arb or icalLinkSc hedule algorithm has nice theoretical prop erties suc h as low computational complexit y , it can b e shown that it may yield a higher num b er o f colors in pr actic e . This leads to lo wer netw ork throughput. W e should p oin t out here that, if w e modif y the ArboricalL inkSche dule algorithm to r euse color s from prev iously colored oriented graphs to colo r the current oriented graph, then the s c hedule length will alwa ys b e lower t ha n the sc hedule length o btained by the 4 An a lgorithm is stro ngly p olynomial if (a ) the num b er of arithmetic op erations (addition, multi- plication, divis ion or co mparison) is polynomially bounded b y the dimension of the input, and (b) the precision of num b ers app e aring in the algorithm is bounded by a polynomial in the dimension and precision of the input. 5 An in-oriented gra ph is a dire c ted gr aph in whic h every vertex ha s at most one outgoing edge. An out-oriented gra ph is a directed graph in which every vertex ha s at most one inc o ming edge. 6 The thic knes s of a graph G ( · ) is the minim um n umber of planar gra phs into whic h G ( · ) can b e partitioned. 2.2. Link Sc h ed uling based on Pr otocol Interference Model 23 Arb oricalLinkSc hedule algorithm. This can lead to higher net work throughput. W e dev elop this idea further in Chapter 3. F urthermore, w e sho w that this can b e ac hiev ed with only a slight increase in computational complexit y . In [26], the a uthors inv estigate throug hput b ounds for a give n wireless n et work and traﬃc w o rkload under the proto col interference mo del. They use a conﬂict graph 7 to represen t in terference constrain t s. The problem o f ﬁnding maximum throughput for a giv en source-destination pair under the ﬂexibilit y of multipath routing is form ulated as a linear program with ﬂo w constrain ts and conﬂict graph constrain t s. They sho w that this problem is NP-hard and describe tec hniques to compute low er and upp er b ounds on throughput. Finally , t he authors n umerically ev aluate thro ug hput b ounds and computation time of their heuristics f or simple net work scenarios and IEEE 802.1 1 MA C (bidirectional MA C). Though the authors pro vide a general framew ork for join t routing and sche duling, they neither deriv e the computat ional complexity of their heuristics nor describe their link sc heduling algorithm explicitly . Recen tly , in [25], t he authors in v estigate joint link sc heduling and routing under the proto col in terference mo del for a wireless mesh netw ork consisting of stat ic mes h routers and mobile client devices. Assuming that l ( u ) de notes the aggregate traﬃc de mand on no de u , they consider the pro blem of maximizing λ , suc h that at least λl ( u ) amo unt of traﬃc can b e routed fr om each no de u to a ﬁxed gatew a y node. Since this problem is NP-hard, the authors prop o se heuristics based on linear programming and re-routing ﬂo ws on the communication gra ph. They deriv e the worst case b ound of their a lgorithm and ev aluate its p erformance via simulations. Though the authors mak e a reasonable attempt to solve the joint routing and sc heduling problem , their algorithm is extremely complex 8 and brute f orce in nature. F urthermore, the aut ho rs hav e not pro vided in tuitiv e argumen ts for their algorithm. Another recen t work whic h join tly inv estigates link sc heduling and routing under 7 Under the proto col interference mo del, the co nﬂict graph F ( V F , E F ) is c o nstructed from the com- m unication gr aph G c ( V , E c ) as follows. Le t l ij denote the co mm unica tion edge v i c → v j . V er tices o f F ( · ) corres p ond to directed edges l ij in E c . In F ( · ), there exists a n edge from v e r tex l ij to v er tex l pq if a ny of the fo llowing is true: (a) D ( i, q ) 6 R i or (b) D ( p, j ) 6 R i . 8 The algor ithm in [2 5] co nsists o f ﬁve steps: solve linea r pr ogram, channel a ssignment, po st pro cess- ing, ﬂow scaling and interference fr e e link scheduling. Mor eov er, the channel assignment step consists of three a lg orithms. 24 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks proto col in t erference mo del is rep orted in [30 ]. The author s consider wireless mesh net works with half duplex and full duplex orthogonal c ha nnels, wherein eac h no de can transmit to at most one no de and/or r eceiv e from at most k no des ( k > 1) during an y time slot. They in v estigate the jo in t problem o f routing and sc heduling to analyze the ac hiev a bilit y o f a giv en rate v ector b et w een m ultiple source-destination pairs. The sc heduling algorithm is equiv alen t to an edge-coloring on a multi-graph represen tation 9 and the corresp onding necessary conditions lead t he ro uting problem to b e for mulated as a linear optimization problem. The authors describ e a p olynomial time approximation algorithm to obtain an ǫ -optimal solution of the routing problem using the primal dual approac h. Fina lly , they ev aluate t he p erfo r mance o f their algorithms via sim ulations. It has b een observ ed that high dat a rates are ac hiev a ble in a wireless mesh netw ork b y a llo wing a no de to transmit to only one neighboring no de at ﬁxed p eak p o wer in an y time slot [30]. W e p oint out here that a similar assumption of uniform transmission p o w er ha s b een made in our system m o del in subsequen t c hapters of the thesis. Algorithms based on the prot o col interference mo del represen t the netw ork b y a comm unication or tw o-tier graph and emplo y a plethora of tec hniques fro m graph theory [35] and appro ximation algo rithms [36], [37] to dev ise heuristics whic h yield a minim um length sc hedule. Consequen tly , suc h a lgorithms hav e the adv an tage of low computational complexit y (in general). How ev er, recen t researc h s uggests that t hese alg orithms result in low net w ork throughput. This asp ect is f urther illustrated in the following section. 2.3 Limitations o f Algorit hms bas e d on Proto col In- terferenc e Mo del Due to its inheren t simplicit y , the proto col interferenc e mo del has b een traditionally emplo ye d to represen t a wide v ariet y of wireless net works . Ho wev er, it leads to lo w net work throughput in wireless mesh net w orks. T o emphasize this p oint, we provide examples to demonstrate that algorithms based on the proto col in terference model can result in sc hedules that yield low net work throug hput. 9 A multi-graph is a directed graph in which m ultiple edg es can emanate from a vertex v i and terminate at a nother vertex v j ( v j 6 = v i ). 2.3. Limitations of Algorithms based on Pr otocol Interference Mod el 25 In tuitive ly , the proto col in terference mo del divides the deploym en t regio n of the STDMA wireless netw ork in to “comm unication zones” and “in t erference zones”. This transforms t he sc heduling problem to an edge coloring problem for the communic ation graph represen tation of the net w ork. How ev er, this simpliﬁcation can res ult in sche dules that do not satisfy the SINR threshold condition (2.7). Sp eciﬁcally , a lgorithms base d on the proto col interference model do not neces sarily maximize t he throug hput of an STDMA wireless net w ork because: 1. They can lead to high cum ulative in t erference at a receiv er, due to hard-thresholding based on comm unication and in t erference ra dii [32], [33]. This is b ecause the SINR at receiv er r i,j decreases with an increase in the num b er of concurrent transmis- sions M i , while the comm unication r a dius R c and the interferenc e r a dius R i ha ve b een deﬁned for a single transmission only . X Y 2 ≡ ( − 450 , 0) 1 ≡ ( − 3 6 0 , 0) 4 ≡ (0 , 0) 3 ≡ (90 , 0) 5 ≡ (360 , 0) 6 ≡ (450 , 0) Figure 2.5: An STDMA wireless net w ork with six no des. v 1 v 5 v 4 v 2 v 3 v 6 Figure 2.6: Tw o- t ier graph mo del of the STDMA wireless netw ork describ ed by Fig ure 2.5 and T able 2.1. v 1 v 5 v 4 v 2 v 3 v 6 Figure 2.7: Subgra ph o f tw o-tier graph sho wn in Fig ure 2.6. 26 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks v 1 v 5 v 4 v 2 v 3 v 6 Figure 2.8: Coloring of subgraph show n in Figure 2.7. X Y 3 4 5 6 2 1 t i, 1 r i, 1 t i, 2 t i, 3 r i, 3 r i, 2 SINR r i, 1 = 21 . 2 6 dB SINR r i, 2 = 18 . 4 2 dB SINR r i, 3 = 19 . 74 dB Figure 2.9: P o in t to p o in t link sc heduling alg o rithms ba sed on proto col in terference mo del can lead to high interfere nce. F or example, consider t he STDMA wireless net w ork whose deplo ymen t is shown in Figure 2.5. The netw ork consists of six lab eled no des whose co ordinates (in meters) are 1 ≡ ( − 360 , 0), 2 ≡ ( − 450 , 0), 3 ≡ (90 , 0), 4 ≡ (0 , 0), 5 ≡ (360 , 0 ) a nd 6 ≡ (450 , 0) . The sy stem parameters are sho wn in T able 2.1, wh ic h yield R c = 100 m and R i = 177 . 8 m. The tw o-tier graph mo del of the STD MA netw ork is sho wn in Figure 2.6; note that in terference edges are absen t . Consider the transmiss ion requests 1 → 2, 3 → 4 and 5 → 6, whic h corresp ond to comm unication edges of the subgraph show n in Figure 2.7. The comm unication edges v 1 c → v 2 , v 3 c → v 4 and v 5 c → v 6 sho wn in Figure 2.7 do not hav e primary or secondary edge conﬂicts. T o minimize the num b er of colors, suc h an algorithm will color these edges with the same color, as sho wn in Fig ure 2.8. Equiv alen tly , transmissions 1 → 2, 3 → 4 and 5 → 6 will b e sc heduled in the same time slot, say time slot i . How ev er, our computations sho w that the SINRs at receiv ers r i, 1 , r i, 2 and r i, 3 are 21 . 2 6 dB, 18 . 42 dB and 19 . 74 dB resp ectiv ely . Figure 2.9 sho ws the no des of the net work along with the lab eled transmitter-receiv er pairs, receiv er- centric comm unication and interference zones and the SINRs a t the receiv ers. F rom the SINR threshold 2.3. Limitations of Algorithms based on Pr otocol Interference Mod el 27 condition (2.6), transmission t i, 1 → r i, 1 is successful, while transmissions t i, 2 → r i, 2 and t i, 3 → r i, 3 are unsucces sful. This leads to low net work throug hput. 2. Moreo v er, these algorithms can b e extremely conserv at ive a nd result in higher n umber o f colors. 1 ≡ (0 , 0) 2 ≡ (50 , 0) Y X 4 ≡ (1 70 , 0) 3 ≡ (220 , 0) Figure 2.10: An STDMA wireless net work with four no des. v 1 v 2 v 4 v 3 Figure 2.11 : Tw o-tier graph mo del of STDMA wireless net w ork des crib ed by Figure 2.10 and T a ble 2.1 . F or example, consider t he STDMA wireless net w ork whose deplo ymen t is shown in F igure 2.1 0. The ne t w ork consists of four labeled no des whose co o rdinates (in meters) are 1 ≡ (0 , 0), 2 ≡ (50 , 0), 3 ≡ (220 , 0 ) and 4 ≡ ( 1 70 , 0). The system parameters are shown in T able 2.1 , whic h lead to R c = 100 m and R i = 177 . 8 m. The tw o-tier graph mo del of the STDMA net work is show n in F igure 2.11 . Consider the tra nsmission re quests 1 → 2 and 3 → 4, whic h correspond to comm unication edges of the subgraph sho wn in F igure 2.12. The communication edges v 1 c → v 2 and v 3 c → v 4 sho wn in Figure 2.12 ha v e secondary edge conﬂicts. Hence, suc h an algorithm will typically color these edges with diﬀeren t colors, a s sho wn in Figure 28 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks v 1 v 2 v 4 v 3 Figure 2.12 : Subgraph of t w o-t ier gr aph shown in Figure 2.11. v 1 v 2 v 4 v 3 Figure 2.13: Colo r ing of subgraph show n in Figure 2.12. 2.13. Equiv alently , a link sc heduling alg o rithm based on the proto col inte rference mo del will sc hedule transmissions 1 → 2 and 3 → 4 in diﬀerent time slots, sa y time slots i and j resp ectiv ely , where i 6 = j . Our computations show that the resulting SINRs at re ceiv ers r i, 1 and r j, 1 are bo t h equal to 32 . 04 dB. Figure 2.14 sho ws the no des of the ne t work along w ith the labeled transm itter-receiv er pairs, receiv er-cen tric communic ation and in terference zones and SINRs at the r eceiv ers. Observ e that, with an a lgorithm based on t he proto col in terference mo del, the SINRs at b oth r eceiv ers are w ell ab ov e the communic ation threshold of 20 dB. Alternativ ely , consider an algorit hm (p erhaps ba sed on the ph ysical inte rference mo del) that sche dules transmissions 1 → 2 and 3 → 4 in the same time slot, sa y time slot i . The corresp onding edge coloring is sho wn in Fig ure 2.15. Our computations show that the resulting SINRs at receiv ers r i, 1 and r j, 1 are b oth equal to 20 . 91 dB, whic h are also ab ov e the comm unication threshold. Figure 2.16 shows the no des o f the net w ork along with the lab eled transmitter-receiv er pairs and SINRs at the receiv ers. In essence, with the alternate algorithm, b oth 2.3. Limitations of Algorithms based on Pr otocol Interference Mod el 29 Y 2 1 3 4 t i, 1 r i, 1 t j, 1 r j, 1 SINR r j, 1 = 32 . 04 dB SINR r i, 1 = 32 . 04 dB X Figure 2 .14: P oin t t o p oin t link sc heduling algorithms based on proto col in terference mo del can lead to higher num b er of colors. v 1 v 2 v 4 v 3 Figure 2.15 : Alternativ e coloring of subgraph sho wn in Figure 2.12. 30 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks 2 1 3 4 t i, 1 r i, 1 t i, 2 r i, 2 X SINR r i, 1 = 20 . 91 dB SINR r i, 2 = 20 . 91 dB Y Figure 2.16: A p o in t to p oin t link sc hedule cor r esponding to F ig ure 2.15 that yields lo wer num b er of colors. transmissions t i, 1 → r i, 1 and t i, 2 → r i, 2 are successful, since signals lev els are so high a t the receiv ers that strong in terferences can b e tolerated. In summary , a p oin t to p oint link sc heduling algorithm based on t he proto col in terference model will typic ally sc hedule the ab o ve transmissions in diﬀeren t slots and yield low er net work throughput compared to the alternate algorithm. 3. Lastly , these a lgorithms are not aw are o f the top olo gy of the net work, i.e., they determine a link sc hedule without b eing cognizant of the exact p ositions of the transmitters and receiv ers. The abov e ex amples demonstrate that sc heduling algorithms base d on the proto col in terference mo del can result in low net w ork throug hput. Observ e tha t algor it hms that construct an appro ximate mo del of the STDMA net work (t w o tier graph or comm uni- cation graph) and fo cus on minimizing the sc hedule length do not necessarily maximize net work throughput. This observ ation is dev elop ed in to a prop osal for an appropriate p erformance metric in Section 2 .6. Since lin k sche duling algorithms based on the proto col interferenc e mo del yield low throughput, researc hers ha v e prop ounded algorithms based on the ph ysical interferenc e mo del to impro ve the throughput of STDMA wireless net w orks. T o ac hiev e higher throughput, one p ossible tec hnique is to mo del the STDMA net w ork by a comm unication graph and c hec k SINR threshold conditions during assignmen t of links to time slots; this is the approach most commonly emplo y ed, for example in [27 ], [32], [38]. The other tec hnique is to incorp orate SINR threshold conditions in to a sp ecial graph mo del of the netw ork; this approa c h is more challenging and ( to the b est of our know ledge) is 2.4. Link Sc h ed uling based on C ommunication Graph Model and SINR Conditions 31 considered only in researc h work suc h as [3 9], [40 ], [41]. Researc h pa p ers whic h emplo y the former approa c h are review ed in Section 2.4, while researc h pap ers whic h emplo y the la t ter appro ac h are review ed in Section 2.5. 2.4 Link Sc heduling based on Comm unicati o n Graph Mo de l and SINR Conditi ons In this section, we examine recen t researc h in link sc heduling based on mo deling the STDMA net work b y a comm unication gra ph and verify ing SINR conditions at the re- ceiv ers. Though algorithms based on this mo del [24], [42], yield higher t hr o ughput, they usually result in higher computationa l complexit y t han algorithms based on the prot o col in terference model. In [27], the autho r s in ve stigate throughput improv emen t in an IEEE 802.11 like wireless mesh netw ork with CSMA/CA channe l access sch eme replaced b y STDMA. F or a successful pac ke t tr a nsmission, they mandate that t w o- w ay commun ication b e success ful, i.e., a pack et transmission is deﬁned to b e succes sful if and only if b oth data and ac kno wledgemen t pac k ets a r e receiv ed successfully . Under this “extended ph ysical in terference mo del”, they presen t a greed y algorithm whic h computes a po in t to p oin t link transmission sc hedule in a cen tralized manner. Assuming uniform random no de distribution a nd using results from o ccupancy theory [43], they deriv e an appro ximation factor for the length of this sc hedule relativ e to the shortest sc hedule. Though the analysis presen ted in [27] is no v el, their mo del is restrictiv e b ecause it is only applicable to wireless netw orks using link-lay er reliabilit y proto cols. The throughput p erfor ma nce of link sc heduling algorithms based o n t wo-tier graph mo del G ( V , E c ∪ E i ) has b een analyzed under phys ical inte rference conditions in [32]. The authors determine the optimal num b er of simultaneous transmissions b y maximizing a low er b ound on the throughput and subsequen tly prop ose T runcated G raph-Based Sc heduling Algorithm (TGSA), an algorithm that pro vides probabilistic g uaran tees for net work throughput. Though the analysis presen ted in [32] is mathematically elegan t and ba sed on the Edm undson-Madansky b ound [44], [45 ], their a lgorithm do es not yield high netw ork thro ug hput. This is b ecause the partitioning of a maximal indep enden t 32 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks set of communication edges into m ultiple subsets (time slots) is arbitrary and not base d on net work top o logy , which can lead to signiﬁcan t interfere nce in certain regions of the net work. This is further elucidated b y the sim ulation results in Chapter 3. The p erformance of alg orithms based on the pro to col in t erference model ve rsus those based on comm unication graph mo del and SINR conditio ns is ev aluated and compared in [33]. T o generate a non-conﬂicting link sc hedule based o n the proto col in terference mo del, the authors use a t w o- t ier g r a ph mo del with certain SINR threshold v alues c hosen based on heuristics and examples. T o gene rate a conﬂict-f ree p oin t to p o int link sch ed- ule based on the ph ysical in terference mo del, the autho rs emplo y a metho d suggested in [46] whic h describ es heuristics ba sed o n t wo path loss mo dels, namely terrain-data based ground w a v e pro pagation mo del and V ogler’s ﬁv e knife-edge mo del. Their sim- ulations r esults indicate that, un der a P oisson arriv al pro cess, algorithms based on the proto col in terference mo del result in higher av erage pac ket dela y tha n algorithms based on commu nication graph mo del and SINR conditions. In [42], the authors inv estigate the tradeoﬀ b et w een the a v erage num b er of concur- ren t tra nsmissions ( spatial reuse) a nd sustained data r a te p er no de for an IEEE 802.11 wireless net w ork. They sho w that spatial reuse dep ends only on the ra t io of transmit p o w er to carrier sens e threshold [6]. Keeping the carrier sense threshold ﬁxed, they pro- p ose a distributed p ow er and rate con trol alg o rithm based on in terference measuremen t and ev a luate its p erforma nce via sim ulations. In [2 4], the a uthors in v estigate mitig ation of in ter-ﬂo w interference in an IEEE 802.11e wireless mesh net w ork from a tempora l- spatial divers it y p erspective . Measure- men ts of receiv ed signal strengths are used t o construct a virtual co ordinate system to iden tify concurrent transmissions with minim um inter-ﬂo w inte rference. Based on this new co ordinate system, one o f the no des, designated as gatew a y node, determines the sc heduling order for do wnlink frames o f diﬀeren t connections. Through extensiv e sim ulations w ith real-life measuremen t traces, the authors demonstrate t hro ughput im- pro veme n t with their algo rithm. Algorithms ba sed on represen ting the netw ork b y a communic ation graph and v eri- fying SINR threshold conditions yield hig her net w ork throughput than algorithms based on the proto col in t erference mo del. How ev er, this is a c hiev ed at the cost of higher com- putational complexit y . F urthermore, the gains in throughput ma y not b e signiﬁcan t 2.5. Link Sc h ed uling based on S INR Graph Mo del 33 enough to justify the increase in computational complexity . This has prompted few researc hers to solv e the link sc heduling problem in a more fundamental manner. These researc hers ha v e prop osed an altogether diﬀeren t mo del of t he net w ork, termed as SINR graph mo del, and deve lop ed heuristics. Suc h algorithms are review ed in the following section. 2.5 Link Scheduling based on SINR Graph Mo del In literature, many aut ho rs refer to algorithms based on comm unication graph mo del and c hec king SINR conditions as “algorithms based on ph ysical in terference mo del”. In this thesis, only algo rithms that embed SINR threshold conditio ns in to an appropriate gra ph mo del of the net w ork are referred to as “algorithms based on the ph ysical interferen ce mo del”. Though the ph ysical interference mo del is more realistic, algor it hms based on this mo del [39], [40], [41] ha v e, in g eneral, higher computationa l complexit y than algorithms based o n the prot o col interfere nce mo del. P oint to p oint link sc heduling for p o wer-con trolled STDMA netw orks under the phys- ical inte rference mo del is a nalyzed in [3 9]. The a uthors deﬁne sc heduling c omplexit y as the minim um n um b er of time s lots req uired fo r strong connectiv it y of the graph 10 con- structed from the p o in t to p oint link sc hedule. They dev elop an algorithm emplo ying non-linear p ow er assignmen t 11 and sho w t hat it s sc heduling complexit y is polylogarith- mic in the n umber of no des. In a related w ork [4 0 ], the authors inv estigate the time complexit y of sc heduling a set of comm unicatio n requests in an arbitrary net w ork. They consider a “generalized phy sical mo del” wherein the actual receiv ed p o we r of a signal can deviate from the theoretically rec eiv ed p o wer by a m ultiplicative factor. Their a lgo- rithm success fully sc hedules all links in t ime prop ortional to t he squared logarithm of the n umber of no des times the static interferen ce measure [47]. Though the authors of [3 9], [40] allo w non-uniform transmission p o w er at all no des and dev elop no v el algorithms, 10 A directed gr aph G ( · ) is strongly connected if there exists a directed path from ev ery v ertex to every other vertex. 11 In uniform p ow er assignment, a ll no des transmit with the s a me transmissio n p ow er. In linear p ow er assignment [39], a no de trans mits with minimum power required to s atisfy the SINR threshold condition at the re c eiv er, i.e., tra nsmission power equals N 0 γ c D β . Non-linear p ow er assig nmen t re fer s to a p ow er assignment scheme tha t is neither unifor m no r linear. 34 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks their algorithms are impractical. This is b ecause wireless devices hav e constraints on maxim um t ransmission p o wer, while the algo rithms in [39], [40] can result in arbitra rily high transmission p o w er at some no des. In [26], the authors pro vide a g eneral fr a mew ork for computatio n of throug hput b ounds for a given wireless net work and traﬃc w orkload. Though t heir w ork primarily fo cuses on the proto col in terference mo del, they brieﬂy allude to the phy sical in terference mo del t o o. Sp eciﬁcally , they desc rib e a tec hnique to construct a w eigh ted conﬂict graph to represen t interferenc e constraints. They brieﬂy describ e metho ds to compute low er and upper b ounds on throughput and the issues in v olved therein. Ho w ev er, the authors do no t describ e sim ulation results under the phys ical interference mo del, p erhaps due to t he t r emendous complexit y incurred in solving linear programs for represen tative net work scenarios. Remark 2.5.1. Under physic al interfer enc e mo del, the weighte d c onﬂict gr aph F ( V F , E F ) [26] is c onstructe d fr om the network as fol lows. L et S ij := P D β ( i,j ) denote the r e c eive d signal p ower at no de j d ue to the tr ansm ission fr om no de i . I n F ( · ) , a vertex c orr esp onds to a dir e cte d link l ij (e quivalently, no de p air ( i, j ) ) pr ovide d S ij N 0 > γ c . F ( · ) i s a p e rfe c t gr aph whe r ein the weight w pq ij of the dir e cte d e dge fr om v e rtex l pq to v e rtex l ij is given by w pq ij = S pj S ij γ c − N 0 . W e should p o in t out here that, ana logous to a conﬂict gra ph, an SINR graph rep- resen tation of an STDMA wireless net w ork has b een prop osed b y us in Chapter 4. F urthermore, the authors of [26] do not prop ose an y speciﬁc link sc heduling algorithm and use the w eighted conﬂict graph only to compute b ounds on netw ork throughput. On the other ha nd, we us e an SINR gra ph represen tatio n of the netw ork under the ph ysical in terference model a nd dev elop a link sc heduling algorithm with low er time complexit y and demonstrably sup erior p erformance. More speciﬁcally , in Chapter 4, w e in ves tigate link sc heduling for STDMA wireles s net works under the phys ical interferenc e mo del. Unlik e [39], [40], we assume that a no de tr a nsmits at ﬁxed p o we r, i.e., w e assume unifo rm p o w er assignmen t. Moreov er, unlik e [39], [40], w e do no t assume a minim um distance of unit y b et w een an y t w o no des. Consequen tly , our system model is more pra ctical than those of [3 9], [40]. Under t hese realistic ass umptions, w e propose a link sc heduling algorithm based o n an SINR graph 2.6. Spatial Reuse as P erformance Metric 35 represen tation of the netw ork. In the SINR g raph 12 , w eigh ts of the edges corresp ond to in terferences b etw een pairs of no des. W e prov e the correctness of the algorithm and deriv e its compu tational complexit y . W e demonstrate that the prop osed algo rithm ac hiev es higher throughput tha n existing alg orithms, without any increase in computa- tional complexit y . So f ar, w e ha ve provided a brief glimpse into three classes of link sc heduling algo- rithms, eac h with its relative merits and demerits. F or example, algorithms based on the proto col in terference mo del hav e lo w computational complexity and are simple to implemen t, but yield low net work throughput. On the other hand, a lgorithms based on SINR graph represen tation hav e higher computatio na l complexit y a nd are mor e cum b er- some to implemen t, but a c hiev e higher net w ork throughput. Also, there exist a lg orithms based on comm unication graph and SINR conditions whose p erformance ch aracteristics lie b et w een these tw o classes. Hence, in general, these three classes of a lg orithms exhibit a tradeoﬀ b et w een complexit y and performance. Finally , algorithms based on the pro- to col in terference mo del are b etter suited t o mo del WLANs, while the latter t w o classes of algorithms a r e better suited to mo del wireless mes h ne t works. F or the se reasons, w e in ve stigate and dev elop algorithms from eac h of these classes in t his thes is. Prior to prop osing eﬃcien t a lg orithms in eac h of these classes, we seek t o address the f o llo wing q uestion: Is sc hedule length an appro priate p erformance metric for an a l- gorithm that considers the SINR threshold condition (2.6) as the criterion for succ essful pac ke t reception? In other w or ds, should algo r ithms based on comm unication graph and SINR conditions and algorithms based on SINR graph represen tation fo cus on minimiz- ing the sc hedule length? W e answ er this imp o rtan t question in detail in the follow ing section. 2.6 Spatial Reu s e as P erfor mance Metric In literature, link sc heduling algo r ithms hav e only f o cused o n minimizing the sc hed- ule length. How ev er, algorithms that minimize the sc hedule length do not necessarily maximize netw ork throughput, as explained in Section 2.3 . Th us, from a p ersp ectiv e 12 The SINR graph is analogous to a line gr a ph [35] constructed from the communication graph representation of the net work. 36 Chapter 2. A F ramewo rk f or Link Sc heduling Algorithms for STDMA Wireless Net wo rks of maximizing net w ork thro ug hput observ ed b y the phys ical la y er, it is imp erativ e to consider a p erformance metric that t ak es into accoun t SINR threshold condition (2.6) as the criterion for successful pac k et reception, i.e., a metric also suitable for the ph ysical in terference mo del. W e prop ose suc h a p erformance metric, spatial reuse, in t his sec- tion. W e show that maximizing spatial reuse directly translates t o maximizing netw ork throughput. Consider an STDMA wireless netw ork that op erates ov er ( k 2 − k 1 + 1) time slots k 1 , k 1 + 1 , . . . , k 2 − 1 , k 2 . The total num b er of su ccessfully sc heduled links from slot k 1 to slot k 2 is τ [ k 1 , k 2 ] = k 2 X i = k 1 M i X j =1 I (SINR r ij > γ c ) . (2.11) So, the num b er of successfully sc heduled links p er time slot fr o m slot k 1 to slot k 2 is η [ k 1 , k 2 ] = P k 2 i = k 1 P M i j =1 I (SINR r ij > γ c ) k 2 − k 1 + 1 . (2.12) W e deﬁne sp atial r euse σ as the limiting v alue of η [ k 1 , k 2 ] (assuming that the limit exists). In other words, spatial reuse is the limiting v alue of η [ k 1 , k 2 ] as the duration of the time interv al b ecomes v ery large. Mathematically , Spatial Reuse := lim | k 2 − k 1 |→∞ η [ k 1 , k 2 ] , ∴ σ = lim | k 2 − k 1 |→∞ P k 2 i = k 1 P M i j =1 I (SINR r ij > γ c ) k 2 − k 1 + 1 . (2 .13) Assuming a constant data ra t e o f R bits p er second on eac h successful link and a slot durat ion of τ s seconds, the (a ggregate) netw ork throughput is giv en b y σ Rτ s bits p er second. Th us, spatial reuse is directly prop ort io nal to net w ork throughput. No t e that spatial reus e is cognizan t of the phy sical in terference mo del, thereb y making it an appropriate p erformance metric for t he comparison of v arious link sc heduling algorithms. The fact that the in terference at a r eceiv er is an increasing function of the n um b er of concurren t transmissions in a time slot limits the v alue of s patial reuse (for a giv en STDMA net work). More sp eciﬁcally , if to o many tr a nsmissions are sc heduled in a single time slot, the inte rference at some receiv ers will b e high enough to drive the SINRs b elo w t he communication threshold, leading to lo w er spatial reuse. Therefore, for a 2.6. Spatial Reuse as P erformance Metric 37 giv en STDMA net work, there a re certain fundamental limits (upp er b ounds) on the spatial reuse. In our system mo del, we only consider static link sc hedules, i.e., the same ﬁxed pattern of slots rep eats cyclically . Hence, for our sy stem mo del, the equation for spatial reuse, (2.13), can b e simpliﬁed to Spatial Reuse = σ = P C i =1 P M i j =1 I (SINR r ij > γ c ) C . (2.14) The essence of STDMA is to hav e a reasonably larg e n um b er of concurren t and success ful transmissions. F or a net work wh ic h is operat ional f or a long p erio d o f time, sa y L time slots, the total num b er of successfully receiv ed pac k ets is Lσ . Thus , a high v alue of spatial reuse directly t r anslates t o hig her net work throughput and the n um b er of colors C is relativ ely unimp ort a n t. Hence, spatial reuse 13 turns out to b e a crucial metric fo r the comparison o f diﬀeren t STDMA algorithms in Chapters 3, 4 a nd 5. 13 Note that spatia l reuse in our net work mo del is a nalogous to sp ectral eﬃciency in digital commu- nication systems. Both p erfor mance metrics cor resp ond to the “ rate of data tr ansfer” and a re upp er bo unded by their respective s ystem pa rameters. Chapter 3 P oi n t to P oin t Li nk Sc heduling based on C om m unication G r a ph Mo del W e b egin our inv estigation in link sch eduling b y critically examining the Arb oricalL- inkSc hedule algorithm prop osed in [16]. The algorithm is based only on the communi- cation graph (prot o col in terference mo del) and seeks to minimize t he sc hedule length. Though Arb oricalLinkSc hedule ha s go o d prop erties suc h as low computational com- plexit y , it can yield higher sc hedule length in practice. T ow ards this end, we pro p ose a no vel m o diﬁcation to Arbo ricalLinkSc hedule that results in lo w er sc hedule length. W e compare the p erformance o f the mo diﬁed algorithm with the Arb oricalLinkSc hedule a l- gorithm and deriv e its run t ime (computational) complexit y in Section 3.1. W e then prop ose t he ConﬂictF reeLinkSc hedule p o in t to p oin t link sc heduling algorithm, whic h is based on comm unication graph mo del and SINR conditions, in Section 3 .2. The p erfor- mance of the prop osed algorithm is compared with existing link sch eduling algorithms under v arious wireless c hannel conditions. W e sho w that the prop osed a lgorithm has p olynomial run time complexit y . Finally , w e summarize the implications of our w ork. 39 40 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del 3.1 Arb o ricalLinkSc hedule Algo rithm Revisi ted In this section, w e propose a mo diﬁcation to the A rb oricalLinkSc hedule p oin t to p oint link sc heduling algorithm. Since b oth the original algorithm and the prop osed mo diﬁ- cation are based on the pro t o col in terference mo del, w e compare their p erf o rmance in terms of a v erage sc hedule length. Finally , w e also deriv e the run time complexit y of the mo diﬁed a lgorithm. Our system mo del a nd notatio n are same as describ ed in Section 2.2 . W e seek an algorithm that determines a minimum length p oin t to p oin t link sc hedule for an STDMA wireless netw ork under the proto col interference mo del. F or consistency with the g raph mo del describ ed in [16], w e ass ume that the STDMA wireless ne t work Φ( · ) is m o deled b y the comm unication g raph G c ( V , E c ) only , i.e., interfere nce edges are absen t ( E = E c ). It is well known tha t, under the proto col interferenc e mo del, the problem of de- termining an optimal sc hedule, i.e., a minim um length sc hedule, is NP-hard [48]. As p oin ted out in Section 2.2.1, this is close ly related to the problem of coloring all e dges of the communic ation graph with minim um n umber o f colors, whic h is also kno wn to b e NP-hard [16]. Consequen tly , the only recourse is to devise appro ximation a lgorithms (heuristics) and show their eﬃciency theoretically and exp erimen tally . One suc h a lg orithm, Arb oricalLinkSc hedule, has b een describ ed in [16]. Fir st, the algorithm uses the lab eler function to lab el all the vertice s of the comm unication graph. Next, it partitions the comm unication graph into edge-disjoint subgraphs, whic h are termed a s “oriented graphs”. Fina lly , the orien ted gra phs are colored in sequence . Sp ecif- ically , the v ertices in eac h oriented graph are scanned in increasing order o f lab el and the unique edge asso ciated with eac h v ertex is colored using the NonConﬂictingEdge function [16]. The la b eler function and the partitioning tec hnique are describ ed later in the section. In [16], the authors a pp ear to hav e missed a delicate p o in t that colors from previously colored or iented graphs can b e used to color the presen t orien ted graph. Sp eciﬁcally , they use a fr esh se t of colo r s t o color eac h successiv e orien ted gr aph. In our opinion, the authors emplo y this metho d to upper b ound the n um b er of colors used b y the algorithm ([16], Lemma 3.4) and thus de riv e the running time c omplexit y of the algorithm ([16], Theorem 3.3). How ev er, suc h a heuristic can p oten tially lead to a higher num b er of 3.1. Arb oricalLinkSc hedule Algorithm Revisited 41 colors (and higher sc hedule length) in practice. Therefore, w e prop ound a mo diﬁcation to the Arb oricalLinkSc hedule algor ithm that r euses colors from previously colo red o rien ted graphs to colors t he current orien ted gr aph. The resulting sche dule length will alw ays b e lower than that of Arb oricalLinkSc hedule, leading to p otentially higher thro ughput. Our prop osed link sc heduling alg orithm is ALSReuseColors, whic h considers the comm unication graph G c ( V , E ) and is describ ed in Alg orithm 1. In Phase 1, we lab el all the v ertices using the lab eler function [16]. The lab eler function is repro duced in Algorithm 2 for con v enience. It is a recursiv e function that assigns a unique lab el (from 1 t o N ) to ev ery v ertex of the comm unication gra ph. Let L ( w ) denote the lab el assigned t o ve rtex w . The notat io n G r \ { u } denotes the graph t ha t results when v ertex u and all its inciden t edges are remov ed from graph G r ( · ). A t ev ery step in the recursion, it ch o oses t he minim um degree ve rtex u in the residual graph G r ( · ) and assigns it the highest lab el that has not b een assigned so far. Note that v ertices with low er degree tend to b e ass igned h igher labels. The lab eler function ensures that , for an y giv en no de, the n um b er of neigh b ors with low er lab els is muc h lo w er than the n umber o f v ertices in G c ( · ). In Phase 2, the comm unication graph G c ( · ) is decomp osed in to what are called as out-orien ted and in-orien ted graphs T 1 , T 2 , . . . , T k , similar to the t ec hnique employ ed in [16]. Recall that an in- orien ted graph is a directed graph in which e v ery verte x has at most one o utgoing edge, while an out- orien ted graph is a directed graph in whic h ev ery v ertex has at mo st one incoming edge. Eac h T i is a forest 1 and eve ry edge of G c ( · ) is in exactly one of the T i ’s. This decomp osition is ac hiev ed by partitioning graph G c ( · ), the undirecte d equiv a lent of G c ( · ), in to undirected forests. The num b er of fo r ests can be minimized by using techniq ues from Matroid t heory ([49], k - forest problem). Ho w ev er, this optimal decomp osition requires extensiv e computatio n. Hence, w e adopt a fa ster alb eit non-optimal approach o f using succes siv e breadth ﬁrst searc hes [50] to decomp ose G c ( · ) in to undirected forests. Eac h undirected f orest is further mapp ed to t w o directed forests. In one forest, the edges in ev ery connected gra ph p oin t aw a y from the ro ot and ev ery v ertex has at most one incoming edge, th us producing an out-oriente d graph. In 1 A gra ph that is a co llection of trees is termed a s a forest. 42 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del the other forest, the edges in ev ery connected g r aph p oint tow ard the ro ot and ev ery v ertex has at most one outgoing edge, th us pro ducing a n in- orien ted graph. In Phase 3, the orien ted graphs are conside red sequen tially . F or eac h orien ted graph, the v ertices are considered in increasing o rder of lab el and the unique edge asso ciated with eac h v ertex is colored using the NCEReuseColors function. The NCEReuseColors function is explained in Algorithm 3. F o r the edge x under consideration, it discards an y color from any orien ted graph that has an edge with a primary or secondary conﬂict with x . It r eturns the least color among the resid ual set of n on-conﬂicting colors from all orien ted g r a phs colored so far. If no non-conﬂicting color from an y oriente d graph is found, it returns a new color. Algorithm 1 ALSReuseColors 1: input: Directed comm unicatio n graph G c ( V , E ) 2: output: A coloring C : E → { 1 , 2 , . . . } 3: n ← lab eler( G c ) { Phase 1 } 4: use suc cessiv e breadth ﬁrst searc hes to partition G c ( · ) in to orien ted gr a phs T i , 1 6 i 6 k { Phase 2 } 5: for i ← 1 to k do { Phase 3 b egins } 6: for j ← 1 to n do 7: if T i is out- orien ted then 8: let x = ( s, d ) b e suc h that L ( d ) = j 9: else 10: let x = ( s, d ) b e suc h that L ( s ) = j 11: end if 12: C ( x ) ← NCEReuseColors( x ) 13: end for 14: end for { Phase 3 ends } 3.1.1 P erformance Results In the sim ulation exp eriment, ev ery no de lo cation is generated randomly , using a uniform distribution for its X and Y co ordinates in the deplo ymen t area. W e assume that the deplo ymen t region is a square of length L . Th us, if ( X j , Y j ) are the Cartesian 3.1. Arb oricalLinkSc hedule Algorithm Revisited 43 Algorithm 2 in teger lab eler( G r ) 1: if G r ( · ) is not empty t hen 2: let u b e a v ertex of G r ( · ) o f minim um degree 3: L ( u ) ← 1 + lab eler ( G r \ { u } ) 4: else 5: return 0 6: end if Algorithm 3 in teger NCEReuseColors( x ) 1: input: Directed comm unicatio n graph G c ( V , E ) 2: output: A non-conﬂicting color 3: C ← set o f existing colors 4: C 1 ← { C ( h ) : h is colored and x and h hav e a p rimary edge conﬂict } 5: C 2 ← { C ( h ) : h is colored and x and h hav e a s econdary edge conﬂict } 6: C nc = C \ {C 1 ∪ C 2 } 7: if C nc 6 = φ t hen 8: return t he least colo r ∈ C nc 9: else 10: return |C | + 1 11: end if 44 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del co ordinates of j th no de, then X j ∼ U [0 , L ] and Y j ∼ U [0 , L ]. The v alues chosen fo r system parameters P , γ c , β and N 0 , a r e proto t ypical v alues of system par ameters in wireless net works [42]. After generating random p ositions for N nodes, w e ha v e complete information o f Φ( · ). Using (2 .4), w e c ompute the comm unication range, a nd then map the STDMA net w ork Φ( · ) to the comm unication g raph G c ( · ). Once the sc hedule Ψ( · ) is computed b y ev ery alg o rithm, we know its sc hedule length | C | . F or a g iv en set of parameters ( N , L, R c ), w e calculate the av erage sc hedule length by a veraging |C | o v er 1000 ra ndo mly generated net w orks. Keeping a ll other parameters ﬁxed, we observ e the eﬀect of increasing the num b er o f no des on t he av erage sc hedule length. In our exp eriments, w e compare the p erformance of the follow ing algorithms: • Arb oricalLinkSc hedule [1 6 ], • Prop osed ALSReuseColors. 100 110 120 130 140 150 160 170 180 190 200 0 50 100 150 200 number of nodes average schedule length L = 750.0 m, R c = 100.0 m ArboricalLinkSchedule ALSReuseColors Figure 3.1: Sche dule length vs. n umber of no des. W e assume that P = 10 mW, β = 4, N 0 = − 90 dBm a nd γ c = 20 dB. F rom (2.4), w e obtain R c = 100 m. W e assume that L = 750 m, and v ary the num b er of nodes from 100 to 200 in steps o f 10. F igure 3.1 plots the a v erage sc hedule length vs. n umber of no des fo r b oth the algor ithms. 3.1. Arb oricalLinkSc hedule Algorithm Revisited 45 F or b oth the algor it hms, we observ e that a v erage sc hedule length increases a lmo st linearly with t he n um b er of no des. The a v erage sc hedule length of ALSR euseC olors is ab out 2 3% low er than that of Arb o r icalLinkSc hedule. Note that an increase in the n um b er of nodes in a giv en geographical area leads to an increase in the n um b er of edges inciden t on a verte x and a subsequen t increase in the num b er of orien ted graphs. Arb oricalLinkSc hedule, whic h is based on using a fresh set of colors for each orien ted gr a ph, requires increasingly higher num b er of colors to color the communic ation gra ph compared to ALSR euseColors. Consequen tly , the g ap b et wee n t he av erage sc hedule le ngths increases with n um b er of nodes in Figure 3 .1 . 3.1.2 Analytical Results W e no w deriv e upp er b ounds on the running time (computational) complexit y o f the ALSReuseColors a lgorithm. With resp ect to the comm unication graph G c ( V , E ), let: e = num b er of edges , v = num b er of v ertices , ρ = maxim um degree of an y vertex , θ = thic kness of the graph := minim um n um b er of planar graphs in t o w hic h the undirected equiv alent of G c ( · ) can b e part itioned , ω = maximu m n um b er o f neigh b ors with lo w er la b els (for any v ertex) . Recall that the mo diﬁed algorithm partitions the comm unicatio n gra ph G c ( · ) in to orien ted graphs T 1 , T 2 , . . . , T k , a nd colors the o r ien ted graphs in that order. T 1 is termed as the ﬁrst oriente d gr aph , while any orien t ed graph T j , where 2 6 j 6 k , is termed as a subse quent oriente d gr aph . Lemma 3.1.1. Supp ose that e ach vertex of the ﬁrst oriente d g r aph T 1 has at m ost ω neighb ors w ith lower lab e ls. Th en, T 1 may b e c ol o r e d using no mor e than O ( ω ρ ) c olo rs . Pr o of. This is similar to the pro of o f L emma 3.1 in [16].  Lemma 3.1.2. A ny subse quent orie n te d gr a ph T j , wher e 2 6 j 6 k , c an b e c ol o r e d using no mor e than O ( ρ 2 ) c olors. 46 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del y n 1 n 2 n 3 n 4 v u y x i x i x x Figure 3.2: Poten tial conﬂicting edges when coloring edge ( u, v ). Pr o of. W e pro ve the lemma fo r an out-oriented graph. A similar pro of holds for an in-orien ted graph. Let G c b e par t it io ned into edge-disjoint oriente d graphs T 1 , . . . , T k . Consider the coloring of edge ( u, v ) in j th orien ted graph T j , where 2 6 j 6 k , as sho wn in Figure 3.2. Now, edges of previously colored orien ted graphs T 1 , . . . , T j − 1 m ust also b e considered fo r p otential edge conﬂicts with edge ( u, v ) of T j . Deﬁne S 1 := n ( v , x ) : ( v , x ) ∈ j [ i =1 T i and ( v , x ) is colored o , S 2 := n ( u, x ) : ( u, x ) ∈ j [ i =1 T i and ( u, x ) is colored o , S 3 := n ( y , x i ) : ( y , x i ) ∈ j [ i =1 T i and ( y , x i ) is colored and ( u, x i ) ∈ G c o , S 4 := n ( x i , y ) : ( x i , y ) ∈ j [ i =1 T i and ( x i , y ) is colored and ( x i , v ) ∈ G c o . An y edge whic h can cause a primary edge conﬂict with ( u, v ) mus t belong to S 1 or S 2 . Also, a ny edge whic h can cause a secondary edge conﬂict with ( u, v ) must b elong to S 3 or S 4 . Let n i = | S i | for i = 1 , 2 , 3 , 4. The lemma r educes to prov ing that n 1 + n 2 + n 3 + n 4 is O ( ρ 2 ). By deﬁnition of maxim um v ertex degree, n 1 6 ρ − 1 and n 2 6 ρ − 1 . Thus , n 1 + n 2 3.1. Arb oricalLinkSc hedule Algorithm Revisited 47 is O ( ρ ). F o r the computation of n 3 , w e m ust also consider secondary edge conﬂicts with edges o f previously colo red oriented graphs, as sho wn in Figure 3.2. The w orst-case v alue of n 3 is ( ρ − 1)( ρ − 1). Thus, n 3 is O ( ρ 2 ). Similarly , by considering secondary edge conﬂicts with edges of previously color ed orien ted gra phs, it follo ws that n 4 is O ( ρ 2 ). Finally , n 1 + n 2 + n 3 + n 4 is O ( ρ 2 ).  Lemma 3.1.3. F or the ﬁrst oriente d gr aph T 1 , the running time o f Phase 3 of AL- SR euseColors is O ( v ω ρ ) . Pr o of. This is similar to the pro of o f L emma 3.2 in [16].  Lemma 3.1.4. F or any subse quent oriente d g r aph T j , wher e 2 6 j 6 k , the running time of Phase 3 of ALSR euseC olors is O ( v ρ 2 ) . Pr o of. F rom Lemma 3.1.2, for an y s ubsequen t orien ted graph T j , the siz e of the set of conﬂicting colors ( C 1 ∪ C 2 ) of function NC EReuseColors is O ( ρ 2 ). Th us, determining a new color for an edge in Phase 3 of ALSReuseColors tak es O ( ρ 2 ) steps. Since this is done f o r ev ery lab el and hence for ev ery v ertex, it follows that the o v erall running time of Phase 3 of ALSReuseColors is O ( v ρ 2 ).  Theorem 3.1.5. F or an arbitr ary gr aph of thickne s s θ and maximum de gr e e ρ , AL- SR euseColors ha s a running time of O ( ev log v + v θ ρ 2 ) . Pr o of. The running time of the lab eler function is O ( e + v log v ) using a Fib onacci Heap [51]. The partitioning metho d of [49] results in a decomp osition of a gra ph of thic kness θ in t o at m ost 6 θ o rien ted graphs in time O ( ev log v ). Th us, k 6 6 θ . F rom Lemma 3.2 in [16 ], the ﬁrst orien t ed g raph T 1 can b e colored in time O ( v ω ρ ). How ev er, consider the coloring of j th orien ted graph T j , where 2 6 j 6 k . F rom Lemma 3.1.4, T j can b e colored in time O ( v ρ 2 ). Hence, the for lo op of ALSReuseC olors runs in time O ( v θ ρ 2 ). Therefore, the ov erall running time of ALSR euseColors is O ( e + v log v + ev log v + v θ ρ 2 ). Since e + v log v < ev log v holds for an y directed gr a ph G c ( · ) tha t mo dels a wireless mesh net work, the o verall running time of ALSReuseColors simpliﬁes to O ( ev log v + v θ ρ 2 ).  3.1.3 Discussion In this section, we hav e considered an STDMA wireless net w o rk with uniform tra nsmis- sion p ow er at all no des and presen t ed an algorithm for p oin t to p oint link sc heduling 48 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del under the proto col in terference mo del. The prop osed algo rithm, whic h is a mo diﬁcation of the Arb oricalLinkSc hedule algorithm in [16], mo dels the net w ork b y a comm unication graph, partitions the comm unicatio n graph into edge-disjoin t oriented graphs a nd colors eac h orien ted graph suc cessiv ely . Ho w ev er, unlik e [16], w e reuse c olors from previously colored orien ted graphs to color the curren t orien ted graph. The prop osed algorithm re- sults in around 26% lo w er sc hedule length than that of [16], alb eit at the cost of slightly higher computational complex it y 2 . Since sc hedules are constructed only o nce oﬄine and then used b y the net w ork for a long p erio d of time, our appro a c h has the p oten tia l of pro viding higher long-term net work throug hput. F or the rest o f this chapter, we consider p oint to p oint link sche duling under the ph ysical in terference mo del. The alg o rithm dev elop ed in t his section will b e further reﬁned to design a link sc heduling algorithm in the next section. 3.2 A High Spatial Reuse Link S c hedu l i ng Algo- rithm In this section, w e prop ose a p oin t to p oint link sc heduling algorithm based on the com- m unication gr aph model of an STDMA wireless net w ork as w ell as SINR computations. W e adopt spatial reuse as the p erformance metric, whic h has b een motiv ated in Sec- tion 2.6. W e compare the p erformance of t he pr o p osed algorithm with link sc heduling algorithms whic h utilize a comm unication graph mo del of the netw ork. W e sho w that the prop osed algo rithm ac hieve s higher spatial reuse compared to existing alg orithms, without an y increase in computational complexit y . 3.2.1 Problem F orm ulation Our syste m mo del and notation are exactly as described in Section 2.2. A link sc hedule is fe asi b le if it satisﬁes the follo wing conditions: 1. Op erational constraint (2 .1). 2 The computationa l co mplexit y of Arb or ic alLinkSchedule is O ( ev log v + vθ 2 ρ ) [16]. 3.2. A High Spatial Reuse Lin k Scheduling Alg orithm 49 2. Range constraint: Ev ery receiv er is within the comm unication range o f its inte nded transmitter, i.e., D ( t i,j , r i,j ) 6 R c ∀ i = 1 , . . . , C ∀ j = 1 , . . . , M i . (3.1) A link sc hedule Ψ( · ) is exhaustive if ev ery pair of no des which ar e within comm u- nication r a nge o ccur exactly twice in the link sc hedule, once with o ne no de b eing the transmitter and the other no de b eing the r eceiv er, and during anot her time slot with the tr a nsmitter-receiv er ro les in terc hanged. Mathematically , D ( j, k ) 6 R c ⇒ j → k ∈ C [ i =1 S i and k → j ∈ C [ i =1 S i ∀ 1 6 j < k 6 N . (3.2) Our aim is to design a low complexit y conﬂict-free STDMA p o int to po int link sc heduling algo r ithm that ac hiev es high spatial reuse, where spatial reuse is g iv en b y (2.14). W e only consider STDMA link sc hedules whic h are feasible and exhaustiv e 3 . Th us, our sche dules satisfy (2.1), (2.7), (3.1) and (3.2). 3.2.2 Motiv ation W e brieﬂy describ e t he essen tial features of STDMA link sc heduling algorithms. An STDMA link sc heduling algo rithm is equiv a len t to assigning a unique color to ev ery edge in the comm unication gra ph, suc h that tra nsmitter-receiv er pairs corresp o nding to comm unication edges with the same colo r are simu ltaneously active in a particular time slot, as describ ed in Section 2.2 .1. The core of a typ ical link sc heduling algorithm consists o f the following functions: 1. An order in whic h c omm unication edges are considered for coloring. 2. A function whic h determines the set of all existing color s whic h can b e assigned to the edge under consideration without violating the problem constraints. 3. A Be s tCo l o r rule to determine whic h color to assign to t he edge under considera- tion. 3 The s et of edges in G c ( · ) to be scheduled is determined by a routing algorithm. F or simplicit y , w e only consider exhaus tiv e schedules, i.e., schedules which assign exactly o ne time slot to ev ery directed edge in G c ( · ). 50 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del The second function considers only op erational and range constrain ts in link sc heduling algorithms based on the prot o col in terference mo del (equiv a lently , based on the com- m unication graph). How ev er, in the link sc heduling alg o rithm that w e prop o se, SINR constrain ts are also ta ken into accoun t. Algorithms based o n the proto col interferenc e mo del are inadequate to design ef- ﬁcien t link sc hedules. This is b ecause the comm unication graph G c ( V , E c ) is a crude appro ximation o f Φ( · ). Ev en the t w o- tier graph G ( V , E c ∪ E i ), w hic h is a b etter approxi- mation of Φ( · ), leads to lo w net work throughput, as argued in Section 2.3. On the other hand, from Φ( · ) and G c ( · ), o ne can exhaustiv ely determine the STDMA sc hedule whic h yields the highes t s patial reuse. Ho we v er, this is a combinatorial optimization problem of prohibitive complexit y ( O ( |E c | |E c | )) and is thus computationally infeasible. T o o vercome these problems, we prop o se a new algo rithm for STDMA link sc heduling under the realistic phy sical in terference mo del. Our algorithm is bas ed on the commu- nication graph mo del G c ( V , E c ) as w ell as SINR computations. Motiv ated b y tec hniques from matroid theory [52], w e dev elop a computationally feasible algorithm with demon- strably high spatial reuse. The essen ce of o ur alg orithm is to partition the set of com- m unication edges into subsets (forests) and color the edges in each subset sequen tia lly . The edges in eac h fo r est are considered in a random or der fo r colo ring, since random- ized algor ithms are kno wn to outp erfo rm deterministic algorithms, esp ecially when the c haracteristics of the input ar e not kno wn a priori [53]. A similar matroid-based net w ork partitioning techniq ue is used in [54] to gener- ate high capacit y subnet works for a distributed thr o ughput maximization pr o blem in wireless mesh net works. T echn iques from matroid theory hav e also b een employ ed to dev elop eﬃcien t heuristics for NP-hard comb inatorial optimization problems in ﬁelds suc h as distributed computer systems [5 5] and linear net work theory [56]. 3.2.3 ConﬂictF reeLinkSc hedule Algorithm W e call the prop osed p oin t to p oint link sc heduling algorithm as ConﬂictF reeLinkSc hed- ule (CFLS). The algorithm considers the comm unication graph G c ( V , E c ) and SINR con- ditions a nd is explained in Algorithm 4. In Phase 1 , w e lab el all the ve rtices randomly . Sp eciﬁcally , if G c ( · ) has v v ertices, w e 3.2. A High Spatial Reuse Lin k Scheduling Alg orithm 51 p erform a random p erm utation of the sequence (1 , 2 , . . . , v ) and assign these lab els to v ertices with indices 1 , 2 , . . . , v respectiv ely . L ( u ) denotes the lab el assigned to ve rtex u . In Phase 2, the communic ation graph G c ( · ) is decomp osed in to what are called out- orien ted and in- o rien ted g raphs T 1 , T 2 , . . . , T k [16]. Eac h T i is a forest and ev ery edge of G c ( · ) is in exactly one of the T i ’s. This decomp osition is ac hiev ed b y partitioning graph G c ( · ), the undirected equiv alent o f G c ( · ), into undirected forests. The num b er of fo rests can b e minimized b y using techniq ues fr om Matroid theory ([49], k - forest problem). Ho wev er, this optimal de comp osition req uires extensiv e computation. Hence, we adopt the faster alb eit non-optimal approa c h of using successiv e breadth ﬁrst searc hes [5 0] to decomp ose G c ( · ) into undirected forests. Eac h undirected fo r est is further ma pp ed to tw o directed for ests. In one forest, t he edges in ev ery connected comp onent p oint a wa y from t he ro ot and ev ery ve rtex ha s at most one incoming edge, th us pro ducing an out-orien ted graph. In the o t her forest, the edges in eve ry connected component point to w ard the ro ot and ev ery ve rtex has at most one outgoing edge, thus pro ducing an in-orien ted gr aph. In Phase 3, the orien ted graphs are conside red sequen tially . F or eac h orien ted graph, v ertices are considered in increasing order b y lab el and the unique edge asso ciated with eac h v ertex is colored using the FirstConﬂictF reeColor (FC F C) function. The F CFC function is explained in Algorithm 5. F or the edge under conside ration x , it discards an y color tha t has an edge with a primary conﬂict with x . Among the residual set of colors, w e c ho ose the ﬁrst color suc h t hat the resulting SINRs at the receiv er of x and the rece iv ers of all co-colored edges ar e no less than the comm unication threshold γ c . If no suc h color is found, w e a ssign a new color to x . Hence, this function guaran tees that the ensuing sche dule is conﬂict-free. 3.2.4 P erformance Results Sim ulation Mo del In the s im ulat io n exp erimen t s, the lo cation of ev ery node is generated randomly , using a uniform distribution for its X and Y co ordinates, in the deplo ymen t area. F or a fair comparison of our algorithm with the T runcated Gra ph-Based Sc heduling Algorithm 52 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del Algorithm 4 ConﬂictF reeLinkSc hedule (CFLS) 1: input: STDMA net work Φ( · ), communication gr a ph G c ( · ) 2: output: A coloring C : E c → { 1 , 2 , . . . } 3: lab el t he ve rtices of G c randomly { Phase 1 } 4: use successiv e breadth ﬁrst searc hes to partition G c in to oriented gra phs T i , 1 6 i 6 k { Phase 2 } 5: for i ← 1 to k do { Phase 3 b egins } 6: for j ← 1 to n do 7: if T i is out- orien ted then 8: let x = ( s, d ) b e suc h that L ( d ) = j 9: else 10: let x = ( s, d ) b e suc h that L ( s ) = j 11: end if 12: C ( x ) ← FirstConﬂictF reeColor( x ) 13: end for 14: end for { Phase 3 ends } Algorithm 5 in teger Fir stConﬂictF reeColor( x ) 1: input: STDMA net work Φ( · ), communication gr a ph G c ( · ) 2: output: A conﬂict-free color 3: C ← set o f existing colors 4: C c ← { C ( h ) : h ∈ E c , h is colored, x and h ha ve a primary edge conﬂict } 5: C cf = C \ C c 6: for i ← 1 to |C cf | do 7: r ← i th color in C cf 8: E i ← { h : h ∈ E c , C ( h ) = r } 9: C ( x ) ← r 10: if SINR at all receiv ers of E i ∪ { x } exceed γ c then 11: return r 12: end if 13: end for 14: return |C | + 1 3.2. A High Spatial Reuse Lin k Scheduling Alg orithm 53 (TGSA) [3 2 ], w e assume that the deploym en t region is a circular region of radius R . Th us, if ( X j , Y j ) a r e t he Cartesian co ordinates of j th no de, j = 1 , . . . , N , then X j ∼ U [ − R, R ] and Y j ∼ U [ − R, R ] sub ject to X 2 j + Y 2 j 6 R 2 . Equiv alently , if ( R j , Θ j ) are the p olar coor dina t es of j th no de, then R 2 j ∼ U [0 , R 2 ] and Θ j ∼ U [0 , 2 π ]. After generating random po sitions for N no des, w e ha ve complete info rmation of Φ( · ). Using (2.4) and (2.5), w e compute the comm unication and in t erference ra dii, and then map the net w ork Φ( · ) to the tw o-tier graph G ( V , E c ∪ E i ). Once the link sche dule is computed by an algorithm, σ is computed using (2.14). System para meters are c hosen based on their protot ypical v alues in wireless mesh net works [4 2]. F or a given set of system parameters, w e calculate the av erage spatial r euse by av eraging σ o v er 1000 randomly generated net works . Keeping all ot her parameters ﬁxed, w e observ e the eﬀect of increasing the n umber o f no des N on the av erage spatia l reuse. In o ur exp erimen ts, we compare the p erformance of t he following algorit hms: • Arb oricalLinkSc hedule (ALS) [1 6], • T runcated G r a ph-Based Sc heduling Algorithm 4 (TGSA) [32], • GreedyPh ysical (GP) [27], • Prop osed ConﬂictF reeLinkSc hedule (CFLS). P erformance Comparison under Path Loss Model In the ﬁr st set of exp erimen ts (Exp erimen t 1), w e a ssume that R = 500 m, P = 10 mW, β = 4, N 0 = − 90 dBm, γ c = 20 dB and γ i = 10 dB [42 ]. Th us, R c = 100 m a nd R i = 177 . 8 m . W e v ary the num b er of no des from 30 to 110 in steps of 5. Fig ure 3.3 plots t he av erage spatial reuse vs. num b er of no des for all the algor it hms. In the second set of exp erimen ts (Ex p erimen t 2), w e ass ume that R = 7 0 0 m, P = 15 mW, β = 4 , N 0 = − 85 dBm, γ c = 15 dB and γ i = 7 dB. Th us, R c = 110 . 7 m and 4 In T runcated Graph-B ased Scheduling Algorithm, for the computation of optimal num b er of tra ns- missions M ∗ , we follow the metho d descr ibed in [32]. Since 0 < ξ < N 0 P , we as s ume that ξ = 0 . 99 99 N 0 P and c ompute succes sive Edmundson-Madansky (EM) upp er b ounds [44], [45], till the diﬀerence b etw een successive EM bo unds is less than 0 . 3 %. W e hav e exp erimentally veriﬁed that only high v alues of ξ lea d to reaso nable v alues for M ∗ , wherea s low v alue s of ξ , say ξ = 0 . 1 N 0 P , lead to the extremely cons erv ative v alue of M ∗ = 1 in most case s. 54 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del 30 40 50 60 70 80 90 100 110 0 0.5 1 1.5 2 2.5 3 number of nodes average spatial reuse R = 500 m, P = 10 mW, β = 4, N 0 = −90 dBm, γ c = 20 dB, γ i = 10 dB ALS TGSA CFLS GP Figure 3.3: Spatial reuse vs. num b er of no des for Exp erimen t 1. R i = 175 . 4 m . W e v ary the num b er of no des from 70 to 150 in steps of 5. Fig ure 3.4 plots t he av erage spatial reuse vs. num b er of no des for all the algor it hms. F or t he ALS algor it hm, w e observ e that spatial reuse increases v ery slowly with increasing n umber of no des. F or the TGSA algo r it hm, w e observ e that spatia l reuse is 18-2 7% lo w er than that of ALS and 3 0 -55% lo w er than t ha t of GP . A pla usible explanation for this b eha vior is as follo ws. The basis for TGSA is the computatio n of M ∗ , the optimal n um b er of transmissions in ev ery slot [32]. M ∗ is determined by maximizing a lo w er b ound on the exp ected n um b er of success ful transmissions in a time slot. Since the par t it ioning o f a maximal indep enden t se t of comm unication arcs into subsets o f cardinalit y at most M ∗ is ar bit r a ry and not g eograph y-based, the re could b e scenarios where the transmissions sc heduled in a subset are in the vicinit y of eac h other, resulting in mo derate to high in terference. In e ssence, maximizing this lo w er bound do es not neces sarily translate to maximizing the n umber of success ful transmissions in a time slot. Also, due to its design, the TGSA alg orithm yields higher num b er of colors compared to ALS and GP . Though the GP a lgorithm is based on comm unication graph and SINR conditions, it yields sligh tly low er spatial r euse than CFLS. A p ossible r eason for this observ ation is as 3.2. A High Spatial Reuse Lin k Scheduling Alg orithm 55 70 80 90 100 110 120 130 140 150 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 number of nodes average spatial reuse R = 700 m, P = 15 mW, β = 4, N 0 = −85 dBm, γ c = 15 dB, γ i = 7 dB ALS TGSA CFLS GP Figure 3.4: Spatial reuse vs. num b er of no des for Exp erimen t 2. follo ws. The GP algorithm colors edges of the comm unication graph in the decreasing order of in terference n umber. The inte rference n umber of edge e is the num b er of edges e i suc h that, if ( e, e i ) are sc heduled sim ultaneously , then the SINR threshold c ondition (2.7) is violated along one or b oth links. Edges with higher in terference num b er tend to b e lo cated to wards the cen ter of the deplo yment region. Since these edges are colored ﬁrst, a large n umber of colors are utilize d in the initial stages of the algorithm, le ad to p oten tia lly higher sc hedule length and low er spatial reuse. A b etter techniq ue would b e success iv ely examine edges at the cen tre and the p eriphery , whic h is ac hiev ed by the partition t echniq ue employ ed by CFLS. F or the pro p osed CFLS algorithm, we observ e that spatial reuse increases steadily with increasing n umber of no des and is ab out 15 % higher tha n the s patial reuse of ALS, TGSA a nd G P . P erformance Comparison under Realistic Conditions In a realistic wirele ss en vironmen t, channe l impairmen ts lik e m ultipa th fa ding a nd shad- o wing aﬀect t he receiv ed SINR at a receiv er [14]. In this section, we compare the 56 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del p erformance of the ALS, T GSA, GP and CFLS algorithms in a wireless c hannel whic h exp eriences Ra yleigh fading and lognormal shado wing. In the absence of fading and shado wing, the SINR at receiv er r i,j is giv en b y (2.2). W e assume that ev ery algorithm (ALS, TGSA, GP and CFLS) considers o nly path loss in the channel prio r to constructing the t wo-tier graph G ( V , E c ∪ E i ) and computing the link sc hedule. Ho wev er, for computing the av erage spatial reuse o f each algor ithm, w e tak e in to ac- coun t fading and shado wing c ha nnel gains b et wee n each pair o f no des. More sp eciﬁcally , for computing the spatial reuse using (2.1 4), the ( actual) SINR at re ceiv er r i,j is giv en b y SINR r i,j = P D β ( t i,j ,r i,j ) V ( t i,j , r i,j )10 W ( t i,j ,r i,j ) N 0 + P M i k =1 k 6 = j P D β ( t i,k ,r i,j ) V ( t i,k , r i,j )10 W ( t i,k ,r i,j ) , (3.3) where ra ndom v ariables V ( · ) and W ( · ) corresp ond to c hannel gains due to Rayleigh fading a nd lognormal shadow ing resp ectiv ely . W e assume t ha t { V ( k , l ) | 1 6 k, l 6 N , k 6 = l } are independent and iden tically distributed (i.i.d.) ra ndo m v ariables with pro babilit y densit y function (p df ) [10 ] f V ( v ) = 1 σ 2 V e − v σ 2 V u ( v ) , ( 3.4) where u ( · ) is the unit step function. Also, { W ( k , l ) | 1 6 k , l 6 N , k 6 = l } are assumed t o b e i.i.d. zero mean Gaussian random v ariables with p df [57] f W ( w ) = 1 √ 2 π σ W e − w 2 2 σ 2 W . (3.5) Random v a r ia bles V ( · ) and W ( · ) a r e indep enden t of eac h ot her and also indep endent of the no de lo cations. The sim ulation mo del and experimen ts are exactly as described b efo r e. In the sim u- lations, w e assume σ 2 V = σ 2 W = 1. F or Exp erimen t 1, Figure 3.5 plots t he a verage spatial reuse vs. n umber of no des for all the algo r it hms. F or Exp erimen t 2, Fig ure 3.6 plots the av erage spatial reuse vs. n umber of no des fo r all the algorithms. F rom Figures 3.3, 3.4, 3.5 a nd 3.6, w e observ e that spatial reuse decreases b y 20- 40% in a c hannel exp eriencing m ultipat h fading and shadowing eﬀects. A plausible explanation for this observ a t io n is as f ollo ws. Since the c ha nnel gains b et w een ev ery pair 3.2. A High Spatial Reuse Lin k Scheduling Alg orithm 57 30 40 50 60 70 80 90 100 110 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 number of nodes average spatial reuse R = 500 m, P = 10 mW, β = 4, N 0 = −90 dBm, γ c = 20 dB, γ i = 10 dB, σ V 2 = 1, σ W 2 = 1 ALS TGSA CFLS GP Figure 3.5: Spat ia l reuse vs. n um b er of no des for Exp erimen t 1 under m ultipath fading and shado wing channe l conditions. 70 80 90 100 110 120 130 140 150 0 0.5 1 1.5 2 2.5 3 number of nodes average spatial reuse R = 700 m, P = 15 mW, β = 4, N 0 = −85 dBm, γ c = 15 dB, γ i = 7 dB, σ V 2 = 1, σ W 2 = 1 ALS TGSA CFLS GP Figure 3.6: Spat ia l reuse vs. n um b er of no des for Exp erimen t 2 under m ultipath fading and shado wing channe l conditions. 58 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del of no des are indep enden t of eac h other, it is reasonable to assume that the interference p o w er a t a t ypical receiv er remains almost the same as in the non-fa ding case. This is b ecause, ev en if the p o w er receiv ed fro m f ew uninte nded transmitters is lo w, the p o w er receiv ed from other unin tended transmitters will b e high (on a ve rage); thus the in terference p ow er remains constant. Consequen tly , the c hang e in SINR is determined b y the c ha nge in receiv ed signal p ow er only . If the receiv ed signal p o w er is higher compared to the non-fading case, the transmission is anyw a y succe ssful and spatial reus e remains unc hanged (see (2.14)). How ev er, if the receiv ed signal p ow er is low er, the transmission is now unsuccessful and spatia l reuse decreases. Hence, on a v erage, the spatial reuse decreases. Finally , from Figures 3.5 and 3 .6, w e o bserv e that the prop osed CFLS algorithm ac hiev es 5-17% higher spatial reuse than the ALS and G P algorithms a nd 40 - 80% higher spatial reuse than the TGSA algorit hm, under realistic wireless c hannel conditions. 3.2.5 Analytical Results In this se ction, w e deriv e upp er b ounds on the running time (computatio na l) complexity of ConﬂictF reeLinkSc hedule a lgorithm. W e use the follo wing notation with resp ect t o the communic ation g raph G c ( V , E c ): e = num b er of comm unication e dges , v = n um b er of v ertices , θ = thic kness of the gra ph := minim um n um b er of planar graphs into whic h the undirected equiv alen t of G c ( · ) can b e partitioned . Before we pro v e our results, it is instructiv e to observ e Figure 3 .7, whic h sho ws the v ariation of θ and e with v for the tw o exp erimen t s describ ed in Section 3.2.4. Since determining the thic kness of a gra ph is NP-hard [58 ], eac h v alue of θ in F igure 3.7 is an upp er b ound on the actual thic kness base d on the nu m b er of forests in to whic h the undirected equiv alent o f the comm unication g r a ph has b een decomp osed using successiv e breadth ﬁrst se arc hes. W e observ e that the graph t hic kness increases v ery slo wly with the n um b er of ve rtices ( θ ≪ v ), while the nu m b er of edges increases sup er-linearly with 3.2. A High Spatial Reuse Lin k Scheduling Alg orithm 59 30 40 50 60 70 80 90 100 110 0 20 40 60 80 100 120 140 160 180 200 number of nodes graph parameters R = 500 m, P = 10 mW, β = 4, N 0 = −90 dBm, γ c = 20 dB, γ i = 10 dB Thickness Number of Edges (a) Exp eriment 1 70 80 90 100 110 120 130 140 150 0 50 100 150 200 250 number of nodes graph parameters R = 700 m, P = 15 mW, β = 4, N 0 = −85 dBm, γ c = 15 dB, γ i = 7 dB Thickness Number of Edges (b) E xper imen t 2 Figure 3.7: Comparison of thic kness and n umber of edges with num b er of v ertices. 60 Chapter 3. P oint to P oin t Link Sc h ed uling based on Comm un ication Graph Mo del the num b er of v ertices. Lemma 3.2.1. An orien te d gr aph T c an b e c olor e d using no mor e than O ( v ) c olors using ConﬂictF r e eLink Sche dule. Pr o of. Since an oriented graph with v v ertices h as at most v edges, the edges of T can b e colo red with at most v colors.  Lemma 3.2.2. F o r a n o rie nte d gr aph T , the running time of ConﬂictF r e eLinkS che dule is O ( v 2 ) . Pr o of. Assuming that an ele men t can b e c hosen randomly and uniformly from a ﬁnite set in unit time ([53], Chapter 1 ), the running time of Phase 1 can b e show n to b e O ( v ). Since there is only one orien ted g raph, Phase 2 runs in time O ( v ). In Phase 3, the unique edge asso ciated with the v ertex under consideration is assigned a color using FirstConﬂictF reeColor. F rom Lemma 3 .2.1, the size o f the set of colors to b e examined |C c ∪ C cf | is O ( v ). In FirstConﬂictF reeColor, the SINR is chec k ed only o nce fo r ev ery colored edge in the set S |C cf | i =1 E i and at most v times fo r t he edge under consideration x . With a careful implemen tation, Fir stConﬂictF reeColor runs in time O ( v ). So, the running time o f Phase 3 is O ( v 2 ). Th us, the total running time is O ( v 2 ).  Theorem 3.2.3. F or an arbitr ary gr aph G , the running time of ConﬂictF r e eLi n kSche d- ule is O ( ev log v + ev θ ) . Pr o of. Assuming that an ele men t can b e c hosen randomly and uniformly from a ﬁnite set in un it time [53], the running time of Ph ase 1 c an b e s ho wn to b e O ( v ). F or Phase 2, the optimal partitioning tec hnique of [49] based on Matroids can b e used to partition the comm unication graph G c in to at most 6 θ orien ted g r aphs in time O ( ev log v ). Thus , k 6 6 θ holds for Phase 3. F rom Lemma 3.2.2, it fo llo ws that the ﬁrst oriente d gra ph T 1 can be colored in time O ( v 2 ). How ev er, consider the coloring of j th orien ted graph T j , where 2 6 j 6 k . When c oloring edge x from T j using Fir stConﬂictF reeColor, conﬂicts can o ccur not only with the colored edges of T j , but also with the edges of the previously colored orien ted g raphs T 1 , T 2 , . . . , T j − 1 . Hence, the w o r st-case size of t he set of colors to b e examined |C c ∪ C cf | is O ( e ). Note that in FirstConﬂictF reeColor, the SINR is c hec k ed o nly once for ev ery colored edge in the set S |C cf | i =1 E i and at most e times for the 3.2. A High Spatial Reuse Lin k Scheduling Alg orithm 61 edge under cons ideration x . With a careful implemen tatio n, FirstConﬂictF reeColor runs in time O ( e ). Hence, an y subsequen t orien ted graph T j can b e colored in time O ( ev ). Th us, the running time of Phase 3 is O ( ev θ ). Therefore, t he ov erall running time of ConﬂictF r eeLinkSc hedule is O ( ev log v + ev θ ).  3.2.6 Discussion In this section, w e hav e dev elop ed ConﬂictF reeLinkSc hedule, a p o int to p o int link sc heduling algorithm for an STDMA wireless mesh net work under the phy sical interfer- ence mo del. The p erformance of the prop osed algo rithm is sup erior to those o f existing link sc heduling algo rithms f o r STDMA wireless net works with uniform p ow er assign- men t. A practical exp erimen tal mo deling shows that, o n av erage, the prop osed alg orithm ac hiev es 20% higher spatial reuse than the Arb oricalLinkSc hedule [16], G r eedyPh ysical [27] and T runcated Graph-Based Schedu ling [32] alg orithms. Since link sc hedules are constructed oﬄine only once and then used by the net work for a long p erio d of t ime, these impro vem en ts in p erformance directly translate to higher long- term net work throug hput. The computational complexit y of ConﬂictF reeLinkSc hedule is comparable to the computational complexit y of Arb oricalLinkSc hedule a nd is m uc h lo w er tha n the com- putational complexit y of GreedyPh ysical and T runcated Graph- Ba sed Sche duling al- gorithms. Th us, in cognizance of spatial r euse as we ll a s computatio nal complexit y , ConﬂictF r eeLinkSc hedule app ears to b e a go o d candidate fo r eﬃcien t STDMA link sc heduling algorithms. Chapter 4 P oi n t to P oin t Li nk Sc heduling based on SI NR Graph Mo del In this chapter, we prop ound a somewhat diﬀeren t approa ch for p oin t to p oin t link sc heduling in an STDMA w ireless net work under the ph ysical in terference mo del. This approac h is based o n SINR gr a ph represen tation of the netw ork wherein w eights of edges corresp ond to in terf erences b etw een pairs of nodes and w eigh ts of v ertices correspond to normalized noise p o w ers at receiving no des. W e dev elop a nov el link sc heduling algo rithm with p o lynomial time complexit y and improv ed p erformance in terms of spatial reuse. The rest of the chapter is org anized as f ollo ws. W e motiv ate our SINR graph approa c h in Section 4.1. W e describ e the prop osed link sc heduling algorithm and provide an illustrativ e example in Section 4.2. W e pr ov e the correctness of the alg o rithm and deriv e its computational complexit y in Section 4 .3. The p erformance of the prop osed algorithm is compared with existing link sc heduling alg o rithms in Section 4.4. W e discuss the implications of our work in Section 4.5. 4.1 Motiv ati o n The system mo del, notation and problem formulation are exactly as describ ed in Section 3.2.1. Speciﬁcally , w e seek a lo w complexit y conﬂict-free p oint to p oint link sc heduling algorithm t ha t a c hiev es high spatial reuse. In general, for the STDMA wireless netw ork Φ( · ), the set of links to b e sc heduled 63 64 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del is determined b y a routing algorithm. F or simplicit y , w e only consider exhaustiv e link sc hedules, i.e., w e consider uniform load o n all links. Note that for p oin t to p oint link sc hedules that are conﬂict-free, i.e., for link sche dules that satisfy ( 2 .7), the equation for spatial reuse (2.14) reduces to Spatial Reuse = σ = e C , (4.1) where e denotes the num b er of directed edges in t he comm unication graph G c ( V , E c ) and C denotes the n um b er of slots in the link sc hedule. Therefore, for conﬂict-free link sc hedules, maximizing spatial reuse is equiv alen t to minimizing the n umber of colors, i.e., minimizing the sc hedule length. T o the b est of our know ledge, there is no kno wn p olynomial time a lgorithm that determines a pro v ably optimal sc hedule (minim um length sc hedule) for an STDMA wireless net w ork with constrained transmission p ow er. Hence, the only r ecourse is to devise heuristics and show their eﬃciency theoretically and exp erimen tally . T o w ar ds this end, w e prop ose a heuristic based on an SINR graph repres en tatio n of the net w o r k. Consider any directed graph G ( V , E ), where V is t he set of v ertices and E is the set of edges. The line graph of G ( V , E ) is the graph G ′ ( V ′ , E ′ ) whose v ertices ar e the edges of G ( · ), i.e., V ′ = E [35]. The SINR graph that w e consider in this c hapter is analo g ous to the concept of line graph in [35]. How ev er, unlik e the line graph, we assum e that the SINR graph is a complete graph, i.e., for an y tw o distinct v ertices v ′ i , v ′ j ∈ V ′ , there is a directed edge fr o m v ′ i to v ′ j in E ′ . The crux of the prop osed link sc heduling algo r it hm can b e understo o d by revisit- ing the condition fo r successful pac k et reception under the ph ysical in terference mo del (Equation 2 .6), i.e., P D β ( t i,j ,r i,j ) N 0 + P M i k =1 k 6 = j P D β ( t i,k ,r i,j ) > γ c . (4.2) Rearranging the t erms in (4.2), w e obtain N 0 γ c P D β ( t i,j , r i,j ) + M i X k =1 k 6 = j γ c D β ( t i,j , r i,j ) D β ( t i,k , r i,j ) 6 1 . (4.3) Dropping t ime slot inde x i for clarit y , w e obtain the “equiv alen t” SINR threshold con- 4.2. SINR GraphLinkSchedule Algorithm 65 dition N 0 γ c P D β ( t j , r j ) + M X k =1 k 6 = j γ c D β ( t j , r j ) D β ( t k , r j ) 6 1 , (4.4) where t j , r j and M can b e interpreted as j th transmitter, j th receiv er and n um b er of concurren t transmissions, resp ectiv ely , in a giv en time slot. The terms a pp earing in (4.4) corresp ond to ve rtex a nd edge w eights in a sp ecial graph represen tatio n o f the STDMA net work, termed as SINR graph. This idea will b e elucidated further in Section 4.2.1. 4.2 SINR GraphLinkS c hedul e Algorith m In this section, w e explain t he pr o p osed link sche duling algorithm based on SINR graph represen tation of the STDMA net work. W e pro vide an illustrativ e example to elucidate the intricacies of the prop o sed algorit hm. 4.2.1 Description The prop osed link sc heduling algorithm under the phys ical in t erference mo del is SINR Gr aphLinkSc hedule (SGLS), whic h considers the comm unication graph G c ( V , E c ). First, w e construct a direc ted complete SINR g raph G ′ ( V ′ , E ′ ) that has the edges of G c ( · ) as its v ertices, i.e., V ′ = E c . Let the edges of G c ( · ) and the corresp onding v ertices of G ′ ( · ) b e labeled 1 , 2 , . . . , e . Let t i and r i denote the transmitter and rece iv er resp ectiv ely of edge i in G c ( · ). F or an y t w o edges i and j in g raph G c ( · ), the interfer enc e weight function w ij is deﬁned as: w ij :=    1 if i and j ha v e a common ve rtex , γ c D ( t j ,r j ) β D ( t i ,r j ) β otherwise . The inte rference w eigh t function w ij indicates the in terference ene rgy at r j due to tr a ns- mission f rom t i to r i scaled with res p ect to the signal energy of t j at r j . Note that the in terference w eigh t f unction app ears as a summand in the equiv alen t SINR threshold condition (4.4). W e then compute the c o-sche dulability weight function w ′ . F or a n y t w o edges i and j in G c ( · ), the w eight of edge e ′ ij in G ′ ( · ) is giv en b y w ′ ij = max { 0 , 1 − w ij } . Since w ij 66 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del Algorithm 6 SINR G raphLinkSc hedule (SGLS) 1: Input: Comm unication graph G c ( V , E c ), γ c , N 0 , P 2: Output: A coloring C : E c → { 1 , 2 , . . . } 3: V ′ ← E c 4: Construct t he directed complete graph G ′ ( V ′ , E ′ ) 5: for all e ′ ij ∈ E ′ do 6: if edges i and j hav e a common v ertex in G c ( · ) t hen 7: w ij ← 1 8: else 9: w ij ← γ c D ( t j ,r j ) β D ( t i ,r j ) β 10: end if 11: end for 12: for all e ′ ij ∈ E ′ do 13: w ′ ij ← max { 0 , 1 − w ij } 14: end for 15: for all v ′ j ∈ V ′ do 16: N ( v ′ j ) ← N 0 γ c P D ( t j , r j ) β 17: end for 18: p ← 0; V ′ uc ← V ′ 19: while V ′ uc 6 = φ do 20: p ← p + 1; c ho ose v ′ ∈ V ′ uc randomly 21: C ( v ′ ) ← p ; V ′ uc ← V ′ uc \ { v ′ } ; V ′ c p ← { v ′ } ; ψ ← 1 22: while ψ = 1 and V ′ uc 6 = φ and max y ′ ∈V ′ uc P x ′ ∈V ′ c p w ′ x ′ y ′ + w ′ y ′ x ′ > 0 do 23: for all u ′ ∈ V ′ uc suc h that P x ′ ∈V ′ c p w ′ x ′ u ′ + w ′ u ′ x ′ > 0 do 24:  ← 1 25: for all v ′ c ∈ V ′ c p do 26: if P v ′ 1 ∈V ′ c p \{ v ′ c }∪{ u ′ } w ′ v ′ 1 v ′ c 6 |V ′ c p | + N ( v ′ c ) − 1 then 27:  ← 0 28: end if 29: end for 30: if  = 1 and P v ′ 2 ∈V ′ c p w ′ v ′ 2 u ′ > |V ′ c p | + N ( u ′ ) − 1 then 31: C ( u ′ ) ← p ; V ′ c p ← V ′ c p ∪ { u ′ } ; V ′ uc ← V ′ uc \ { u ′ } 32: else 33:  ← 0 34: end if 35: end for 36: if  = 0 then 37: ψ ← 0 38: end if 39: end w hile 40: end while 4.2. SINR GraphLinkSchedule Algorithm 67 and w j i represen t in t erferences a mong links i a nd j in the STDMA net w ork Φ( · ), w ′ ij and w ′ j i in tuitively represen t the co- sc hedulabilit y of links i and j in Φ( · ) (equiv alen tly , co-sc hedulability of v ertices i and j in G ′ ( · )). F or example, if w ij is greater tha n or equal to 1, t hen the interference at the receiv er of link j from the transmitter of link i is v ery high and these links cannot be sc heduled sim ultaneously . This will result in w ′ ij b eing equal to 0 indicating that vertice s i and j in G ′ ( · ) a r e not co-sc hedulable. On the other hand, if w ij is slightly greater than 0 (0 < w ij ≪ 1), w ′ ij will b e sligh tly less than 1 indicating that the v ertices i and j in G ′ ( · ) ar e co- sche dulable. Note that for the SINR graph G ′ ( · ), the w eigh t of an edge refers to the v alue o f co- sche dulabilit y function for that edge. Next, w e determine the normalized no ise pow er at the receiv er of each link of Φ( · ). This is ta ntamoun t to computing the normalized noise p o w er for eac h edge of G c ( · ), i.e., at eac h v ertex of G ′ ( · ). Note that the normalized noise p ow er function app ears as a term in t he equiv alen t SINR threshold condition (4.4). Our ob jectiv e is to color the v ertices of G ′ ( · ) (equiv alently , edges of G c ( · )) using minim um n um b er of colors under the ph ysical in terference mo del, i.e., sub ject to the condition that the SINR a t the receiv er of ev ery link in Φ( · ) is no less than the commu- nication threshold γ c . Equiv alen tly , for an y V ′ cc ⊆ V ′ , the coloring of all vertice s v ′ i ∈ V ′ cc with the same color is deﬁned to b e f e a s ible if P D ( t v ′ i ,r v ′ i ) β N 0 + P v ′ j ∈V ′ cc \{ v ′ i } P D ( t v ′ j ,r v ′ i ) β > γ c ∀ v ′ i ∈ V ′ cc . (4.5) In the SINR graph G ′ ( · ), this conditio n translates to the sum of we igh t s of edges incoming to a v ertex from all co-colored v ertices b eing greater than the sum o f the nu m b er of r emaining co-colored v ertices a nd the nor malized noise p ow er minus a constan t factor (unit y); this will b e prov ed in Theorem 4.3.1. Finally , we color v ertices of G ′ ( · ), i.e., edges o f G c ( · ), according to the following pro cedure. L et V ′ uc denote the set of uncolored v ertices of G ′ ( · ). Initially , V ′ uc includes all v ertices of G ′ ( · ). First, w e choose a vertex randomly from V ′ uc . This is assigne d a new color, sa y p . Then, w e consider ev ery v ertex u ′ from V ′ uc suc h that the sum of w eigh t s of all the edges b et w een u ′ and the vertice s colored with p is p ositiv e. Next, for each ve rtex colored with p , w e c hec k if the sum of weigh ts of all incoming e dges is greater than the 68 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del sum of the n um b er of vertice s colored with p and the nor ma lized noise p ow er at tha t v ertex min us a constan t factor (unit y). If this inequalit y is satisﬁed, we further c heck if the sum of we igh ts of all e dges incoming to u is greater tha n the s um of the num b er of v ertices colo red with p and the nor malized noise p ow er at u ′ min us unity . If this inequalit y is also satisﬁed, then v ertex u ′ is colored with p . If a n y of these inequalities are not satisﬁe d, ve rtex u ′ is colored with a new color. The algorithm exits when a ll the v ertices are colored. The pseudo co de of the algorithm is pro vided in Alg o rithm 6 . 4.2.2 Example Consider the STDMA wireless netw ork Φ( · ) whose deplo ymen t is sho wn in Figure 4.1. It consists of four lab eled nodes whose co ordinates (in meters) are 1 ≡ ( − 40 , 5 ) , 2 ≡ ( 0 , 0), 3 ≡ (95 , 0) a nd 4 ≡ (135 , 0 ). W e use typic al v alues of sys tem parameters in wireless net works [42]. These v alues are sho wn in T able 4 .1, which lead to R c = 100 m. 1 ≡ ( − 40 , 5) X 3 ≡ (95 , 0) 2 ≡ (0 , 0) 4 ≡ (135 , 0) Y Figure 4.1: An STDMA wireless net work with four no des. P arameter Sym b ol V alue transmission p o w er P 10 mW path loss exp onent β 4 noise p ow er sp ectral densit y N 0 -90 dBm comm unication thres hold γ c 20 dB T able 4.1 : System parameters for the STDMA netw ork sho wn in Figure 4.1. The comm unication graph mo del of the STDMA net work is sho wn in Figure 4.2. The comm unication graph G c ( V , E c ) consists of four v ertices and six directed edges. The 4.2. SINR GraphLinkSchedule Algorithm 69 v ertex and edge sets are giv en b y V = { v 1 , v 2 , v 3 , v 4 } , (4.6) E c = { (1 , 2) , (2 , 1 ) , (2 , 3) , (3 , 2) , (3 , 4 ) , (4 , 3) } . (4.7) v 3 v 2 v 4 v 1 (2 , 3) (1 , 2) (3 , 4) (2 , 1) (3 , 2) (4 , 3) Figure 4.2: Comm unication gra ph mo del of STDMA netw ork describ ed b y Figure 4.1 and T a ble 4.1 . The SINR g raph mo del of the communic ation graph G c ( V , E c ) is sho wn in Figure 4.3. The SINR graph G ′ ( V ′ , E ′ ) is a complete graph and consists of six vertice s and thirty directed edges. The v ertex set of the SINR g raph is giv en b y V ′ = { (1 , 2) , (2 , 1 ) , (2 , 3) , (3 , 2) , (3 , 4 ) , (4 , 3) } . (4.8) The edge set E ′ of t he SINR graph is en umerated in T able 4.2, along with the in terference w eight function w ij and co-sc hedulability w eigh t function w ′ ij for eac h edge i → j ∈ G ′ ( · ). The normalized noise p ow ers at ve rtices of the SINR gra ph are en umerated in T able 4.3. The truncated SINR graph G ′ t ( V ′ , E ′ t ) is sho wn in Figure 4 .4 . The truncated SINR graph consists of all v ertices of the SINR gra ph and only those edges whose co-sc hedulability w eight func tion is p ositiv e, i.e., E ′ t = { ( i, j ) : i, j ∈ E c and w ′ ij > 0 } . The v alues of t he co-sc hedulability w eight functions f or all edges and the normalized noise p ow ers at all v ertices are a lso show n in the ﬁgure. W e use the truncated SINR graph to ex plain the SGLS algorithm, sinc e edges ha ving zero w eigh t in the SINR graph do not pla y an y role in the SGLS algorithm. Not e that, in the truncated SINR graph, the w eigh t of an edge refers to t he v alue of the co- sc hedulabilit y we igh t function for that edge. Initially , the set of uncolored v ertices is V ′ uc = { (1 , 2) , ( 2 , 1) , (2 , 3) , (3 , 2) , ( 3 , 4) , (4 , 3) } . In the ﬁrst iteration, w e randomly c ho ose v ′ = (1 , 2) and assign it Color 1 (sa y , red). So, C (1 , 2) = 1. The s et of uncolored v ertices is V ′ uc = { (2 , 1 ) , (2 , 3) , (3 , 2) , (3 , 4 ) , (4 , 3) } and 70 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del 1 , 2 2 , 1 2 , 3 3 , 2 3 , 4 4 , 3 Figure 4.3: SINR graph mo del of comm unication graph sho wn in Figure 4.2 . 1 , 2 2 , 1 2 , 3 3 , 2 3 , 4 4 , 3 0.8145 0.0256 0.0256 0.0264 0.0264 0.8145 0.2072 0.2293 0.2050 0.2314 0.7189 0.7275 Figure 4.4: T runcated SINR graph derive d from SINR graph show n in Figure 4.3 and w eight v alues giv en in T a bles 4 .2 and 4.3. 4.2. SINR GraphLinkSchedule Algorithm 71 Edge i → j of Edge i → j of SINR g raph G ′ ( V ′ , E ′ ) w ij w ′ ij SINR g raph G ′ ( V ′ , E ′ ) w ij w ′ ij (1 , 2) → (2 , 1) 1 0 (3 , 2) → (1 , 2) 1 0 (2 , 1) → (1 , 2) 1 0 (1 , 2) → (3 , 4) 0.2725 0.727 5 (2 , 1) → (2 , 3) 1 0 (3 , 4) → (1 , 2) 3.2420 0 (2 , 3) → (2 , 1) 1 0 (2 , 1) → (3 , 2) 1 0 (2 , 3) → (3 , 2) 1 0 (3 , 2) → (2 , 1) 1 0 (3 , 2) → (2 , 3) 1 0 (2 , 1) → (3 , 4) 0.7707 0.229 3 (3 , 2) → (3 , 4) 1 0 (3 , 4) → (2 , 1) 0.7928 0.207 2 (3 , 4) → (3 , 2) 1 0 (2 , 1) → (4 , 3) 3.1430 0 (3 , 4) → (4 , 3) 1 0 (4 , 3) → (2 , 1) 0.2811 0.718 9 (4 , 3) → (3 , 4) 1 0 (2 , 3) → (3 , 4) 1 0 (4 , 3) → (1 , 2) 0.7950 0.2050 (3 , 4) → (2 , 3) 1 0 (1 , 2) → (4 , 3) 0.7686 0.2314 (2 , 3) → (4 , 3) 1 0 (1 , 2) → (2 , 3) 1 0 (4 , 3) → (2 , 3) 1 0 (2 , 3) → (1 , 2) 1 0 (3 , 2) → (4 , 3) 1 0 (1 , 2) → (3 , 2) 1 0 (4 , 3) → (3 , 2) 1 0 T able 4.2: In terference and c o-sc hedulabilit y weigh t functions for edges of SINR graph sho wn in F ig ure 4.3. V ertex v ′ j of SINR g raph G ′ ( V ′ , E ′ ) N ( v ′ j ) (1,2) 0.0264 (2,1) 0.0264 (2,3) 0.8145 (3,2) 0.8145 (3,4) 0.0256 (4,3) 0.0256 T able 4.3 : Normalized noise p ow ers a t vertice s of SINR graph sho wn in Figure 4 .3. 72 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del the set of v ertices colored 1 is V ′ c 1 = { (1 , 2) } . F rom the set o f uncolored v ertices V ′ uc , w e consider ev ery v ertex u ′ suc h that the sum of weigh ts of ed ges from the presen tly colored v ertex (1 , 2 ) to u ′ and fro m u ′ to the presen tly colored ve rtex is p ositiv e. F rom Figur e 4.4, w e obt a in t w o candidates: u ′ = (3 , 4) and u ′ = (4 , 3). W e ﬁrst examine the candidate v ertex (3 , 4). W e chec k if the w eigh t of t he edge from (3 , 4 ) to the presen tly color ed v ertex (1 , 2) is no greater than the n um b er of v ertices colored with the presen t color (red) plus the normalized noise p ow er at the colored verte x minus unity . Our calculations sho w that inequality holds (0 < 0 . 0 264) and candidate v ertex (3 , 4) cannot b e assigned Color 1. W e next ex amine the candidate v ertex (4 , 3 ). W e c hec k if the weigh t of the edge from the candidate v ertex to (1 , 2) is no greater than the n umber o f v ertices colored with the presen t color plus the normalized noise p o wer a t (1 , 2 ) minus unit y . Our calculations sho w tha t inequalit y do es not hold (0 . 2050 6 6 0 . 026 4 ). F urthermore, w e chec k if the w eight of the edge from the presen tly colored v ertex ( 1 , 2) to the candidate v ertex ( 4 , 3) is greater than the num b er o f v ertices colored red plus the normalized noise p ow er at (4 , 3 ) min us unit y . The inequalit y holds and hence the candidate v ertex (4 , 3) is assigned Color 1 (red). So, C (4 , 3) = 1. The set of uncolored v ertices is V ′ uc = { (2 , 1) , ( 2 , 3) , (3 , 2) , (3 , 4) } and the set o f v ertices colored 1 is V ′ c 1 = { (1 , 2) , ( 4 , 3) } . Again, from the set of uncolored v ertices V ′ uc , w e consider ev ery v ertex u ′ suc h that the sum of w eigh ts of edges from the presen tly colored vertice s { (1 , 2) , ( 4 , 3) } to u ′ and from u ′ to the presen tly colored vertice s is p ositiv e. F rom Figure 4.4, the candidate vertices are (2 , 1) and (3 , 4). Consider the candidate vertex (2 , 1). F or ev ery v ertex v ′ c colored 1, we c hec k if the sum of w eights of edges from remaining co-colored v ertices and the candidate v ertex to the colored v ertex is no greater than the num b er of co-colored v ertices and the nor malized no ise p o wer at the colored v ertex minus unit y . F or the colored v ertex (1 , 2), our calculations sho w that inequalit y holds (0 . 205 0 6 1 . 0264). So, w e discard (2 , 1), consider the next candidate v ertex (3 , 4) and p erform an a nalogous comparison with v ′ c = (1 , 2). Since inequalit y holds in this case to o (0 . 2050 6 1 . 0 264), we discard (3 , 4) and pro ceed to the next iteration. The set of v ertices colored so far is show n in Figure 4.5. In the second iteration, w e ra ndomly c ho ose v ′ = (2 , 3) and assign it Color 2 (sa y , blue). So , C (2 , 3) = 2. The set of uncolored v ertices is V ′ uc = { (2 , 1) , ( 3 , 2) , (3 , 4) } and the set of v ertices colored 2 is V ′ c 2 = { (2 , 3) } . F rom the set o f uncolored v ertices V ′ uc , w e consider ev ery v ertex u ′ suc h that the sum of weigh ts of ed ges from the presen tly colored 4.2. SINR GraphLinkSchedule Algorithm 73 1 , 2 2 , 1 2 , 3 3 , 2 3 , 4 4 , 3 0.8145 0.0256 0.0256 0.0264 0.0264 0.8145 0.2072 0.2293 0.2050 0.2314 0.7189 0.7275 Figure 4.5: Coloring of v ertices of truncated SINR graph after ﬁrst iteration of SG LS algorithm. v ertex (2 , 3) to u ′ and from u ′ to the presen tly colored vertex is p ositiv e. F rom Figure 4.5, no suc h v ertex exists. So, w e pro ceed to the next iteration. The vertice s colored so far a r e shown in Figure 4.6. In the third iteration, w e randomly choose v ′ = (3 , 4) and assign it Color 3 (say , green). So, C (3 , 4) = 3. The set of uncolored v ertices is V ′ uc = { (2 , 1) , ( 3 , 2) } and the set of vertic es colored 3 is V ′ c 3 = { (3 , 4) } . F rom the set of uncolored vertice s V ′ uc , w e consider ev ery v ertex u ′ suc h that the sum of w eights of e dges from the presen tly colored v ertex (3 , 4) to u ′ and from u ′ to the presen tly colo red v ertex is p o sitiv e. F rom Figure 4.4, w e obtain u ′ = (2 , 1) as the only candidate v ertex. Next, w e c hec k if the w eight o f the edge fro m the candidate v ertex (2 , 1) to the presen tly colored v ertex (3 , 4) is no greater than the n um b er of v ertices c olored w ith the presen t color (g reen) plus the normalized noise p o w er at the colored ve rtex minus unity . Our calculations sho w that inequalit y do es not hold (0 . 2293 6 6 0 . 0256). So, w e further c hec k if the w eigh t of the edge f r o m the presen tly colored ve rtex (3 , 4) to the candidate v ertex (2 , 1) exceeds the num b er of v ertices colored with the presen t color plus the normalized no ise p ow er at the candidate 74 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del 1 , 2 2 , 1 2 , 3 3 , 2 3 , 4 4 , 3 0.8145 0.0256 0.0256 0.0264 0.0264 0.8145 0.2072 0.2293 0.2050 0.2314 0.7189 0.7275 Figure 4.6: Color ing of v ertices of truncated SINR graph after second it era t io n of SGLS algorithm. v ertex min us unity . Since the ineq ualit y holds ( 0 . 2072 > 0 . 0264), the candidate v ertex (2 , 1) is assigned Color 3 ( green). So, C (2 , 1) = 3. The set of uncolored v ertices is V ′ uc = { (3 , 2 ) } a nd the set of ve rtices colored green is V ′ c 3 = { (3 , 4) , (2 , 1) } . Next, from the set of uncolored v ertices V ′ uc , w e choose that uncolored ve rtex u ′ suc h that the sum of w eigh ts of edges from u ′ to the set of presen tly colored v ertices { (3 , 4) , ( 2 , 1) } and from { (3 , 4 ) , (2 , 1) } to u ′ is p ositiv e. F rom Figure 4.6, no suc h v ertex u ′ exists. So, w e pro ceed to the next iteration. Fig ur e 4.7 sho ws the set o f v ertices colored so far. In the fourt h iteration, (3 , 2) is the only uncolored vertex . So, w e c ho ose v ′ = (3 , 2 ) and a ssign it Color 4 (say , pink). The set of v ertices colored 4 is V ′ c 4 = { (3 , 2) } a nd the set of uncolored v ertices is V ′ uc = φ . So, the algo rithm ends. The ﬁnal color ing of v ertices of the truncated SINR graph by SGLS a lgorithm is shown in Figure 4.8. The output of the SG LS algorithm is enume rated in T able 4.4 and is also sho wn pictorially in Figure 4.9. The resulting link sc hedule is denoted by Ψ( S 1 , S 2 , S 3 , S 4 , S 5 ), 4.2. SINR GraphLinkSchedule Algorithm 75 1 , 2 2 , 1 2 , 3 3 , 2 3 , 4 4 , 3 0.8145 0.0256 0.0256 0.0264 0.0264 0.8145 0.2072 0.2293 0.2050 0.2314 0.7189 0.7275 Figure 4.7: Coloring of v ertices of truncated SINR gr aph af t er third iterat io n of SGLS algorithm. where S 1 = { 1 → 2 , 4 → 3 } , S 2 = { 2 → 3 } , S 3 = { 3 → 4 , 2 → 1 } , S 4 = { 3 → 2 } . Finally , we c hec k if the link sche dule en umerated in T able 4.4 is conﬂict-free, i.e., if Time slot Color Activ e (transmitter, receiv er) pair s 1 red ( 1,2), (4,3 ) 2 blue (2,3) 3 green (3,4), ( 2 ,1) 4 pink (3,2) T able 4.4 : Output of SGLS algorithm for STDMA net w ork describ ed b y Figure 4.1 and T able 4.1 . 76 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del 1 , 2 2 , 1 2 , 3 3 , 2 3 , 4 4 , 3 0.8145 0.0256 0.0256 0.0264 0.0264 0.8145 0.2072 0.2293 0.2050 0.2314 0.7189 0.7275 Figure 4.8: Coloring of vertice s of truncated SINR graph after complete execution of SGLS alg o rithm. v 3 v 2 v 4 v 1 Figure 4.9: Output o f SG LS a lgorithm for STDMA netw ork describ ed by Fig ur e 4.1 and T able 4.1 . 4.3. Analytical Results 77 the SINR threshold condition (2.7) is satisﬁe d at eve ry rec eiv er for the STDMA netw ork described b y F igure 4.1 and T able 4.1. Only o ne transmitter-receiv er pair is activ e during time slots 2 and 4 . Since the receiv er is within the comm unication range of its corresp onding tr a nsmitter for eac h of these t ime slots, the SINR threshold condition is satisﬁed trivially for time slots 2 and 4. Tw o transmitter-receiv er pairs are activ e during time slots 1 and 3 . In time slot 1, the a ctiv e transmitter-receiv er pair s are (1 , 2) and (4 , 3). Our computations sho w that the SINRs at Receiv ers 2 and 3 are 20 . 85 dB and 21 dB, b oth of whic h exceed the comm unication threshold of 20 dB. In time slot 3, the activ e transmitter-receiv er pairs are (2 , 1) and (3 , 4). Our computations s ho w that the SINRs at Receiv ers 1 and 4 are 20 . 87 dB and 20 . 99 dB resp ectiv ely , b oth o f whic h exceed the comm unication threshold. This v eriﬁes tha t the SG LS algo rithm yields a conﬂict-free link sc hedule for the net w ork describ ed b y Figure 4.1 and T able 4.1. Note that, fro m (4.1), the spatial reuse of SGLS algorithm for this net work is 1 . 5. 4.3 Analytical Results In this section, w e pro v e the correctness of t he SGLS algorithm and deriv e its running time (computatio nal) complexit y . W e follow the notation o f Algorithm 6. Theorem 4.3.1. F or any V ′ cc ⊆ V ′ , if X v ′ 2 ∈V ′ cc \{ v ′ 1 } w ′ v ′ 2 v ′ 1 > |V ′ cc | + N ( v ′ 1 ) − 2 ∀ v ′ 1 ∈ V ′ cc , (4.9) then t he c oloring o f al l vertic es of V ′ cc with the same c olor is fe asible. Pr o of. Recall that w ′ v ′ 2 v ′ 1 = 0 or 1 − w v ′ 2 v ′ 1 and that 0 6 w ′ v ′ 2 v ′ 1 6 1. Supp ose w ′ v ′ 3 v ′ 1 = 0 for some v ′ 1 , v ′ 3 ∈ V ′ cc , v ′ 1 6 = v ′ 3 , then X v ′ 2 ∈V ′ cc \{ v ′ 1 } w ′ v ′ 2 v ′ 1 = X v ′ 2 ∈V ′ cc \{ v ′ 1 ,v ′ 3 } w ′ v ′ 2 v ′ 1 , 6 X v ′ 2 ∈V ′ cc \{ v ′ 1 ,v ′ 3 } 1 , = | V ′ cc \ { v ′ 1 , v ′ 3 }| , = | V ′ cc | − 2 , whic h con tra dicts the h yp othesis since N ( v ′ 1 ) > 0. So, an edge connecting an y t w o v ertices in V ′ cc m ust hav e p ositiv e w eigh t. Th us, 0 < w ′ v ′ 2 v ′ 1 6 1 ∀ v ′ 1 , v ′ 2 ∈ V ′ cc , v ′ 1 6 = v ′ 2 . 78 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del Equiv alently , 0 < 1 − w v ′ 2 v ′ 1 6 1 ∀ v ′ 1 , v ′ 2 ∈ V ′ cc , v ′ 1 6 = v ′ 2 . If t wo v ertices v ′ 1 , v ′ 2 ∈ V ′ cc (equiv alently , edges v ′ 1 , v ′ 2 ∈ G c ( · )) ha v e a common vertex in G c ( · ), then w v ′ 2 v ′ 1 = 1, whic h is a con tradiction. So, no t wo v ertices in V ′ cc ha ve a comm on v ertex in G c ( · ). F rom the h yp othesis, X v ′ 2 ∈V ′ cc \{ v ′ 1 } w ′ v ′ 2 v ′ 1 > |V ′ cc | + N ( v ′ 1 ) − 2 ∀ v ′ 1 ∈ V ′ cc , ⇔ X v ′ 2 ∈V ′ cc \{ v ′ 1 } (1 − w v ′ 2 v ′ 1 ) > |V ′ cc | + N ( v ′ 1 ) − 2 ∀ v ′ 1 ∈ V ′ cc , ⇔ |V ′ cc \ { v ′ 1 }| − X v ′ 2 ∈V ′ cc \{ v ′ 1 } w v ′ 2 v ′ 1 > |V ′ cc | + N ( v ′ 1 ) − 2 ∀ v ′ 1 ∈ V ′ cc , ⇔ |V ′ cc | − 1 − X v ′ 2 ∈V ′ cc \{ v ′ 1 } w v ′ 2 v ′ 1 > |V ′ cc | + N ( v ′ 1 ) − 2 ∀ v ′ 1 ∈ V ′ cc , ⇔ X v ′ 2 ∈V ′ cc \{ v ′ 1 } w v ′ 2 v ′ 1 + N ( v ′ 1 ) < 1 ∀ v ′ 1 ∈ V ′ cc , ⇔ X v ′ 2 ∈V ′ cc \{ v ′ 1 } γ c D ( t v ′ 1 , r v ′ 1 ) β D ( t v ′ 2 , r v ′ 1 ) β + N 0 γ c P D ( t v ′ 1 , r v ′ 1 ) β < 1 ∀ v ′ 1 ∈ V ′ cc , ⇔ P D ( t v ′ 1 ,r v ′ 1 ) β N 0 + P v ′ 2 ∈V ′ cc \{ v ′ 1 } P D ( t v ′ 2 ,r v ′ 1 ) β > γ c ∀ v ′ 1 ∈ V ′ cc . Therefore, the SINR t hreshold c ondition (4 .5) is satisﬁed at the receiv ers of all ve rtices of V ′ cc .  With r esp ect to (w.r.t.) the comm unicatio n graph G c ( V , E c ), let: e = num b er of edges , v = n um b er of v ertices . Theorem 4.3.2. The running time c omplex ity of SGLS algorithm is O ( e 2 ) . Pr o of. |V ′ | = |E c | = e . Since G ′ ( · ) is a directed complete graph, |E ′ | = e ( e − 1) = O ( e 2 ). Since the computation of w ij for given edges i and j of G ′ ( · ) takes unit time, the computatio n of inte rference we igh t func tions for all edges of G ′ ( · ) ta k es O ( e 2 ) time. Similarly , the computation of co-sc hedulabilit y w eight functions for all edges o f G ′ ( · ) requires O ( e 2 ) time. The computation of no rmalized noise p ow ers at all v ertices of G ′ ( · ) tak es O ( e ) time. 4.4. P erf orm ance Results 79 In G ′ ( · ), let C denote the total num b er of colors used to color all vertic es and let N i denote the n umber of vertices assigned color i , i.e., N i = |V ′ c i | . Since C can nev er exceed the n um b er of v ertices in G ′ ( · ), i.e., the n um b er of edges in G c ( · ), C is O ( e ). The time required b y Line s 2 0-21 is O (1), let it b e k 1 , where k 1 is a constant. With a careful implemen tation o f storing P v ′ 1 ∈V ′ c p \{ v ′ c }∪{ u ′ } w ′ v ′ 1 v ′ c ∀ v ′ c ∈ V ′ c p , Lines 26-28 tak e O (1) time. Thus , Lines 25- 2 9 tak e O ( |V ′ c p | ) time, let it b e equal to k 2 |V ′ c p | , where k 2 is a constan t. Along similar a rgumen ts, Lines 30-34 take O (1 ) time, let it b e equal to k 3 , where k 3 is a constant. The time required by Lines 36-38 is k 4 , where k 4 is a constant. Th us, the total running time o f the coloring phase is τ = C X i =1  k 1 + N i X j =1  |V ′ uc | ( k 2 |V ′ c i | + k 3 ) + k 4   . Since V ′ uc , V ′ c i ⊆ V ′ , it follo ws that |V ′ uc | , |V ′ c i | 6 | V ′ | = e . F urthermore, for an y color i , V ′ uc ∪ V ′ c i ⊆ V ′ . Th us, |V ′ uc ||V ′ c i | 6 e 2 4 . Therefore τ 6 C X i =1 k 1 + C X i =1 N i X j =1 k 2 e 2 4 + C X i =1 N i X j =1 k 3 e + C X i =1 N i X j =1 k 4 = k 1 C + k 2 e 2 4 ( e ) + k 3 e ( e ) + k 4 ( e ) = k 1 C + k 3 e 2 + k 2 4 e 3 + k 4 e = O ( e 3 ) . Hence, t he tot a l running time complexit y of SGLS algorithm is O ( e 3 ).  4.4 P erformance Results In this section, w e demonstrate the eﬃcacy of SGLS algorithm via simulations. T o the b est of our knowle dge, f o r an STDMA net w ork with constrained transmission p o wer, there is no existing w ork on link sc heduling that utilizes an SINR graph represen tation of the net work. How ev er, for completeness, w e compare the p erformance of SGLS algorithm with the CFLS algo rithm prop osed in Chapter 3. Note that SGLS is ba sed o n SINR graph while CFLS is based on comm unicatio n graph and v erifying SINR conditions. In the sim ulation exp erimen ts, the lo cation of ev ery no de is generated randomly using a uniform distribution fo r it s X and Y coordina t es. W e assume that the deplo ymen t 80 Chapter 4. P oin t to P oin t Link Sc h eduling based on SINR Gr ap h Mo del area is a circular region of radius R . The v alues c hosen for system para meters P , γ c , β and N 0 are prototypical v alues of system parameters in wireless ne t works [42]. After generating random p o sitions for N no des, we ha v e complete information of Φ( · ). Once the link sc hedule Ψ( · ) is computed b y ev ery algorithm, σ is c omputed using (4.1). F or a given set of system parameters, we calculate the a v erage sp atial reuse b y av eraging σ o ver 100 0 r a ndomly gene rated net w orks. Keeping all other parameters ﬁxed, w e observ e the eﬀect of increasing the n um b er of no des N on the a v erage s patial reuse. In the ﬁrst exp erimen t (Exp erimen t 1), we assum e tha t R = 500 m, P = 10 mW, β = 4 , N 0 = − 90 dBm and γ c = 2 0 dB. Thu s, R c = 100 m. W e v ary the n umber of no des fro m 30 to 110 in steps of 5 . F ig ure 4.10 plots the av erage spatial reuse vs. n umber o f no des for b oth the algorithms. 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 number of nodes average spatial reuse R = 500 m, P = 10 mW, β = 4, N 0 = −90 dBm, γ c = 20 dB CFLS SGLS Figure 4.10: Spa t ia l reuse vs. n um b er of no des fo r Exp erimen t 1. In the second exp erimen t (Exp erimen t 2), w e assume that R = 7 00 m, P = 15 mW, β = 4, N 0 = − 85 dBm and γ c = 15 dB. Thus, R c = 1 10 . 7 m. W e v a ry the num b er of no des from 70 to 150 in steps of 10. Figure 4.11 plots the av erage spatial reuse vs. n umber o f no des for b oth the algorithms. F rom the ﬁgures, w e observ e that SGLS a c hiev es 5 - 10% hig her spatial reuse than CFLS. How ev er, this impro v emen t in p erformance is obtained at the cost of higher 4.5. Discussion 81 70 80 90 100 110 120 130 140 150 0 1 2 3 4 5 number of nodes average spatial reuse R = 700 m, P = 15 mW, β = 4, N 0 = −85 dBm, γ c = 15 dB CFLS SGLS Figure 4.11: Spa t ia l reuse vs. n um b er of no des fo r Exp erimen t 2. computational complexit y . 4.5 Discuss ion In this c hapter, w e hav e pr o p osed a nov el p o int to p oint link sc heduling algorithm based on an SINR graph r epresen tation of an STDMA wireless net w ork under the phy sical in terference mo del. Our results demonstrate that the spatial reuse fo r the prop osed algorithm is higher than that of the ConﬂictF reeLinkSc hedule a lgorithm. This is due to the fact that we ha v e em b edded in terference c onditions b et w een pairs of no des in t o the edge w eights and normalized noise p o wers at receiv er no des in to v ertex w eights of the SINR gra ph and consequen tly determined a conﬂict-free sc hedule. Our approac h has the p oten tial to scale with the n um b er of no des in the netw ork. Chapter 5 P oi n t to Multip oin t L ink Sc heduling: A H y brid Approac h In this c hapter, w e in v estigate p oin t to m ultip oint link sch eduling in STDMA wireless net works . W e generalize the deﬁnition of spatial reuse in tro duced in Chapter 2 for p oin t to mu ltip oint link sc heduling. W e prop ose a “hybrid” link sc heduling a lgorithm based on a comm unication g raph represen t ation of the net work and SINR conditio ns. W e demonstrate that the propo sed algorithm ac hiev es higher sp atial reuse than existing algorithms, without an y increase in running time c omplexit y . The rest of this c ha pter is organized as follo ws. In Section 5.1, w e describ e our system mo del. W e describ e p oin t to multipoint link sc heduling based on the proto col in terference mo del in Section 5.2 and describ e it s limitations in Section 5.3. In Sec tion 5.4, w e in tro duce spatial reuse as our p erfo r mance metric a nd form ulate the problem. In Section 5.5, w e describ e the prop osed link sc heduling algorithm. W e ev aluate its p erformance in Section 5.6 and deriv e its computational complex it y in Section 5.7. W e discuss the implications of our work in Section 5.8. 5.1 System Mo del Our sys tem mo del and notations are exactly as describ ed in Section 2.1. Ho wev er, w e redeﬁne a nd in tro duce terms that are applicable to p oin t to m ultip oin t link sche duling. If no de k is within no de j ’s comm unication range, then k is deﬁned as a neig hb or of 83 84 Chapter 5. P oint to Multipoint Link Sc heduling: A Hyb rid Approac h j , since k can deco de j ’s pac k et cor r ectly (sub ject to Equation 2.6). Note that if no de k is outside no de j ’s comm unication range, then it can nev er dec o de j ’s pack et correctly (from Equation 2.6). The num b er of neigh b or s of no de j is denoted b y η ( j ). A p oint to m ultip o in t link sc hedule for an STDMA wireless net work Φ( · ) is a mapping from the set of no des to time slots. Let C denote the nu m b er of time slots in a p oint to m ultip oint link sc hedule. F or a given time slot i , j th p oin t to m ultip oin t transmission is denoted b y { t i,j → { r i,j, 1 , r i,j, 2 , . . . , r i,j,η ( t i,j ) }} , where t i,j denotes the index of the no de whic h tra nsmits a pac k et and r i,j, 1 , r i,j, 2 , . . . , r i,j,η ( t i,j ) denote the indices of neighboring no des (neigh b ors of t i,j ) that receiv e the pack et. Note that r i,j,k denotes k th receiv er of j th transmission in time slot i . Let M i denote the n um b er of concurren t p oint to m ultip oin t transmissions in time slot i . A p oin t to multipoint link sc hedule for an STDMA net w ork Φ( · ) is denoted by Ω( B 1 , · · · , B C ), where B i := { t i, 1 → { r i, 1 , 1 , r i, 1 , 2 , . . . , r i, 1 ,η ( t i, 1 ) } , · · · , t i,M i → { r i,M i , 1 , r i,M i , 2 , . . . , r i,M i ,η ( t i,M i ) }} = set of concurren t point to m ultip oin t transmissions in t ime slot i. Ev ery p oint to m ultip oin t sc hedule Ω( · ) m ust satisfy the follo wing: 1. Op erational constraints: (a) A no de cannot tra nsmit and receiv e in the same time s lot, i.e., { t i,j } ∩ { r i,k , 1 , . . . , r i,k ,η ( t i,k ) } = φ ∀ i = 1 , . . . , C ∀ j 6 = k . (5.1) (b) A no de cannot receiv e from mu ltiple transmitters in the same time slot, i.e., { r i,j, 1 , . . . , r i,j,η ( t i,j ) } ∩ { r i,k , 1 , . . . , r i,k ,η ( t i,k ) } = φ ∀ i = 1 , . . . , C ∀ j 6 = k . (5.2) 2. Range constraint: Ev ery receiv er is within the comm unication range o f its inte nded transmitter, i.e., D ( t i,j , r i,j,k ) 6 R c ∀ i = 1 , . . . , C ∀ j = 1 , . . . , M i ∀ k = 1 , . . . , η ( t i,j ) . (5.3) F or an example, consider the STDMA wireless net w ork Φ( · ) shown in Figure 5.1(a). It consists of six no des whose co o r dina t es (in meters) are 1 ≡ ( − 40 , 5), 2 ≡ (0 , 0), 5.1. System Mod el 85 Y X 1 ≡ ( − 40 , 5) 2 ≡ (0 , 0) 3 ≡ (95 , 0) 4 ≡ (135 , 0) 5 ≡ ( − 75 , 0) 6 ≡ (0 , − 75) (a) An STDMA wireless netw ork with six nodes. 2 4 3 5 1 1 1 → { 2 , 5 , 6 } 4 → { 3 } 2 → { 1 , 3 , 5 , 6 } poi n t t o multip ointpoint l i nk schedule · · · · · · 6 → { 1 , 2 } time 5 → { 1 , 2 } 3 → { 2 , 4 } 1 → { 2 , 5 , 6 } 4 → { 3 } poi n t t o multip oint tr ansmissions time sl ots (b) A point to m ultip oint link schedule for the netw ork shown in Figure 5 .1(a). Figure 5.1: Example of STDMA net work and p o in t to m ultip oin t link sc hedule. 86 Chapter 5. P oint to Multipoint Link Sc heduling: A Hyb rid Approac h 3 ≡ (95 , 0 ), 4 ≡ (135 , 0) , 5 ≡ ( − 75 , 0) and 6 ≡ (0 , − 75). One o f the p o ssible p oin t to multipoint link sc hedules for this STDMA net work is sho wn in Figur e 5.1(b). The sc hedule length is C = 5 time slots and the sc hedule is deﬁned b y Ω( B 1 , B 2 , B 3 , B 4 , B 5 ), where B 1 =  t 1 , 1 → { r 1 , 1 , 1 , r 1 , 1 , 2 , r 1 , 1 , 3 } , t 1 , 2 → { r 1 , 2 , 1 }  =  1 → { 2 , 5 , 6 } , 4 → { 3 }  , B 2 =  t 2 , 1 → { r 2 , 1 , 1 , r 2 , 1 , 2 }  =  6 → { 1 , 2 }  , B 3 =  t 3 , 1 → { r 3 , 1 , 1 , r 3 , 1 , 2 , r 3 , 1 , 3 , r 3 , 1 , 4 }  =  2 → { 1 , 3 , 5 , 6 }  , B 4 =  t 4 , 1 → { r 4 , 1 , 1 , r 4 , 1 , 2 }  =  3 → { 2 , 4 }  , B 5 =  t 5 , 1 → { r 5 , 1 , 1 , r 5 , 1 , 2 }  =  5 → { 1 , 2 }  . After 5 time slots, the sc hedule rep eats p erio dically , as sho wn in F igure 5.1(b). A p oin t to multipoint link sche duling algorithm is a set of rules that is used to determine a sc hedule Ω( · ). Ty pically , a sc heduling algorithm is required to satisfy certain ob jectiv es. Consider k th receiv er of j th transmission in time slot i , i.e., receiv er r i,j,k . The p o w er receiv ed at r i,j,k from its inte nded transmitter t i,j (signal p ow er) is P D β ( t i,j ,r i,j,k ) . The p ow er receiv ed at r i,j,k from its unin tended t r a nsmitters (in terference p o w er) is P M i l =1 l 6 = j P D β ( t i,l ,r i,j,k ) . Th us, the SINR at receiv er r i,j,k is given by SINR r i,j,k = P D β ( t i,j ,r i,j,k ) N 0 + P M i l =1 l 6 = j P D β ( t i,l ,r i,j,k ) . (5.4) According to the phys ical in terference mo del [15], receiv er r i,j,k can successfully de- co de the pac k et transmitted b y t i,j if the SINR at r i,j,k is no less tha n the communic ation threshold γ c , i.e., SINR r i,j,k > γ c . (5.5) 5.2. Equiv alence of Link S c hedu ling and Graph V ertex Coloring 87 A link sc hedule Ω( · ) is exhaustive if ev ery tw o no des j , k who are neigh b ors of each other are included in the sc hedule exactly twice , once with j b eing a transm itter and k b eing one o f its receiv ers, and d uring anot her time slot with k b eing a transmitter and j b eing one of its receiv ers. 5.2 Equiv alen ce of Link Sc h eduling and Graph V er- tex C o loring In this section, we describ e the equiv alence b etw een a p oint to multipoint link sc hedule for a n STDMA wireless netw ork and the coloring of v ertices of the comm unication gra ph represen tation ( see Section 2.2.1) of the net work. P arameter Sym b ol V alue transmission p o w er P 10 mW path loss exp onent β 4 noise p ow er sp ectral densit y N 0 -90 dBm comm unication thres hold γ c 20 dB in terference threshold γ i 10 dB T able 5.1: System para meters fo r STDMA net works sho wn in Figures 5.1(a) and 5.4. Consider the STDMA wireless netw ork Φ( · ) whose deplo ymen t is show n in Figure 5.1(a). The s ystem parameters for this net work are giv en in T able 5.1. F rom (2.4), we obtain R c = 100 m. The corresp onding comm unication graph represen tatio n G c ( V , E c ) is sho wn in F ig ure 5.2. The comm unication graph comprises of 6 v ertices a nd 14 directed comm unication e dges. The verte x a nd comm unication edge sets are giv en b y V = { v 1 , v 2 , v 3 , v 4 , v 5 , v 6 } , (5.6) E c = { v 1 c → v 2 , v 2 c → v 1 , v 1 c → v 5 , v 5 c → v 1 , v 1 c → v 6 , v 6 c → v 1 , v 2 c → v 5 , v 5 c → v 2 , v 2 c → v 6 , v 6 c → v 2 , v 2 c → v 3 , v 3 c → v 2 , v 3 c → v 4 , v 4 c → v 3 } . (5.7) Giv en the ab ov e represen tation of an STDMA net w ork, a p oin t to multipoint link sc hedule Ω( · ) can be considere d as equiv alent to assigning a uniq ue color to eve ry v ertex 88 Chapter 5. P oint to Multipoint Link Sc heduling: A Hyb rid Approac h v 3 v 4 v 2 v 5 v 1 v 6 Figure 5.2: Comm unication g raph mo del of STDMA netw ork describ ed by F ig ure 5.1 ( a ) and T a ble 5.1 . v 3 v 4 v 2 v 5 v 1 v 6 Figure 5.3: V ertex coloring of comm unication graph show n in Figure 5.2 corresp onding to the link sch edule show n in Figure 5.1(b). 5.3. Limitations of Algorithms based on Pr otocol Interference Mod el 89 in the commu nication graph, suc h that no des w ith the same color transmit sim ultane- ously in a particular time slot. F or the example net work considered, the link sc hedule sho wn in Figure 5.1(b) correspo nds to the coloring of the v ertices of the comm unication graph shown in Figure 5.3. Time slots 1, 2, 3, 4 and 5 in Ω( · ) corr esp ond to colors red, blue, green, magen ta a nd y ellow in V resp ectiv ely . Note that a coloring algorithm that uses the least num b er of colors a lso minimizes t he sc hedule length. Algorithms for assigning no des to time slots (equiv a lently , colors) require that tw o v ertices v i , v j can b e color ed the same if a nd only if: 1. edge v i c → v j 6∈ E c and edge v j c → v i 6∈ E c , i.e., there is no primary v e rtex c onﬂict, and 2. there is no v ertex v k suc h that v i c → v k ∈ E c and v j c → v k ∈ E c , i.e., there is no se c ondary vertex c onﬂict . These criteria are based on the op erational constrain ts (5.1) and (5.2). Algorithms based on t he proto col interferenc e mo del represen t the netw ork by a com- m unication graph and utilize v arious graph coloring metho do logies to devise heuristic s whic h yield a minim um length sc hedule. Hence, suc h algorithms ha v e the merit of lo w computational complexit y . How ev er, recen t researc h suggests that these algorit hms yield lo w netw ork throughput. This asp ect is elab o rated in the following section. 5.3 Limitations o f Algorit hms bas e d on Proto col In- terferenc e Mo del In this section, we illustrate tha t a lgorithms based on the proto col in terference mo del can r esult in sc hedules that yield low net work throughput. Note that the limitations of p oin t to multipoint link sc heduling algorithm are similar to those of p oint to p oin t link sc heduling algorithms describ ed in Section 2.3. With the inte n t of maximizing the t hro ughput of an STDMA net w ork, alg o rithms based on the proto col in terference mo del tra nsfor m the sc heduling problem to a v er- tex coloring problem for the comm unication graph represen ta tion of t he net work. F or example, the BroadcastSch edule algorithm [1 6] w o r ks in t w o phases. In Phase 1, the 90 Chapter 5. P oint to Multipoint Link Sc heduling: A Hyb rid Approac h v ertices of the comm unication graph are lab eled using the labeler function (Algorithm 2, Section 3.1). In Phase 2, v ertices are considered in incre asing order of la b el. F or the v ertex u under consideration, it discards an y color that leads to primary or se condary v ertex conﬂic ts with u . The least c olor among the residual set o f no n-conﬂicting color s is used to color v ertex u . If no non-conﬂicting color exists, vertex u is colored with a new color. The simpliﬁcation of the link sc heduling problem in a wireless netw ork as a v ertex coloring problem on t he comm unication graph can result in sc hedules that viola te the SINR threshold c ondition (5 .5). Sp eciﬁcally , algorithms based on the proto col in terfer- ence mo del do no t nec essarily maximize the throughput of an STDMA net work b ecause: 1. They can result in high cum ulativ e in terference at a receiv er, due to hard-thresholding based o n comm unication radius. This is b ecause the SINR at receiv er r i,j,k de- creases with a n inc rease in the num b er of concurren t transmissions M i , while R c has b een deﬁned for a single transmission only . Y 1 ≡ (0 , 0) 2 ≡ ( − 80 , 0) 3 ≡ (90 , 0) 5 ≡ (200 , 0 ) 4 ≡ (280 , 0 ) X 6 ≡ (370 , 0 ) Figure 5.4: An STDMA wireless net w ork with six no des. v 2 v 1 v 5 v 4 v 6 v 3 Figure 5.5: Comm unication gra ph mo del of STDMA netw ork describ ed b y Figure 5.4 and T a ble 5.1 . F or example, consider t he STDMA wireless net w ork whose deplo ymen t is shown in Fig ure 5.4. The netw ork consists of six no des whose co ordinates (in meters) 5.3. Limitations of Algorithms based on Pr otocol Interference Mod el 91 v 2 v 1 v 5 v 4 v 6 v 3 Figure 5.6: Coloring of v ertices v 1 and v 4 of gr a ph sho wn in Figure 5.4. 2 1 3 4 5 6 t i, 1 r i, 1 , 1 r i, 1 , 2 t i, 2 r i, 2 , 1 r i, 2 , 2 SINR r i, 1 , 2 = 12 . 4 2 dB SINR r i, 1 , 1 = 2 1 . 85 dB SINR r i, 2 , 1 = 15 . 2 7 dB SINR r i, 2 , 2 = 19 . 9 7 dB X Y Figure 5.7: Poin t to m ultip o int link sche duling algor it hms based on proto col in terference mo del can lead to high interfere nce. 92 Chapter 5. P oint to Multipoint Link Sc heduling: A Hyb rid Approac h are 1 ≡ ( 0 , 0), 2 ≡ ( − 80 , 0), 3 ≡ (90 , 0), 4 ≡ (280 , 0), 5 ≡ (200 , 0) and 6 ≡ (370 , 0). The system parameters are shown in T able 5.1, whic h yie lds R c = 100 m. The comm unication graph mo del of the STDMA net work is sh o wn in Figure 5 .5 . Consider the tra nsmission requests 1 → { 2 , 3 } and 4 → { 5 , 6 } , whic h corresp ond to v ertices v 1 and v 4 of the graph show n in Figure 5.5. Note that ve rtices v 1 and v 4 do not hav e primary or secondary vertex conﬂicts. So, to minimize the n umber of colors, suc h an algor it hm will color these vertice s with the same color, as sho wn in Figure 5 .6. Equiv a len tly , transmissions 1 → { 2 , 3 } and 4 → { 5 , 6 } will b e sc heduled in the same time s lot, sa y time slot i . How ev er, our computations sho w that the SINRs at receiv ers r i, 1 , 1 , r i, 1 , 2 , r i, 2 , 1 and r i, 2 , 2 are 2 1 . 85 dB, 12 . 42 dB, 15 . 27 dB and 19 . 97 dB resp ectiv ely . Figure 5.7 shows the no des of the netw ork along with the lab eled transmitter-receiv ers sets, receiv er-cen tric comm unication zones and SINRs at the receiv ers. F rom the SINR threshold condition (5.5), transmission t i, 1 → r i, 1 , 1 is successful, while tra nsmiss ions t i, 1 → r i, 1 , 2 , t i, 2 → r i, 2 , 1 and t i, 2 → r i, 2 , 2 are unsucces sful. This leads to low net work throug hput. 2. Moreo v er, these algorit hms are not a w are of the to p ology of the netw ork, i.e., they determine a link sc hedule without b eing cognizant of the exact p ositions of the transmitters and receiv ers. As argued ab ov e, p oint to m ultip oin t link sc heduling alg orithms based on the pro- to col in terference mo del can result in lo w net work throughput. In essence , a lg orithms that construct an approximate mo del of an STDMA net work (commun ication graph) and concen tr a te on minimizing the sche dule length do not necessarily maximize netw ork throughput. This observ ation is dev elop ed in to a prop osal for an appro pr ia te p erfor- mance metric in Section 5.4. 5.4 Problem F orm u l ation In this section, w e m otiv ate the need for a performance metric that tak es in to accoun t the SINR thr eshold condition (5.5) as the criterion for successful pack et reception. Anal- ogous to the notion of spatial reuse, we prop o se a p erformance metric for point to m ulti- p oin t link sc heduling, whic h is also termed as spatial reuse. W e argue that spatial reuse 5.4. Problem F ormulatio n 93 is directly prop ortional to t he n umber of succes sful p oint to m ultip o in t transmissions. Finally , w e form ulate the sc heduling problem from a p erspective of maximizing s patial reuse. Algorithms based on the proto col interferenc e mo del are inadequate to design eﬃcien t p oin t to multipoint link sc hedules. This is b ecause thes e algorithms are en tirely based on the comm unication graph G c ( V , E c ), whic h is a crude appro ximation of Φ( · ), and can lead to low netw ork throughput, as a r gued in Section 5.3. On the other hand, from Φ( · ) and G c ( · ), one can exhaustiv ely determine the link sc hedule Ω( · ) whic h yields highest net work thro ughput according to the ph ysical interferenc e mo del. Ho we v er, this is a com binator ia l optimization problem of prohibitiv e complexit y ( O ( |V | |V | )) and is th us computationally infeasible. T o ov ercome these problems, w e pro p ose a p oin t t o multipoint link sc heduling algo- rithm for STDMA wireless net w orks under the phy sical interferenc e mo del. Our a lgo- rithm is base d on the comm unication graph mo del G c ( V , E c ) as w ell as SINR computa- tions. T o ev aluate the p erfo rmance of our algorithm a nd compare it with existing link sc heduling algorithms, we deﬁne the notio n of spatial reuse. Consider the p oint to m ul- tip oin t link sche dule Ω( · ) for the STDMA net w ork Φ( · ). Under the ph ysical in terference mo del, transmission t i,j → r i,j,k is successful if and only if (5.5) is satisﬁed. The sp atial r euse of the link sche dule Ω( · ) is deﬁned as the av erage n umber of successful p oint to m ultip oint transmissions p er time slot. Th us Spatial Reuse = ς := P C i =1 P M i j =1 P η ( t i,j ) k =1 I (SINR r i,j,k > γ c ) η ( t i,j ) C , (5.8 ) where I ( A ) denote the indicator function for ev en t A , i.e., I ( A ) = 1 if even t A o ccurs, I ( A ) = 0 if eve n t A do es not o ccur. Note that in ( 5 .8), the num b er of no des that success fully receiv e a transmitted pack et is normalized by the n umber of neighbors of the transmitting no de. A high v alue of spatial reuse corresp onds to high net w ork throughput. The essence of STDMA is to ha v e a reasonably large num b er of simultaneous a nd success ful t r a nsmissions . F or an STDMA wireless net work which is op erationa l for a long p erio d of time, sa y L time slots, the total n umber of succes sful p oint to multipoint transmissions is Lς . Th us, a high v alue of spatial reuse directly t r anslates to higher 94 Chapter 5. P oint to Multipoint Link Sc heduling: A Hyb rid Approac h net work throughput and the n um b er of colors C is relatively unimp ortant. Hence, spatial reus e turns out to b e a crucial metric f o r the comparison of v arious STDMA link sc heduling algorithms. Our goal is t o design a lo w complexit y p oin t to m ultip oin t link sc heduling algo rithm that ac hiev es high spatial reuse, where spatial reuse is giv en by (5.8). W e only consider link sc hedules t ha t are feasible and exhaustiv e. 5.5 MaxAv erageSINRSc hedule Algorit hm Our prop o sed p oin t to m ult ip oin t link sc heduling sche duling algor it hm under the phy s- ical in terference mo del is MaxAve rageSINRSc hedule (MASS), whic h considers the com- m unication graph G c ( V , E c ) and is describ ed in Algor it hm 7. Algorithm 7 MaxAv erageSINRSc hedule (MASS) 1: input: STDMA wireless net work Φ( · ), communication g r a ph G c ( V , E c ) 2: output: A coloring C : V → { 1 , 2 , . . . } 3: lab el t he ve rtices of G c randomly { Phase 1 } 4: for j ← 1 to n do { Phase 2 b egins } 5: let u b e suc h that L ( u ) = j 6: C ( u ) ← MaxAv erageSINR Color ( u ) 7: end for { Phase 2 ends } In Phase 1, w e lab el all the v ertices randomly 1 . Sp eciﬁcally , if G c ( · ) has v v ertices, w e p erform a random p ermutation of the sequence (1 , 2 , . . . , v ) and assign these labels to v ertices with indices 1 , 2 , . . . , v resp ectiv ely . Let L ( u ) denote the lab el assigned to v ertex u . In Phase 2, the v ertices a re examined in increasing order o f lab el 2 and t he MaxAv er- ageSINR Color (MASC) function is us ed to assign a color to the v ertex under considera- tion. The MAS C function is explained in Algorithm 8. It b egins b y discarding all colors that ha v e a primary or secondary ve rtex conﬂict with u , the ve rtex under conside ration. 1 Randomized a lg orithms are known to outp erfor m deter ministic alg orithms, esp ecially when the characteristics of the input a re not known a priori [53]. 2 In essence, the v er tices are scanned in a random o rder, since lab eling is random. 5.6. P erf orm ance Results 95 Algorithm 8 in teger MaxAv erageSINR Color ( u ) 1: input: STDMA wireless net work Φ( · ), communication g r a ph G c ( V , E c ) 2: output: A non-conﬂicting color 3: C ← set o f existing colors 4: C p ← { C ( x ) : x is colored and is a n eigh b or of u } 5: C s ← { C ( x ) : x is colored and is tw o hops aw a y from u } 6: C nc = C \ {C p ∪ C s } 7: if C nc 6 = φ t hen 8: r ← color in C nc whic h results in maximum av erage SINR at neighbors of u 9: if maxim um a ve rage SINR > γ c then 10: return r 11: end if 12: end if 13: return |C | + 1 Among the set of non- conﬂicting colors C nc , it c ho oses that c olor for u whic h results in the maximum v alue of av erage SINR at the neigh b ors of u , provided this v alue exceeds the comm unication threshold. In t uitively , the av erage SINR is also a measure of the a verage distance of ev ery neigh b or of u from all co-colored tra nsmitters. The higher the a verage SINR, the higher is this av erage distance. W e c ho ose tha t color whic h results in the maximum av erage SINR at the neighbor s of u , so tha t the additiona l in terference at the neigh b or s of all co-colored transmitters is k ept lo w. If no suc h color is fo und, it assigns a new color to u . 5.6 P erformance Results In this section, we describ e our sim ulat io n model. W e compare the p erformance of t he prop osed algorithm with existing p oin t to m ultip oin t link s c heduling algorithms. In our sim ulation exp erimen ts, t he lo cation of ev ery no de is generated randomly in a circular region of radius R . If ( X j , Y j ) are the Cartesian co ordinates of no de j , then X j ∼ U [ − R, R ] and Y j ∼ U [ − R, R ] sub ject to X 2 j + Y 2 j 6 R 2 . Equiv alen tly , if ( R j , Θ j ) a r e the p olar co ordinates of no de j , then R 2 j ∼ U [0 , R 2 ] and Θ j ∼ U [0 , 2 π ]. 96 Chapter 5. P oint to Multipoint Link Sc heduling: A Hyb rid Approac h Using (2.4) and (2 .5), w e compute R c and R i , and then map t he STDMA netw ork Φ( · ) to the comm unication g raph G ( V , E c ). Once the link sc hedule Ω( · ) is computed b y ev ery alg orithm, the spatial reuse ς is computed using (5 .8). W e use tw o sets of protot ypical v alues of system parameters in wireless netw orks [42]. F or a given set o f system parameters, w e calculate the a v erage spatial reuse b y av eraging ς ov er 1000 randomly generated net w orks. Keeping all other parameters ﬁxed, w e observ e the eﬀect of increasing t he num b er of no des o n the av erage spatial reuse. In o ur exp erimen ts, we compare the p erformance of t he following algorit hms: 1. BroadcastSc hedule (BS) [1 6] 2. MaxAv erageSINRSc hedule (MASS) In our ﬁrst set of exp erimen ts (Exp erimen t 1), we assume that R = 500 m, P = 10 mW, β = 4, N 0 = − 90 dBm, γ c = 20 dB and γ i = 1 0 dB. Th us, R c = 100 m and R i = 177 . 8 m . W e v ary the num b er of no des from 30 to 110 in steps of 5. Fig ure 5.8 plots t he av erage spatial reuse vs. num b er of no des for b oth the algo rithms. 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 number of nodes average spatial reuse R = 500 m, P = 10 mW, α = 4, N 0 = −90 dBm, γ c = 20 dB, γ i = 10 dB BroadcastSchedule MaxAverageSINRSchedule Figure 5.8: Ave rage spatial reuse vs. n um b er of no des fo r Exp erimen t 1. In o ur second set of exp erimen ts (Exp erimen t 2), we assume that R = 700 m, P = 15 mW, β = 4 , N 0 = − 85 dBm, γ c = 15 dB and γ i = 7 dB. Th us, R c = 110 . 7 m and 5.6. P erf orm ance Results 97 R i = 175 . 4 m . W e v ary the num b er of no des from 70 to 150 in steps of 5. Fig ure 5.9 plots t he av erage spatial reuse vs. num b er of no des for b oth the algo rithms. 70 80 90 100 110 120 130 140 150 0 1 2 3 4 5 6 number of nodes average spatial reuse R = 700 m, P = 15 mW, α = 4, N 0 = −85 dBm, γ c = 15 dB, γ i = 7 dB BroadcastSchedule MaxAverageSINRSchedule Figure 5.9: Ave rage spatial reuse vs. n um b er of no des fo r Exp erimen t 2. F rom Figures 5.8 and 5.9, we observ e that a v erage spatial reuse increases with the n umber of no des for b o th the algorithms. The MASS a lg orithm consisten tly yields higher av erage s patial reuse compared to BS. The spatial reuse o f MASS is a b out 15% higher than BS in Exp erimen t 1 and 4% higher in Exp erimen t 2. This improv emen t in p erformance translates to substan tially hig her net w ork throughput. Also, an increase in the n um b er o f no des in a give n g eographical area leads t o an increase in the num b er o f ve rtices having a primary or secondary verte x conﬂict with a giv en v ertex. Hence, the n umber of non-conﬂicting colors for a give n v ertex also de- creases. F rom this reduced set of no n- conﬂicting colors, BroadcastSc hedule c ho oses a color randomly , while MaxAv erag eSINRSc hedule c ho oses a color ba sed on SINR condi- tions. Since spatial reuse tak es SINR threshold conditions in to accoun t , the gap b et w een a verage spatial reuse v alues increases with num b er of no des in Figures 5.8 and 5.9 . 98 Chapter 5. P oint to Multipoint Link Sc heduling: A Hyb rid Approac h 5.7 Analytical Result In this section, w e deriv e an upp er b ound on the running time ( computational) com- plexit y of the MaxAv erageSINRSc hedule algorithm. Let v denote the n um b er of v ertices of the commu nication gra ph G c ( V , E c ). Theorem 5.7.1. The running time of MaxA ve r ageSINRSche dule is O ( v 2 ) . Pr o of. Assuming that an elemen t can b e c hosen randomly and unifo rmly fro m a ﬁnite set in unit t ime ([53], Chapter 1), the running time of Phase 1 can b e sho wn to be O ( v ). In Phase 2, the vertex under consideration is a ssigned a color using MaxAv erageSINR Color. The w orst-case size of the set of colors to b e e xamined |C nc ∪ C p ∪ C s | is O ( v ). With a careful implemen tation, MaxAv erageSINRC olor runs in time pro p ortional to |C nc | , i.e., O ( v ). Th us, the running time o f Phase 2 is O ( v 2 ). Finally , the ov erall running time of MaxAv erageSINRSc hedule is O ( v 2 ).  5.8 Discuss ion In this c hapter, w e ha ve de v elop ed a p oint to m ultip oin t link sc heduling a lgorithm for STDMA wireless net w o rks under the ph ysical interfere nce mo del, namely MaxAv era- geSINRSc hedule. The p erfo rmance o f our algorithm is sup erior to existing algorithms. A practical exp erimen ta l mo deling sho ws that, o n an av erage, our algorithm achie v es 15% higher s patial reuse than the BroadcastSc hedule algo rithm [16]. Since link s c hed- ules are constructed oﬄine only once a nd then used by the net work for a long p erio d o f time, this impro v emen t in p erformance directly translates to higher netw ork throug hput. The computational complexit y of MaxAv erageSINRSc hedule is also c omparable to the computational complexit y of BroadcastSc hedule. Therefore, MaxAv erageSINRSc hedule is a go o d candidate for eﬃcie n t STDMA p o int to m ultip oint link sc heduling algorit hms. Chapter 6 A Review of Random Access Algorithms for Wireless Net w orks The MAC problem or m ultip oint to po in t problem is presen t in all comm unication net works , b o th wired and wireless. Multiple no des ( users) can access a single c hannel sim ultaneously to comm unicate with eac h other or a common rec eiv er – the c hallenge is to design eﬃcie n t c hannel acces s algorithms to a chiev e the desired p erformance in terms of throughput and delay . Sev eral solutions to the MA C problem hav e b een prop osed dep ending on source traﬃc c haracteristics, c ha nnel mo dels and Quality of Service (QoS) requiremen ts o f the users. MA C prot o cols can b e broadly classiﬁed into tw o t yp es: ﬁxed r esource allo cation proto cols and random access proto cols. Fixed resource allo catio n pr o to cols suc h a s Time Division Multiple Access (TDMA), F requency Division Multiple Access (FDMA) and Co de Division Multiple Access (CDMA) assign orthogonal or near-or t hogonal channels to ev ery user and are mostly implemen ted in v oice-dominan t wireless cellular net w orks. These pr o to cols typic ally require the presence of a cen tr al en tity (base station) to p erform c hannel allo cation and admission con trol, i.e., they are highly cen tralized. Though ﬁxed resource allo cation proto cols are con ten tion-free and can m ultiplex users with similar traﬃc characteristics easily , they suﬀer from lo w throughput and high channel access dela y when the traﬃc is burst y and there are large n umber of users. On the other hand, in random a ccess proto cols, users v ary their transmission probabilities or transmission times based on limited c hannel feedbac k, i.e., random access proto cols are highly distributed. 99 100 Chapter 6. A Review of Random Access Algo rithms for Wireless Net w orks Random access proto cols are more suitable for scenarios wherein many users with v aried traﬃc requiremen ts hav e to b e m ultiplexed, i.e., the traﬃc is burst y . Random access algo rithms f o r satellite comm unicatio ns, multidrop telephone lines and m ultitap bus (“traditiona l ra ndom access algorithms”) ha ve b een w ell studied for the past four decades. These algorithms can b e broadly classiﬁed into three cat ego ries: ALOHA [17 ], [59], Carrier Sense Multiple Access [6 0] and tree ( o r stack or splitting) algorithms [18]. T raditional ra ndom access a lgorithms ha v e b een implemen ted in practi- cal systems. F or example, ALOHA is used in most cellular net works to request ch annel access and also in satellite communic ation net w or ks. Carr ier Sense Multiple Ac cess with Collision Detection (CSMA/CD) is used t o resolv e conten tions in Lo cal Area Net works (LANs). On the other hand, random access a lgorithms that incorp orate ph ysical la yer c har- acteristics suc h as SINR and ch annel v ariations hav e only b een studied recen tly . These algorithms, whic h ha v e b een primarily prop osed fo r wirele ss net works , can b e broadly classiﬁed in t o three categories: algorithms base d on signal processing and div ersity tec h- niques, c hannel-aw are ALOHA algorithms based on adapting the retransmission prob- abilities of conte nding users and “t ree-lik e” algorithms based o n adapting the set of con tending users. Existing r a ndom access algorithms, suc h as Carrier Sense Multiple Access with Collision Av oidance (CSMA/CA ), are not channel-a w ar e and c an lead to lo w thro ughput. Th us, the design of ph ysical lay er a w are random a ccess algorithms can b e a p otential step to w ards ac hieving higher data r a tes in future wireless net w orks. The organization of this c hapter is as follows. Section 6.1 provide s a summary o f t r a - ditional ra ndo m a ccess algorithms along with the canonical system mo del, p erformance metrics and w ell-kno wn random access tec hniques suc h as ALOHA and t r ee alg o rithms. This helps us understand channel-a w a r e generalizations of these algorithms. In Section 6.2, w e rev iew researc h pap ers whic h employ signal pro cessing and div ersity tec hniques to correctly decode pac kets in random access wireless net works. W e critically review some of the researc h which fo cus on channe l-aw are ALOHA and tree-like algorithms for wire less net w or ks in Sections 6.3 and 6.4 resp ectiv ely . Finally , w e motiv ate the use of v ariable transmission p ow er to increase the throughput of random access wireless net works in Section 6.5. 6.1. T raditional Random Acce ss Algorithms 101 6.1 T raditional Random Access Algori thms In this section, we describ e the idealized slotted multiacc ess mo del, which can b e used to represen t v a r io us m ultiaccess media suc h as satellite c hannels, multidrop telephone lines and m ultitap bus. W e explain traditional random access algo rithms such as ALOHA and t r ee algorit hms. W e also describe the p erfor ma nce metrics used to analyze and ev a lua te random a ccess algor it hms, namely , throughput, delay and stabilit y . Consider an idealized slotted multiacc ess system with m tr a nsmitting no des and one receiv er. The assumptions of the mo del are [22]: 1. Slotted system: All transmitted pack ets hav e the same length and eac h pack et requires o ne time unit, called a slot, for transmission. 2. One of the follow ing is usually assumed: (a) Poiss on arriv als: P ac kets arrive a t eac h of the m no des according to an inde- p enden t P oisson pro cess. Let λ b e the o v erall arriv al rate to the system a nd let λ m b e t he arriv al rate at eac h transmitting node. (b) Backlogged mo del: Ev ery no de alwa ys has a pack et to transmit. Once a no de transmits a pac k et successfully , a new pac ket is generated and aw aits transmission. 3. Collision or p erfect reception: If t wo o r more no des transmit a pac k et in a giv en slot, then there is a collision and the receiv er obtains no information ab out the con tents or the sources of tra nsmitted pa c k ets. If only one no de transmits a pac k et in a give n slot, the pac k et is correctly receiv ed. 4. { 0 , 1 , e } immediate feedbac k: At t he end of eac h slot, ev ery no de obtains feedback from the r eceiv er sp ecifying whether 0 pac k et, 1 pack et or more than one pac ke t ( e denotes error) were transmitted in that slot. 5. Retransmission of collisions: Eac h pac k et in v olved in a collision m ust b e retrans- mitted in some later slot, with further suc h retransmissions un til the pac ket is success fully rece iv ed. A node with a pac k et that must b e retransmitted is s aid to b e ba c klogged. 102 Chapter 6. A Review of Random Access Algo rithms for Wireless Net w orks 6. Only one of the follow ing is a ssumed: (a) No buﬀering: If o ne pac k et a t a no de is curren tly w aiting for tr ansmission or colliding with ano t her pack et during tra nsmission, new arriv als at that no de are discarded and nev er tr a nsmitted. (b) Inﬁnite set of no des: The system ha s an inﬁnite set o f no des and eac h new pac ke t arriv es at a new no de. F or the a nalysis and p erformance ev aluation of ra ndo m access a lg orithms, the metrics of interes t are: 1. Dela y: Index pack ets as 1 , 2 , 3 , . . . according to their arriv al instants. Let D j denote the delay exp erienced by j th pac ke t. Then the av erage pack et delay is deﬁned as D = lim m →∞ E " 1 m m X j =1 D j # . (6.1) 2. Throughput: The following are the tw o most common deﬁnitions of throughput: (a) Throughput is the suprem um of input pac k et arriv al rates λ suc h that the pac ke t de la y remains bounded, i.e., T 1 = sup D < ∞ λ. (6.2) (b) Let n ( t ) denote t he n um b er of pack ets successfully transmitted in [0 , t ]. Deﬁne T ( λ ) =    lim t →∞ E h n ( t ) t i if D < ∞ , 0 otherwise . Throughput is then deﬁned as T 2 = sup λ T ( λ ) . (6.3) 3. Stabilit y: A random access algorithm is stable if the throughput T > 0 a nd unstable if T = 0. The researc h of random acc ess algorithms bega n w ith the unslotted ALOHA (pure ALOHA) algorithm prop osed b y Abramson [17]. Each no de, up on receiving a pac ket, transmits it immediately rather t han waiting for a slot bo undar y . If a pac k et is inv olv ed 6.1. T raditional Random Acce ss Algorithms 103 in a collision, it is retransmitted aft er a random dela y . It can b e sho wn that unslotted ALOHA ac hieve s a maxim um throughput of 1 2 e ≈ 0 . 1839 [22]. An a dv antage of unslotted ALOHA is that it can b e used with v ariable-length pack ets. Slotted ALOHA is a v ariation b y Ro b erts [59] of the original uns lotted ALOHA pro- to col prop osed b y Abramson. Each no de simply tra nsmits a newly a rriving pack et in the ﬁrst slot after the pac k et arriv al. When a collision o ccurs, ev ery no de sending a colliding pac k et disco v ers the collision at the end of the slot and b ecomes bac klogged. Bac klogged no des w ait for a random n um b er of slots before re transmitting. The maxi- m um throughput of slotted ALOHA can b e sho wn to b e 1 e ≈ 0 . 3 6 78 [22]. D rift-analytic 1 metho ds rev eal that slotted ALOHA is unstable. T o stabilize ALOHA, some tec hniques estimate n or p r , so as to main tain the attempt rate G ( n ) at 1, resulting in a maxim um stable throughput o f 1 e [61], [62]. Unlik e unslotted ALOHA, slotted ALOHA cannot be easily used with v ariable-sized pack ets. In slotted ALOHA, long pac k ets m ust be brok en up t o ﬁt into slots and short pack ets m ust b e padded out to ﬁll up slots. Keeping the random a ccess spirit of the ALOHA proto col, researc hers a ttempted to design more eﬃcien t proto cols. A hig hly successful approac h consists of impro ving the con tro l o f the c ha nnel b y carrier sensing, i.e., the Carrier Sense Multiple Access (CSMA) tec hnique. In [22], the autho r s sho w that CSMA outp erforms ALOHA. Researc h has sho wn that CSMA based proto cols can ac hiev e a t hro ughput close to 0.9 [63]. The Ethernet proto col, whic h is used to connect computers on a wired LAN, utilize s Carrier Sense Multiple A ccess with Collision D etection (CSMA/CD). In splitting algorithms, the set of colliding no des splits into subsets, one of whic h transmits in the next s lot. F or a giv en colliding node, the c hoice of it s subset de p ends on a pre-determined rule suc h as, the outcome of tossing an un biased coin, a function of its a rriv al time o r a function of its no de iden tiﬁer. If the collision is not resolve d, a further splitting in to subsets takes place. The alg o rithm pro ceeds recursiv ely un til all collisions are resolv ed. In the Basic T ree Algor it hm (BT A) [18], when a collision o ccurs, sa y in k th slot, all no des not inv olv ed in the collision go into a w aiting mo de, a nd all those in v olved in the collision split in to tw o subsets, according to the pr e- determined rule. The ﬁrst subset 1 Drift in sta te n is deﬁned as the e x pected change in ba c klog ov er one time-slo t, starting in state n . 104 Chapter 6. A Review of Random Access Algo rithms for Wireless Net w orks 1 2 3 4 5 6 7 8 9 10 ti me Collisio n Resoluti on P erio d (CRP) 1 0 1 1 0 R LR L LL LRR LRRR LRL LRRL e e e e abc abc a bc bc b c Figure 6.1: Basic T ree Alg o rithm fo r three no des a , b and c . Slot T ransmitting set W aiting sets F eedbac k 1 U φ e 2 L R e 3 LL LR, R 1 4 LR R e 5 LRL LRR, R 0 6 LRR R e 7 LRRL LRRR, R 1 8 LRRR R 1 9 R φ 0 T able 6.1 : T ransmitting and w aiting sets for basic t ree algor it hm sho wn in Figure 6.1 . 1 2 3 4 5 6 7 8 9 10 ti me abc abc a bc bc bc bc b c c Figure 6.2: Stack represen tation of transmitting and w aiting no des for ba sic tree algo- rithm sho wn in F ig ure 6.1. 6.1. T raditional Random Acce ss Algorithms 105 (“left” subset) transmits in slot k + 1, and if that slot is idle o r successful, the second subset (“right” subse t) transmits in slot k + 2. Alternatively , if another collision o ccurs in slot k + 1, the ﬁrst of the tw o subs ets splits again, and the second subset w aits for the resolution of that collision. Figure 6.1 exempliﬁes the op eration of BT A for three no des a , b a nd c . Observ e the binar y tree structure of the sets of transmitting and w aiting no des in the ﬁgure. The transmitting and w ait ing se ts in terms of subtrees o f this binary tree are show n in T able 6.1, where U = { a, b, c } denotes the set of all no des that w ere in volv ed in the initial collision. The lab eling of the subtrees is recursiv e; for example, LR denotes the right subtree of the left subtree of the orig ina l binary tree. The transmission order corresponds to that of a stac k, as sho wn in Figure 6.2. In eac h slot, the stac k is p opp ed and all the no des that w ere at the top of the stac k tr a nsmit their pac k ets. In case of a collision, the stack is pushed with no des that join the right subset and then pushed a g ain with nodes that join the left subset. In cas e of a suc cess or idle, no push op erations are p erformed on the stack . A Collision Res olution P erio d (CRP) is deﬁned to be complete d when a succe ss or idle o ccurs and there a re no remaining elemen ts on the stack. In Figure 6.1, the length of the CRP is 9 slots. During the op eration of BT A, man y new pac kets migh t arriv e while a collision is b eing r esolv ed. T o s olv e this problem, at the end of a CRP , the set of no des with new arriv als is immediately split in to j subse ts, where j is c hosen so that the exp ected num b er of pac k ets p er subset is sligh tly g reater than 1. The maxim um throughput, optimized o ver the c hoice of j as a f unction o f exp ected n umber of w aiting pack ets, is 0.43 pac kets p er slot [18]. 1 2 3 4 5 6 7 8 9 10 ti me Collisio n Resoluti on P erio d (CRP) 1 0 R LR L LL LRL e e e abc abc a bc 0 1 1 b c LRRL LRRR Figure 6.3: Mo diﬁed T ree Algorithm for three no des a , b and c . 106 Chapter 6. A Review of Random Access Algo rithms for Wireless Net w orks Slot T ransmitting set W aiting sets F eedbac k 1 U φ e 2 L R e 3 LL LR, R 1 4 LR R e 5 LRL LRR, R 0 6 LRRL LRRR, R 1 7 LRRR R 1 8 R φ 0 T able 6.2: T ransmitting and w aiting sets for mo diﬁed tree algorithm sho wn in Figure 6.3. 1 2 3 4 5 6 7 8 9 10 ti me abc abc a bc bc bc c b c Figure 6.4: Stac k represen t a tion o f t ransmitting and waiting no des for mo diﬁed tree algorithm shown in Figure 6.3. 6.1. T raditional Random Acce ss Algorithms 107 There exist v aria nts of BT A that yield higher throughput. F or example, in Mo diﬁed T ree Algorithm (MT A), if a collision in slot k is follo w ed b y an idle in slot k + 1, then no des whic h collided in slot k refrain from transmitting in slot k + 2. Instead, t hey further split in to t w o sub sets, one of whic h transm its in slot k + 2 . As a n example, the op eration of M T A for three nodes a , b and c is sho wn in Figure 6.3. Observ e that the length of t he CRP is 8 slots. F or this example, the transmitting and w aiting sets of subtrees are sh o wn in T able 6 .2, while t he corres p onding stac k represen t a tion is sho wn in F igure 6.4. If an idle o ccurs in the curren t slot and a collision o ccurred in the previous slot (see Slot 5 in Fig ure 6.4), then the stack is p opp ed a second t ime but the no des a t the top of the stac k are not transmitted. Instead, these no des split in to righ t and left subsets and these subsets are pushed o n the stack. This leads t o few er collisions and higher throughput compared to BT A. The maxim um s table throughput o f MT A is 0 .46 pac ke ts p er slot [64]. In First Come First Serve ( FCFS) s plitting algorithm [22], no des inv olv ed in a col- lision split into t wo subsets based on the arriv al t imes o f collided pac k ets. Using this approac h, each subset consists of all pac k ets that arrived in some given in terv al, and when a collision occurs, that in terv al will b e split in to t w o smaller interv als. By alw a ys transmitting pack ets that arrived in the earlier interv al ﬁrst, the algorithm tr ansmits suc- cessful pac kets in the order of their a r r iv al. The FC FS alg o rithm is stable fo r λ < 0 . 4871 [22]. Conﬂict resolution proto cols based on tr ee algorithms hav e prov able stability prop- erties [65]. W e should p oin t out tha t the random access algorithm prop osed in Chapter 7 has a “tree structure” analo g ous to that of MT A. Th e detailed exp lanations of BT A a nd MT A pro vide a basic bac kground to understand the dynamics of the prop osed algorithm. So far, w e ha v e summarize d the metho dolo gy of tr aditional ra ndom access a lg o- rithms. In subsequen t sections, we will fo cus on random access algorithms that are b etter suited for wireless net works suc h as WLANs and Wireless Metrop olitan Ar ea Net works (WMANs). 108 Chapter 6. A Review of Random Access Algo rithms for Wireless Net w orks 6.2 Signal Pro ces sing in Random A c cess The collis ion mo del (Section 6.1, As sumption 4) is simple in that the on us of sc heduling pac ke ts is left en tirely to the MA C la yer. On the con tra ry , ph ysical la y er t ec hniques lik e m ultipac ket reception, capture and net w ork-a ssisted div ersit y are able to corr ectly deco de pack ets from collisions b y means suc h a s co ding and signal pro cessing. These tec hniques are p o t en tial steps tow ards allev iating the burden of deco ding pac k ets from the MA C la y er to the phys ical lay er [66]. In this section, w e review represen tativ e re- searc h w ork whic h exploits signal pro cessing and div ersity tec hniques to cor r ectly deco de the receiv ed pack ets in random access wireless netw orks. With the adv en t of multiacces s tec hniques such as CDMA and Multius er D etection [67], t he ﬁrst fundamen tal c hange in the collision mo del has b een prop ounded in [68 ]. The authors oﬀer the generalization that, in the presence of sim ultaneous tr ansmissions , the reception c an b e describ ed b y conditional p robabilities instead of deterministic failure. They pro p ose the MultiP ac ke t Rece ption (MPR) mo del deﬁned by the matrix C =      C 10 C 11 C 20 C 21 C 22 . . . . . .      , (6.4) where C ij is the conditional probability that, giv en i use rs transmit, j out of i transmis- sions are successful. Giv en k users transmit at the same time, the a verage num b er of success fully rece iv ed pac k ets is giv en by C k = k X j =0 j C k j . (6.5) They sho w that ALOHA under MPR achie v es stable throughput lim k →∞ C k assuming that the lim it exists. The stability a nd dela y of ﬁnite- user slotted ALOHA with m ulti- pac ke t reception has b een analyzed in [69]. In [70], the authors analyze the probabilit y of capture in a m ultip oint to p oint wire- less netw ork. Analogous t o t he ph ysical in terference mo del, the captur e mo del assumes that if a u ser’s SINR exceeds a threshold γ , t hen that user’s pack et will be successfully receiv ed. They consider a realistic multiplicativ e propagatio n mo del in whic h the re- ceiv ed p ow er is obtained by m ultiplying the transmitted p o w er b y indep enden t random 6.3. Channel-Aw are ALOHA Algorithms 109 v ariables represen ting fading, shado wing and path loss eﬀects. T o mo del the near-far eﬀect, they assume that the distance r of a mobile station from t he base station is a random v ariable with distribution function F R ( r ). They show that, under broad condi- tions, the roll-oﬀ parameter δ of the distribution of p o wer receiv ed from a mobile station is determined by the path loss exp onen t a nd F R ( r ). Additio na lly , δ is insensitiv e to other eﬀects suc h as Ra yleigh or Rician fading and log - normal shado wing. Finally , they sho w that in the limit of a large num b er o f transmitters, the probability of capture is determined b y the p o w er capture threshold γ and δ . Though t he analysis provided in [70] is ma t hematically robust, the authors do not describe an y m ultiple access algorithm whic h ac hiev es high throughput in wireless net w orks under the capture mo del, i.e., their result is more existen tial than constructiv e. In [71], the authors prop o se Net work-Assis ted D iv ersit y Multiple Access (NDMA), a tec hnique for resolving collisions in wirele ss net works . They conside r a wireless slot- ted random access netw ork with Rayle igh fading and Additiv e White Gaussian Noise (A WGN). In NDMA, if k users collide in a give n slot, they rep eat their transmissions k − 1 times so that k c opies of the collided pac k ets are receiv ed. Using signal sep aration principles, the receiv er resolv es a k × k source mixing problem to extract the signals of individual users, without incurring any penalty in throughput. The proto col has b een extended to blind us er detec tion [72] and has pro v able stability [73]. A go o d review of NDMA proto cols is give n in [74]. An alternativ e to em plo ying signal processing tec hniques in random acces s wireless net works is to appropriately mo del the wireless c hannel and mo dify the w ell-researc hed ALOHA proto col. W e review suc h re searc h w ork in the next section. 6.3 Channel-A w are ALOHA Algorithms In this section, we review represen tativ e researc h w ork whose cen tral theme is to a dapt the retransmission probabilities o f users in r a ndom access wireless netw orks. In other w ords, w e review researc h work whic h dev elops c hannel-aw are ALOHA algorithms for wireless net works . In [75], the authors dev elop a c hannel-a w are ALOHA proto col for wire less net works . They assume a slotted s ystem, blo c k fading, { 0 , 1 , e } feedbac k and a bac klogged m o del 110 Chapter 6. A Review of Random Access Algo rithms for Wireless Net w orks (Section 6.1, Assumption 2b). They dev elop a distributed random access proto col in whic h eac h no de only has kno wledge of its own c hannel gain and no des ha ve long- term p o w er constraints. A no de t r a nsmits only if its c ha nnel gain exceeds H 0 . F or a system with n no des, the author s show that the optimum transmission pro babilit y is α ( n ) n , where α ( n ) ∈ (0 , 1] and α ( n ) → 1 as n → ∞ . Asymptotically , the ratio of the throughput of c hannel-a ware ALOHA to the throughput of a cen tralized sc heduler (whic h has knowle dge of channe l gains of a ll no des) is sho wn to b e 1 e . Opp ortunistic ALOHA algo rithms f or wireless netw orks hav e b een studied in [76 ]. The authors consider a general reception mo del whic h encompasses { 0 , 1 , e } feedbac k, capture as w ell as m ultipack et reception. Under the assumption that the Channel State Information (CSI) is know n t o eac h user, they prop ose a v ariant of slotted ALOHA, where the transmission proba bility is allo w ed to b e a function o f the CSI. The maxim um throughput for the ﬁnite-user inﬁnite-buﬀer mo del is deriv ed. Finally , t he theory is applied to CDMA netw orks with Linear Minim um Mean Square Error (LMMSE) receiv er and matc hed ﬁ lters. The performance of slotted ALOHA in a wireless net w ork with m ultiple des tinations under the ph ysical interference mo del is ev aluated in [19]. A pac k et is succ essful only if it is captured a t t he receiv er of its intended destination. The authors assume P oisson pac ke t generation, { 0 , 1 , e } feedbac k and circularly symmetric Gaussian distribution of users around eac h destination. They use a mo diﬁed v ersion of Riv est’s pseudo-Ba y esian estimator [22] to estimate the bac klog . Their sim ulation results demonstrate the eﬀect of arriv al rate, capture threshold, v ariance of user distribution a nd num b er of destinations on the throughput and energy eﬃciency p er destination. In [77], the author analyzes the throughput of slotted ALOHA in a multipoint to p oin t wireless ad ho c net w o rk und er the ph ysical in terference mo del. The cluster head emplo ys rev erse link p ow er con trol, similar to IS-95 CDMA systems [78], to ensure that equal p o w er is receiv ed from all no des who attempt transmission in a time slot. The wireless c hannel is mo deled as a m ultipack et reception c hannel. Assuming that one new pac ke t arrives at eac h no de in ev ery time slot, the state of the sy stem is c haracterized b y a discrete time Marko v c hain with a steady state distribution. Finally , the author describes a tec hnique to compute the net work throughput. In [7 9], the authors in tro duce spatial reuse slotted ALOHA, a random access proto- 6.4. Splitting Algo rithms 111 col f o r random h omogeneous mobile wireless netw orks. The occurrence of a collision is determined b y the SINR at a receiv er, i.e., the authors employ the ph ysical interfere nce mo del. They assume that no des are ra ndo mly placed in a tw o-dimensional plane accord- ing to a P oisson p oint pro cess and each no de c ho oses a rando m destination at some ﬁnite distance. The p ow ers at which stations can transmit are assumed to b e indep enden t and iden tically distributed (i.i.d.) and the wireless c hannel is c haracterized b y its propaga- tion path lo ss. No des mov e according to the random w a yp oin t mobility mo del [80]. The authors c haracterize the in terference pro cess using to ols from sto chastic geometry . Sub- sequen tly , t hey determine the probabilit y of c hannel a ccess that maximizes the exp ected pro jected distance tra v ersed p er hop to w a rds a destination, termed as “ spatial dens it y of progr ess”. Under the assumption that there is some non-degenerate no de mobility , the a ut ho rs sho w that the spatial densit y of progress is proportio nal to the square-ro ot of the dens it y of the no des. Though the authors presen t a distributed ALOHA proto col and address certain implemen ta tion issues, their mo del do es not represen t real- w orld scenarios. Practical deploy men ts of wireless no des are b etter mo deled by a uniform distribution in a ﬁnite plane rather than a P o isson p oint pro cess in an inﬁnite plane. Also, most of their results do not hold for static wireless net w orks (say , wireless mesh net works ) since ergodicity ass umptions no longer hold. F ina lly , their prop o sed ro ut ing proto col requires ev ery no de to hav e kno wledge of lo cations and MA C states (receiv er or transmitter) of all other no des, whic h requires a lot of message passing betw een no des (esp ecially with mobile no des) and is thus not scalable. Instead of adapting the transmission probabilities of users in random access wireless net works , o ne can also adapt the tra nsmission times of users based on the c hannel state and feedbac k from the receiv er. Suc h tec hniques, which can b e broadly termed as splitting a lgorithms o r tree-lik e algorithms for wireless net works , are review ed in the next section. 6.4 Splittin g Algo rithms In this section, w e review represen tativ e research work on random a ccess alg orithms whose main idea is to adapt the set of con tending users based on feedbac k from the c hannel or the common receiv er. In suc h w ork, the aut ho rs dev elop a nd analyze splitting 112 Chapter 6. A Review of Random Access Algo rithms for Wireless Net w orks (or tree or stack) algorithms for v arious mo dels of the wireless c hannel and ev aluate the p erformance of their algo rithms via sim ulations. In [8 1], the authors prop ose an opp ortunistic splitting algorithm for a m ultip oin t to p oin t wireless netw ork. They assume a slotted system, block fading channel and { 0 , 1 , e } feedbac k. Assuming that eac h user only knows its ow n channel gain and the n um b er o f bac klogged users, the authors pro p ose a distributed splitting a lgorithm to determine the user with the b est c hannel gain ov er a sequence of mini-slots. The alg o rithm determines a low er threshold H l and a higher threshold H h for each mini-slot, suc h that only users whose c hannel gains lie b etw een b et w een H l and H h are allow ed to transmit their pack ets. Based on r esults from “partitioning a sample with binar y type questions” [82], they sho w that the a v erag e n umber of mini-slots required to determine the use r with the b est c hannel is 2.5, independen t of the n um b er of users and the fading distribution. How ev er, their algorit hm is impractical b ecause it assumes that ev ery user can accurately estimate the num b er of bac klogged users. In [83], the authors consider a random access netw ork with inﬁnite users, Poisson arriv als and { 0 , k , e } immediate feedbac k, where k is an y p ositive in teger. In con trast to standard tree algorithms (BT A, MT A, F CFS) that discard collided pac k ets (Section 6.1, Assumption 4), they pro p ose an algorithm that stores collided pac k ets. The receiv er extracts information from the collided pac kets b y relying on succes siv e in terference can- cellation tec hniques ([67 ], Chapter 7) and the tree structure o f a collision resolution algorithm. Though their algo rithm achie v es a stable thro ughput of 0 .6 93, it requires inﬁnite storage and increased input v olta ge range at the receiv er, which are not feasible in pr a ctical systems. In [84 ], the author considers a mu ltip oin t to p oint wireless c hannel with and without capture and MPR. The c hannel pro vides Empt y(E)/Non-Empty(NE ) feedbac k to all activ e users and ‘success’ feedbac k to successful users only . The users do not need to kno w the starting times and ending times of collision resolution p erio ds. F or suc h a c hannel with E/NE binary feedbac k, the autho r prop oses and analyzes a stac k m ultiple access algo r ithm that is limited sensing and do es not r equire an y fra me sy nc hronizatio n. The author considers t w o mo dels for capture, namely Ray leigh fading with incoheren t and coherent com bining of joint in terference p ow er. F or MPR, the author assumes a maxim um of tw o successes during a collision. The maxim um throughput of the algo r it hm 6.5. T o wards P o w er Cont rolled Random Access 113 is n umerically ev aluated b e to 0.6548 when capture and MPR are presen t, and 0.28 9 1 when b oth eﬀects are absen t. Though a nov el splitting alg o rithm is pro p osed in [8 4 ], the a ut ho r do es not tak e in to a ccoun t throughput gains p ossible by v arying tra nsmission p o w ers of users. So far, we ha ve review ed researc h pap ers that either utilize signal pro cessing tec h- niques or adapt transmission pro babilities or transmission times to increase the t hr o ugh- put in random access wireless netw orks. The throughput can b e further increased by allo wing users to use v ariable transmission p o wers . W e review researc h pap ers which emplo y t his idea in the next section. 6.5 T o w ards P ow er C on tro l led Random Access In this s ection, we review represen tat iv e researc h pap ers whic h fo cus on pow er con trol tec hniques in random access wireless net w orks. W e then motiv ate the use of v ariable transmission p o w er to increase the throughput in ra ndo m access wireless net works . In [85], the author considers a time-slotted CD MA-ba sed wireles s net w ork wherein a ﬁnite n um b er of no des comm unicate with a common receiv er. The author formulates the problem o f determining the set o f no des that can transmit in eac h slot along with their corresp onding tra nsmission p o wers, sub j ect to constrain ts on ma ximum t r ansmission p o w er and the SINRs of all transmissions exceeding the comm unication threshold. Due to its NP-hard nature, the pro blem is relaxed to a case wherein a no de transmits with a certain proba bilit y in eac h slot. Equiv alently , the pro blem of joint p ow er con trol and link sch eduling is transfor med to a problem of p ow er con trolled ra ndo m access, wherein the ob jectiv e is t o determine the proba bilit y of transmission ∆ i and tra nsmission p o w er P i for eac h no de i , sub ject to constrain ts o n maxim um t ransmission p o w er and the “exp ected SINR” excee ding the comm unication threshold. The a uthor seeks to minimize a w eigh ted sum of the maximum transmiss ion p ow er and maxim um recipro cal probabilit y , i.e., minimize (max i P i + λ max i 1 ∆ i ). This con v ex optimization problem is solv ed using tec hniques from geometric programming [86 ]. Finally , the author derive s the proba bilit y of outage 2 and delay distribution of buﬀered pac ke ts and demonstrates 2 An outage o ccur s on a link if the received (a ctual) SINR on the link is less than the communication threshold. 114 Chapter 6. A Review of Random Access Algo rithms for Wireless Net w orks the eﬃcacy of the sc hemes via sim ulations. In [87], the authors inv estigate transmission p ow er con tr o l and rate adapta t io n in random access wireless net w o rks using game theoretic tec hniques. They consider mul- tiple transmitters sharing a time-slott ed channe l to commu nicate equal-length pack ets with a common receiv er. A user’s pac k et is success fully receiv ed if the SINR at the re- ceiv er is no les s than the comm unication threshold, i.e., the authors employ the ph ysical in terference mo del. The random access problem is formulated as a game wh erein eac h user selects its strategy (tra nsmit or wait) at eac h stag e of the game in a non-co op erative (indep enden t) or co op erativ e manner. The autho r s ev aluate equilibrium strategies for non-co op erative and co op erat ive symmetric random access ga mes. Finally , t he authors describe distributed p o wer con trol and ra te adaptation games for non-co op erat ive users for a collision c hannel with p ow er-based capture. Their numeric al results demonstrate impro ved exp ected user utilities when p o w er control and rate adaptation a re incorp o- rated, at the expense of increased computational complexit y . Though the authors pro- p ose a distributed ra ndom access algorithm based on game theoretic tec hniques, t heir algorithm is impractical b ecause it assumes tha t eve ry user kno ws n , the n umber of bac klogged users, in eac h slot. Ho w ev er, in practice, n can only b e estimated using tec hniques suc h as Riv est’s pseudo-Bay esian algorithm [62]. Though researc hers ha v e addressed the problem o f ra ndo m access in wireless net w orks b y considering v arious c hannel mo dels, diﬀerent ty p es of feedbac k and realistic criteria for success ful pac ket r eception, only few of them exploit the idea that throughput gains are ac hiev able in a ra ndom access wireless net w o rk b y v arying transmiss ion p o w ers of users . In general, v arying the transmission p ow ers of users leads to higher long-term a v erage p o w er. Ho w ev er, there exist wireles s netw orks whose users do not ha v e stringen t energy requiremen ts. F or suc h scenarios, it w ould b e useful to in v estigate the throughput gains ac hiev able in the net w ork b y v arying the transmission p o wers of users. W e en visage dev eloping a p ow er con trolled random a ccess algorithm for wireless net- w orks under t he ph ysical in t erference mo del. W e seek an algorithm that yields higher throughput than traditional random a ccess algorithms. In cog nizance of these require- men ts, we prop ose a p o w er con t r olled splitting algorithm for wireless netw orks in Chapter 7. The alg o rithm is so designed that success ful pac k ets are tr ansmitted in the order of their a rriv als, i.e., in an F CFS manner. 6.5. T o wards P o w er Cont rolled Random Access 115 In the system mo del considered in Chapter 7, if m ultiple transmissions o ccur, the receiv er can deco de a ce rtain user’s pac ke t correctly only if the r eceiv ed SINR excee ds a threshold, i.e., w e consider a ch annel w ith pow er-based capture. The notion of c apture has b een addres sed previously , though in diﬀerent con t exts [19], [84], [88]. How ev er, in Chapter 7, we motiv ate the idea that a user can transmit at v ariable p ow er lev els to increase the c hances of capture. Moreov er, unlik e [19], [84], [88], w e assume { 0 , 1 , c, e } feedbac k, where 0, 1 and e denote idle, success and error resp ectiv ely (Section 6.1, Assumption 4), and c denotes capture in the presence of m ultiple transmissions. Note that the system mo del considered in Chapter 7 is diﬀeren t from t ho se considered in existing w orks on splitting algorithms for wireless netw orks. F o r example, in [84], the author pro p oses a nov el splitting a lg orithm, but do es not tak e into accoun t thro ug hput gains p ossible b y v arying the transmission p ow er. Though the authors of [81] prop ose a splitting algorithm to de termine the user with the b est c hannel gain, their algorithm is impractical because it assumes that eve ry user can accurately estimate the n umber of bac klogged us ers. T o t he b est of our knowledge , there is no existing w ork on v ariable p ow er splitting al- gorithms f or a wire less netw ork under the ph ysical in terference mo del. The speciﬁcation of the prop osed algo r it hm along with its performance analysis and ev alua t io n constitute the sub ject mat t er o f the next c hapter. Chapter 7 P o w er Con trolled F CFS Splitting Algorithm for Wireless Net w ork s In this c hapter, w e prop ound a random access algorithm that incorp ora tes v ariable trans- mission p ow ers in a m ultip oint to p oint wireless net work. Sp eciﬁcally , we inv estigate random access in wireless netw orks under the ph ysical interferenc e mo del wherein the receiv er is capable of p ow er-based capture, i.e., a pack et can b e deco ded correctly in the presence o f m ultiple transmissions if the receiv ed SIN R exceeds the comm unication threshold. W e prop ose an inte rv al splitting algo rithm that v aries the transmission p ow - ers of use rs based on c hannel feedback . W e deriv e the maxim um stable throughput of the prop osed algorithm and demonstrate that it ac hiev es b etter p erformance than the F CFS splitting alg orithm [22] with uniform transmission p ow er. The rest of the c hapter is org anized as follows. W e describe our system mo del in Section 7.1 and motiv ate v ariable con tro l of tr a nsmission p ow ers o f con tending users in Section 7 .2. W e describ e the prop osed random a ccess algorithm and provide tw o illustrativ e examples in Section 7.3. W e mo del the algorithm dynamics b y a Mark ov c hain and deriv e its maxim um stable thro ughput in Section 7 .4. The p erformance of the prop osed algorithm is ev aluated in Section 7.5. W e conclude in Section 7.6. 7.1 System Mo del Consider a multipoint to p oint wireless net w ork. W e assume the follow ing: 117 118 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks 1. Slotted sy stem: Users (no des) transmit ﬁxed-length pac k ets to a common r eceiv er o ver a time-slotted channel. All users are sync hronized suc h that the reception of a pack et starts at an integer t ime and ends b efore the next integer time. 2. P oisson arriv als: The pack et arriv al pro cess is Poisson distributed with ov erall rate λ , a nd eac h pack et arrives to a new user tha t has neve r b een assigned a pack et b efore. After a user successfully transmits its pac k et, that user ceases to exist and do es no t conte nd for c ha nnel access in future time slots. 3. Channel mo del: The wireless channe l is mo deled by the path loss propagatio n mo del. The receiv ed signal p o wer at a distance D from the transmitter is giv en b y P D β , where P is the transmission p ow er and β is the path loss factor. W e do not consider fading a nd shadowing eﬀects. 4. P ow er-based capture: According to the phys ical in terference model [1 5 ], a pack et transmission from transmitter t i,j to receiv er r in i th time slot is successful if and only if the SINR at receiv er r is greater than or equal to the comm unication threshold γ c 1 , i.e., P i,j D β ( t i,j ,r ) N 0 + P M i k =1 k 6 = j P i,k D β ( t i,k ,r ) > γ c , (7.1) where M i = n um b er of concurren t transm itters in i th time slot , t i,j = j th transmitter in i th time slot ( j = 1 , 2 , . . . , M i ) , D ( t i,j , r ) = Euclidean distance b et w een t i,j and r , P i,j = transmission p ow er of t i,j , N 0 = thermal noise p ow er sp ectral densit y . 5. { 0 , 1 , c, e } immediate feedbac k: By the end of eac h slot, use rs are informed of the feedbac k from the receiv er immediately and without an y erro r. The feedback is one o f: 1 In literature, γ c is also referred to as capture ratio [8 4], capture threshold [19] and p ow er ratio threshold [70]. 7.2. Motiv ation and Pr oblem F orm ulation 119 (a) idle (0): when no pack et transmission o ccurs, (b) p erfect reception (1): when one pa c k et transmission o ccurs and is receiv ed success fully , (c) capture ( c ): when m ultiple pack et transmissions o ccur and only one pac k et is receiv ed su ccessfully , or (d) collision ( e ): when multiple pack et transmissions o ccur and no pac ket recep- tion is successful. The receiv er can distinguish b etw een 1 a nd c b y using energy dete ctors [8 3], [89]. Th us, b y the end of ev ery slot, only tw o bits are require d to provide feedbac k from the receiv er to all users. Note that tw o bits are required to pro vide feedbac k ev en for the class ical { 0 , 1 , e } feedbac k model. Th us, our { 0 , 1 , c, e } immediate feed bac k assumption do es not increase the nu m b er of bits required for feedbac k.. 6. Gated Channel Acces s Algorithm (CAA): New pack ets are t ransmitted in the ﬁrst a v ailable slot aft er previous conﬂicts are r esolv ed. The time in t erv al fr o m the slot where an initial collision o ccurs up to and including the slot in which all users recognize that all pac k ets inv olv ed in the collision hav e b een successfully receiv ed, is called a Collision R esolution P erio d (CRP). Thus , new ar r iv als a re inhibited from tra nsmiss ion during the CRP . 7. Equal distances: W e assume that each user is at the same distance D fro m the common receiv er. 7.2 Motiv ati o n and Problem F orm ul ati on The maxim um stable throug hput o f the w ell-known F CFS splitting a lgorithm is 0 .4871 [22], whic h is the highest throughput amongst a wide class of random access algorithms for wired net w orks. How ev er, in a wireless net work, t r a nsmission p ow er of a node pro- vides a n extra degree of freedom, and higher throughputs are ac hiev able. Consider a scenario wherein all contending no des transmit with equal pow er P in a giv en time slot. When only o ne no de transmits, its pack et is succes sfully receiv ed if the 120 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks SINR t hreshold conditio n (7.1) is satisﬁed, i.e., P > γ c N 0 D β . (7.2) When M no des transmit concurren tly with equal p o wer P , where M > 2 , the SINR corresp onding to i th transmission is g iven by SINR i = P D β N 0 + ( M − 1) P D β , (7.3) a quantit y whic h is alwa ys less tha n 1. Since γ c > 1 for all practical narro wband comm unication receiv ers [70], SINR i < γ c ∀ i and all M transmissions a r e unsuccess ful 2 . Th us, when m ultiple no des transmit with equal p ow er, a collision o ccurs irresp ectiv e of the tr a nsmission p ow er P . Ho wev er, the ab o v e situation can be circum v en ted b y v arying transmiss ion pow ers of users in some sp ecial case s. With relative ly small attempt ra tes, when a collis ion o ccurs, it is most likely b etw een o nly tw o pack ets [22]. In this case, if the receiv er is capable of p o wer-based capture, a collision b et w een tw o no des can b e a voide d by using diﬀeren t transmission p ow ers. Sp eciﬁcally , one of the no des, say N 1 , transmits with minimu m p o w er P 1 suc h that, if it w ere the only no de tra nsmitting in that time slot, then its pac ke t transmission will b e successful. F rom (7.1), the required no minal p o wer is P 1 = γ c N 0 D β . (7.4) The other no de, sa y N 2 , transmits with minim um p ow er P 2 suc h tha t if there is exactly one other no de tra nsmitting at nominal p ow er P 1 , then the pac k et transmitted b y N 2 will b e successful. F rom (7.1) a nd (7 .4 ), w e obtain P 2 D β N 0 + P 1 D β = γ c , P 2 = γ c ( N 0 D β + P 1 ) , ∴ P 2 = γ c (1 + γ c ) N 0 D β . (7.5) Note that P 2 P 1 = 1 + γ c . W e do not conside r more tha n t wo pow er lev els for the follo wing reasons: 2 F or a spread sp ectrum CDMA system with pr oc essing gain L , (7.3 ) gets mo diﬁed to SINR i = P D β N 0 + I L ( M − 1) P D β [87]. F or such a wideband sys tem, γ c < 1, and mo re than o ne pack et ca n be deco ded correctly in the pr esence of multiple trans missions. How ever, in this thesis , we consider narr owband systems only . 7.3. PCF C FS In terv al Splitting Algorithm 121 1. it complicates the p ow er-con t r ol a lgorithm, and 2. most mobile wireless devices hav e constrain ts on p eak tra nsmission p ow er. Note that the ab o ve p o w er con trol techn ique con verts some collisions into “captures”. Th us, it has the p oten tial of increasing the thro ughput of random access algorithms emplo ying u niform transmission p ow er. W e seek to design a distributed algo r it hm incorp orating this p o w er control tec hnique, while still ensuring that the a lg orithm transmits successful pack ets in the o r der of their arriv al, i.e., in an FCFS manner 3 . 7.3 PCF CFS Interv al Spl i tting Algorith m In this section, we presen t an algo r ithmic description of the pro p osed P ow er Con trolled First Come Fir st Serv e (PC F CFS) splitting algorithm. W e also explain the b eha vior of the pro p osed algo rithm by pro viding tw o illustrative examples. 7.3.1 Description W e ﬁrst describ e the no tation. Slo t k is deﬁned to b e the time in terv al [ k , k + 1). A t eac h in teger time k ( k > 1 ), the alg o rithm speciﬁes the pac k ets to b e transmitted in slot k to b e the set of all pack ets that arriv ed in an earlier in terv al [ T ( k ) , T ( k ) + φ ( k )), whic h is deﬁned as the al lo c ation interval for slot k . The maxim um size of the allo catio n interv al is denoted b y φ 0 , a parameter whic h will b e optimized for maxim um throughput in Section 7.4. P ac kets are indexed as 1 , 2 , . . . in the order of the ir arriv al. Since the arriv al times are P oisson distributed with rate λ , t he in ter-arriv al times are exponentially dis tributed with mean 1 λ . Let a i denote the arriv al time of i th pac ke t. Using the memoryless prop ert y of the exp onen tial distribution (and without loss of generality), we assume that a 1 = 0. The transmiss ion p ow er of i th pac ke t in slot k is denoted by P i ( k ), where P i ( k ) ∈ { 0 , P 1 , P 2 } . No te that, if P i ( k ) = 0 , then i th pac ke t is not transmitted in slot k . 3 Since success ful pack ets are t ransmitted in an FCFS manner, the delay exp erienced by a pack et will not b e sig niﬁcant ly hig her than the av erage pack et delay . Thus, f rom a Qo S p ersp ective, FCFS transmission of pack ets not only guarantees av erage dela y bounds, but also ensures fairness of user pack ets. 122 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks Algorithm 9 describ es the prop osed P ow er Con trolled First Come First Serv e (PCF CFS) splitting algorithm, whic h is the set of rules by whic h the users compute allo cation in- terv al parameters { T ( k + 1) , φ ( k + 1) , σ ( k + 1) } a nd transmission p o w er P i ( k + 1) f or slot k + 1 in terms of the f eedbac k and allo cation in t erv al parameters for slot k . In our algor it hm, ev ery allo cat io n in t erv al is tagged as a “left” ( L ) or “right” ( R ) inter- v al. σ ( k ) denotes the tag ( L or R ) of allo cation in terv al [ T ( k ) , T ( k ) + φ ( k )) in slot k . Moreo ver, whenev er a n allo cation inte rv al is split, it is alwa ys split into t wo equal-sized subin terv als, a nd these subin terv als ( L , R ) are said to c orr esp ond to eac h other. In Phase 1 of the algorithm, w e initialize v arious quan tities. τ denotes the n um b er of slots for whic h the algorithm op erates; ideally τ → ∞ . By con v en tion, the initial allo cation in terv al is [0 , min( φ 0 , 1)), whic h is a rig h t inte rv al ( R ). The initial channel feedbac k is assumed to b e idle (0). In Phase 2 of t he algorit hm, w e determine p o wer lev els, obtain channel feedbac k and compute allo cation interv al parameters for eac h success iv e slot k . In Phase 2a , all users whose a r r iv al times lie in the left ha lf of the curren t allo cation interv al tra nsmit with higher p o wer P 2 , while all users whose arr iv al times lie in the righ t half of the current allo cation in terv al transmit with nominal p ow er P 1 . Ho wev er, if a capture o ccurred in the previous slot k − 1, all users in the current allo catio n in terv al tra nsmit with nominal p o w er P 1 . Therefore, our algorithm alw ays transmits succes sful pack ets in an F CFS manner. In Phase 2b, the allo cation in terv al parameters are mo dulated based on the c hannel feedbac k. More sp eciﬁcally , if a collision o ccurs, then the left half of the curren t allocation in terv al b ecomes the new allo cation in terv al. If a capture o ccurs, then the right half of the current allo cation interv al b ecomes the new allo cation in terv a l. If a succ ess o ccurs and the curren t allo catio n in terv al is tagged as a left in terv al, t hen the correspo nding righ t in terv al b ecomes the new allo cation in terv a l. If an idle o ccurs and the curren t allo cation in terv al is tagged as a left interv al, then the left half of the corresp onding rig h t in terv al b ecomes the new allo cation inte rv al. Otherwise, if a success or an idle o ccurs and the curren t allo catio n in terv al is tag ged as a righ t in terv al, the w aiting in terv al truncated to length φ 0 b ecomes the new allo cation in t erv al, and a ne w Collision Resolution P erio d (CRP) b egins in the next time slot k + 1. Note that the transmit p o w er lev els in PCF CFS a r e v ariable and based on c hannel feedbac k, i.e., they are adaptiv e. 7.3. PCF C FS In terv al Splitting Algorithm 123 Algorithm 9 PCF CFS s plitting algorithm 1: input: φ 0 , P 1 , P 2 , arriv als a 1 , a 2 , a 3 , . . . in [0 , τ ) { Phase 1 b egins } 2: T (1) ← 0 3: φ (1) ← min( φ 0 , 1) 4: σ (1) = R 5: feedbac k = 0 { Phase 1 ends } 6: for k ← 1 t o τ do { Phase 2 b egins } 7: if feedbac k 6 = c then { Phase 2a b egins } 8: for all i suc h that T ( k ) 6 a i 0 , (7.6) ∴ G i = 1 2 G i − 1 ∀ i > 1 . (7.7) W e view ( R , 0) as the starting state as w ell as t he ﬁnal state. F or brevit y in notation, the tra nsition probability fro m state ( A, i ) to state ( B , j ) is denoted by P A i ,B j , where A, B ∈ { L, L ′ , R, R ′ , C } a nd i, j ∈ { 0 } ∪ Z + (see Figure 7.5). F or example, the transition probabilit y from ( L, 1) to ( C , 2) is denoted by P L 1 ,C 2 . LL LR ( x LL pack ets) RR RL L R ( x LR pack ets) ( x RL pack ets) ( x RR pack ets) ( x L pack ets) ( x R pack ets) Figure 7.6: Nota tion for n um b er of pack ets in left and rig h t subin terv als of the original allo cation in terv al. P R 0 ,R 0 is the probability of an idle or success in the ﬁrst slot of the CRP . Since the n um b er of pac k ets in t he initial allo cation in terv al is P oisson with mean G 0 , the probabilit y of 0 or 1 pac ket is P R 0 ,R 0 = (1 + G 0 ) e − G 0 . (7.8) 7.4. Throughpu t Analysis 131 P R 0 ,C 1 is the probability of capture in the ﬁrst slot of a CRP . Let x L and x R denote the n umber of pack ets in the left and rig ht ha lves of the original allo cation in terv al resp ectiv ely , as s ho wn in Figure 7.6. Capture occurs if a nd only if x L = 1 and x R = 1. x L and x R are indep enden t P oisson r.v.s o f mean G 1 eac h. Th us P R 0 ,C 1 = Pr( x L = 1 , x R = 1) , = Pr( x L = 1) Pr( x R = 1) , = G 2 1 e − 2 G 1 , ∴ P R 0 ,C 1 = G 2 0 4 e − G 0 . (7.9) State ( L, 1) is entered after collision in state ( R, 0 ) . Using (7.8) and (7.9), this o ccurs with pro ba bilit y P R 0 ,L 1 = 1 − P R 0 ,R 0 − P R 0 ,C 1 , ∴ P R 0 ,L 1 = 1 −  1 + G 0 + G 2 0 4  e − G 0 . (7.10) Since a capture is alw ays follow ed by a deterministic success, P C i ,R 0 = 1 ∀ i > 1 . (7.11) Lemma 7.4.1. T he o utgoing tr ansition pr ob abi l i ties fr om ( L, i ) , wher e i > 1 , ar e given by P L i ,R i = (1 − e − G i − G i e − G i ) G i e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 , (7.12) P L i ,L ′ i +1 = (1 − e − G i − G i e − G i ) e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 , (7.13) P L i ,C i +1 = G 2 i 4 e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 , (7.14) P L i ,L i +1 = 1 − (1 + G i + G 2 i 4 ) e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 . (7.15) Pr o of. Refer to Fig ure 7.5. F or i = 1, ( L, i ) is en tered only via a collision in ( R , i − 1). F or i = 2, ( L, i ) is entered only via a collision in ( L, i − 1) or ( R, i − 1). F or i > 3, ( L, i ) is en tered only via a collision in ( L ′ , i − 1), ( L, i − 1), ( R, i − 1) or ( R ′ , i − 1). In ev ery case, 132 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks a subin terv al Y is split into Y L a nd Y R , and Y L becomes the ne w allo cation in terv al. Let x Y L and x Y R denote the n um b er of pac k ets in Y L and Y R resp ectiv ely . A priori, x Y L and x Y R are indep enden t P oisson r.v.s of mean G i eac h. The ev en t that a collision o ccurred in the previous state is { x Y L + x Y R > 2 } ∩ { x Y L = x Y R = 1 } c =: C Y . Note that x Y L + x Y R = x Y is a P oisson r.v. of mean G i − 1 . F rom (7 .7), G i = 1 2 G i − 1 ∀ i > 1. The probabilit y of success in ( L, i ) is the probability that x Y L = 1 conditional on C Y , i.e., P L i ,R i = Pr( x Y L = 1 |C Y ) , = Pr( C Y | x Y L = 1) Pr( x Y L = 1) Pr( C Y ) , = Pr( { x Y R = 1 } ∩ { x Y R = 1 } c ) Pr( x Y L = 1) Pr( C Y ) , = Pr( x Y R > 2) Pr( x Y L = 1) Pr( { x Y L + x Y R > 2 } ∩ { x Y L = x Y R = 1 } c ) , = Pr( x Y R > 2) Pr( x Y L = 1) Pr( x Y > 2) − Pr ( x Y L = 1) Pr( x Y R = 1) , = (1 − e − G i − G i e − G i ) G i e − G i 1 − e − G i − 1 − G i − 1 e − G i − 1 − G 2 i e − 2 G i , ∴ P L i ,R i = (1 − e − G i − G i e − G i ) G i e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 . (7.16) The probability of idle in ( L, i ) is the pr o babilit y that x Y L = 0 conditional on C Y , i.e., P L i ,L ′ i +1 = Pr( x Y L = 0 |C Y ) , = Pr( C Y | x Y L = 0) Pr( x Y L = 0) Pr( C Y ) , = Pr( { x Y R > 2 } ∩ { x Y R = 1 } c ) Pr( x Y L = 0) Pr( C Y ) , = Pr( x Y R > 2) Pr( x Y L = 0) Pr( { x Y L + x Y R > 2 } ∩ { x Y L = x Y R = 1 } c ) , = Pr( x Y R > 2) Pr( x Y L = 0) Pr( x Y > 2) − Pr ( x Y L = 1) Pr( x Y R = 1) , = (1 − e − G i − G i e − G i ) e − G i 1 − e − G i − 1 − G i − 1 e − G i − 1 − G 2 i e − 2 G i , ∴ P L i ,L ′ i +1 = (1 − e − G i − G i e − G i ) e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 . (7.17) Let x Y LL and x Y LR denote the n umber of pa c k ets in Y LL a nd Y LR resp ectiv ely . x Y LL and x Y LR are indep enden t Pois son r.v.s of mean G i +1 eac h, and x Y LL + x Y LR = x Y L . 7.4. Throughpu t Analysis 133 The probability of capture in ( L, i ) is the probabilit y that x Y LL = 1 and x Y LR = 1 conditional on C Y , i.e., P L i ,C i +1 = Pr( x Y LL = 1 , x Y LR = 1 |C Y ) , = Pr( C Y | x Y LL = 1 , x Y LR = 1) Pr( x Y LL = 1 , x Y LR = 1) Pr( { x Y L + x Y R > 2 } ∩ { x Y L = x Y R = 1 } c ) , = Pr( C Y | x Y L = 2) Pr( x Y LL = 1) Pr( x Y LR = 1) Pr( { x Y L + x Y R > 2 } ∩ { x Y L = x Y R = 1 } c ) , = Pr( x Y R > 0) Pr( x Y LL = 1) Pr( x Y LR = 1) Pr( x Y > 2) − Pr( x Y L = 1) Pr( x Y R = 1) , = 1 .G 2 i +1 e − 2 G i +1 1 − e − G i − 1 − G i − 1 e − G i − 1 − G 2 i e − 2 G i , ∴ P L i ,C i +1 = G 2 i 4 e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 . (7.18) F rom (7.16 ), (7.17) and (7.18), w e obtain P L i ,L i +1 = 1 − P L i ,R i − P L i ,L ′ i +1 − P L i ,C i +1 , ∴ P L i ,L i +1 = 1 −  1 + G i + G 2 i 4  e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 . (7.19)  Lemma 7.4.2. The outgoing tr a nsition pr ob abilities fr om ( R, i ) ar e given by P R i ,C i +1 = G 2 i 4 e − G i 1 − (1 + G i ) e − G i ∀ i > 1 , (7.20) P R i ,L i +1 = 1 −  1 + G i + G 2 i 4  e − G i 1 − (1 + G i ) e − G i ∀ i > 1 . (7.21) Pr o of. Refer to Figure 7.5. F or i > 1, ( R , i ) is ente red only via a success in ( L, i ). Recall that ( L, i ) was e n t ered only via a collision fro m a previous state. W e us e the notation in tro duced in the pro of of Lemma 7.4.1. Deﬁne the ev en t S Y L := C Y ∩ { x Y L = 1 } , = { x Y L + x Y R > 2 } ∩ { x Y L = x Y R = 1 } c ∩ { x Y L = 1 } , = { x Y R > 1 } ∩ { x Y R = 1 } c ∩ { x Y L = 1 } , ∴ S Y L = { x Y R > 2 } ∩ { x Y L = 1 } . (7.22) 134 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks Let x Y RL and x Y RR denote the n um b er of pac k ets in Y RL and Y RR resp ectiv ely . x Y RL and x Y RR are indep enden t P oisson r.v.s o f mean G i +1 eac h. Since x Y R > 2, a success or an idle can nev er occur in state ( R, i ). Note that x Y R = x Y RL + x Y RR . The probabilit y of capture in state ( R, i ) is the pro babilit y that x Y RL = 1 and x Y RR = 1 conditional on S Y L , i.e., P R i ,C i +1 = Pr( x Y RL = 1 , x Y RR = 1 | x Y R > 2 , x Y L = 1) , = Pr( x Y RL = 1 , x Y RR = 1 | x Y R > 2) , = Pr( x Y R > 2 | x Y RL = 1 , x Y RR = 1) Pr( x Y RL = 1 , x Y RR = 1) Pr( x Y R > 2) , = Pr( x Y RL + x Y RR > 2 | x Y RL = 1 , x Y RR = 1) Pr( x Y RL = 1 , x Y RR = 1) Pr( x Y R > 2) , = 1 . Pr( x Y RL = 1) Pr( x Y RR = 1) Pr( x Y R > 2) , = G 2 i +1 e − 2 G i +1 1 − e − G i − G i e − G i , ∴ P R i ,C i +1 = G 2 i 4 e − G i 1 − (1 + G i ) e − G i . (7.23) F rom (7.23 ), w e obtain P R i ,L i +1 = 1 − P R i ,C i +1 , = 1 − G 2 i 4 e − G i 1 − (1 + G i ) e − G i , ∴ P R i ,L i +1 = 1 −  1 + G i + G 2 i 4  e − G i 1 − (1 + G i ) e − G i . (7.24)  Lemma 7.4.3. The outgoing tr a nsition pr ob abilities fr om ( L ′ , i ) ar e given by P L ′ i ,R ′ i = (1 − e − G i ) G i e − G i 1 − (1 + G i − 1 ) e − G i − 1 ∀ i > 2 , (7.25) P L ′ i ,L ′ i +1 = (1 − e − G i − G i e − G i ) e − G i 1 − (1 + G i − 1 ) e − G i − 1 ∀ i > 2 , (7.26) P L ′ i ,C i +1 = G 2 i 4 e − G i 1 − (1 + G i − 1 ) e − G i − 1 ∀ i > 2 , (7.27) P L ′ i ,L i +1 = 1 −  1 + G i + G 2 i 4  e − G i 1 − (1 + G i − 1 ) e − G i − 1 ∀ i > 2 . (7.28) 7.4. Throughpu t Analysis 135 Pr o of. Refer to F ig ure 7.5. F or i = 2, ( L ′ , i ) is entered only b y an idle in ( L, i − 1). F or i > 3, state ( L ′ , i ) is en tered by an idle in ( L ′ , i − 1 ) or an idle in ( L, i − 1). In ev ery case, a res idual righ t subin terv al, sa y Z , is split in t o Z L and Z R , and Z L b ecomes the new allo cation in terv al. Note that ( L ′ , i ) can b e en t ered if and only if there is a collision (in some time slot) follow ed b y one or more idles. Therefore, Z m ust con tain at least tw o pac ke ts. Let x Z L and x Z R denote the num b er o f pac k ets in Z L and Z R r espectiv ely . A priori, x Z L and x Z R are indep enden t P oisson r.v.s of mean G i eac h. Let x Z denote the n umber of pac kets in Z . Th us x Z = x Z L + x Z R , x Z is a Poiss on r.v. of mean G i − 1 and x Z > 2. The probability of success in ( L ′ , i ) is the proba bility that x Z L = 1 conditional on x Z > 2, i.e., P L ′ i ,R ′ i = Pr( x Z L = 1 | x Z > 2) , = Pr( x Z > 2 | x Z L = 1) Pr( x Z L = 1) Pr( x Z > 2) , = Pr( x Z L + x Z R > 2 | x Z L = 1) Pr( x Z L = 1) Pr( x Z > 2) , = Pr( x Z R > 1) Pr( x Z L = 1) Pr( x Z > 2) , ∴ P L ′ i ,R ′ i = (1 − e − G i ) G i e − G i 1 − (1 + G i − 1 ) e − G i − 1 . (7.29) The proba bility of idle in ( L ′ , i ) is the probabilit y that x Z L = 0 conditio na l on x Z > 2, i.e., P L ′ i ,L ′ i +1 = Pr( x Z L = 0 | x Z > 2) , = Pr( x Z > 2 | x Z L = 0) Pr( x Z L = 0) Pr( x Z > 2) , = Pr( x Z L + x Z R > 2 | x Z L = 0) Pr( x Z L = 0) Pr( x Z > 2) , = Pr( x Z R > 2) Pr( x Z L = 0) Pr( x Z > 2) , ∴ P L ′ i ,L ′ i +1 = (1 − e − G i − G i e − G i ) e − G i 1 − (1 + G i − 1 ) e − G i − 1 . (7.30) Let x Z LL and x Z LR denote the num b er of pack ets in Z LL and Z LR respectiv ely . A priori, x Z LL and x Z LR are indep enden t Poisson r.v.s of mean G i +1 eac h. The probabilit y of capture in ( L ′ , i ) is the probability that x Z LL = 1 and x Z LR = 1 conditional on x Z > 2, i.e., 136 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks P L ′ i ,C i +1 = Pr( x Z LL = 1 , x Z LR = 1 | x Z > 2) , = Pr( x Z > 2 | x Z LL = 1 , x Z LR = 1) Pr( x Z LL = 1 , x Z LR = 1) Pr( x Z > 2) , = 1 . Pr( x Z LL = 1) Pr( x Z LR = 1) Pr( x Z > 2) , = G 2 i +1 e − 2 G i +1 1 − e − G i − 1 − G i − 1 e − G i − 1 , ∴ P L ′ i ,C i +1 = G 2 i 4 e − G i 1 − (1 + G i − 1 ) e − G i − 1 . (7.31) F rom (7.29 ), (7.30) and (7.31), w e obtain P L ′ i ,L i +1 = 1 − P L ′ i ,R ′ i − P L ′ i ,L ′ i +1 − P L ′ i ,C ′ i +1 , (7.32) ∴ P L ′ i ,L i +1 = 1 −  1 + G i + G 2 i 4  e − G i 1 − (1 + G i − 1 ) e − G i − 1 . (7.33)  Lemma 7.4.4. The outgoing tr a nsition pr ob abilities fr om ( R ′ , i ) ar e given by P R ′ i ,R 0 = G i e − G i 1 − e − G i ∀ i > 2 , (7.34) P R ′ i ,C i +1 = G 2 i 4 e − G i 1 − e − G i ∀ i > 2 , (7.35) P R ′ i ,L i +1 = 1 −  1 + G i + G 2 i 4  e − G i 1 − e − G i ∀ i > 2 . (7.36) Pr o of. Refer to Figure 7.5. F or i > 2 , state ( R ′ , i ) is entered if and only if a succe ss o ccurs in state ( L ′ , i ). When ( L ′ , i ) w as en tered, a residual right subin terv al Z w as split in to Z L and Z R , and Z L b ecame the new allo cation in terv al. R ecall that x Z > 2, since ( L ′ , i ) can only b e en tered after a collision follow ed b y one or mor e idle s. A succe ss in ( L ′ , i ) implies x Z L = 1. Hence, ( R ′ , i ) is entere d if and only if b oth these eve n ts o ccurs, i.e., x Z > 2 a nd x Z L = 1. Therefore, ( R ′ , i ) can b e entered if and only if x Z R > 1. Note that there c an nev er b e an idle from ( R ′ , i ). The probability of succe ss in ( R ′ , i ) is the probability that x Z R = 1 c onditional o n 7.4. Throughpu t Analysis 137 x Z R > 1, i.e., P R ′ i ,R 0 = Pr( x Z R = 1 | x Z R > 1) , = Pr( x Z R > 1 | x Z R = 1) Pr( x Z R = 1) Pr( x Z R > 1) , ∴ P R ′ i ,R 0 = G i e − G i 1 − e − G i . (7.37) Let x Z RL and x Z RR denote the num b er o f pa c k ets in Z RL and Z R R resp ectiv ely . Note t ha t x Z R = x Z RL + x Z RR . x Z RL and x Z RR are indep enden t P oisson r.v.s of mean G i +1 eac h. The probabilit y of capture in state ( R ′ , i ) is the probability t ha t x Z RL = 1 and x Z RR = 1 conditional o n x Z R > 1, i.e., P R ′ i ,C i +1 = Pr( x Z RL = 1 , x Z RR = 1 | x Z R > 1) , = Pr( x Z R > 1 | x Z RL = 1 , x Z RR = 1) Pr( x Z RL = 1 , x Z RR = 1) Pr( x Z R > 1) , = 1 . Pr( x Z RL = 1) Pr( x Z RR = 1) Pr( x Z R > 1) , = G 2 i +1 e − 2 G i +1 1 − e − G i , ∴ P R ′ i ,C i +1 = G 2 i 4 e − G i 1 − e − G i . (7.38) F rom (7.37 ) and (7.38), w e obtain P R ′ i ,L i +1 = 1 − P R ′ i ,R 0 − P R ′ i ,C i +1 , ∴ P R ′ i ,L i +1 = 1 −  1 + G i + G 2 i 4  e − G i 1 − e − G i . (7.39)  In summary , Fig ure 7.5 is a D TMC and the tra nsition probabilities are given by (7.8), ( 7 .9), (7.10 ) and (7.11), and Lemmas 7.4.1, 7.4.2, 7.4.3 and 7.4.4. W e no w analyze the D TMC in Figure 7.5. Observ e that no state can b e en tered more than once before the return to ( R, 0). Let Q X i denote the probabilit y that state ( X , i ) is entered b efore re turning to ( R, 0), where X ∈ { L, L ′ , R, R ′ , C } and i ∈ Z + . In other w ords, Q X i denotes the probability of hitt ing ( X , i ) in a CRP given that w e start from ( R, 0). No te that Q C 1 = P R 0 ,C 1 and Q L 1 = P R 0 ,L 1 . The probabilities Q X i can b e 138 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks calculated iterativ ely from the initial state ( R , 0) as follows: Q C 1 = G 2 0 4 e − G 0 , (7.40) Q L 1 = 1 −  1 + G 0 + G 2 0 4  e − G 0 , (7.41) Q C 2 = Q L 1 P L 1 ,C 2 + Q R 1 P R 1 ,C 2 , (7.42) Q L ′ 2 = Q L 1 P L 1 ,L ′ 2 , (7.43) Q L 2 = Q L 1 P L 1 ,L 2 + Q R 1 P R 1 ,L 2 , (7.44) Q L i = Q L ′ i − 1 P L ′ i − 1 ,L i + Q L i − 1 P L i − 1 ,L i + Q R i − 1 P R i − 1 ,L i + Q R ′ i − 1 P R ′ i − 1 ,R i ∀ i > 3 , (7.45) Q L ′ i = Q L ′ i − 1 P L ′ i − 1 ,L ′ i + Q L i − 1 P L i − 1 ,L ′ i ∀ i > 3 , (7.46) Q R i = Q L i P L i ,R i ∀ i > 1 , (7.47) Q R ′ i = Q L ′ i P L ′ i ,R ′ i ∀ i > 2 , (7.48) Q C i = Q L ′ i − 1 P L ′ i − 1 ,C i + Q L i P L i ,C i + Q R i − 1 P R i − 1 ,C i + Q R ′ i − 1 P R ′ i − 1 ,C i ∀ i > 3 . (7.4 9 ) Let ra ndom v ariable K denote the n um b er of slots in a CRP . Th us, K equals the n umber of states visited in the Mark ov c hain, including the initial state ( R , 0), b efore the return to ( R , 0). Th us E [ K ] = 1 + ∞ X i =1 ( Q L i + Q L ′ i + Q R i + Q R ′ i + Q C i ) , (7.50) where w e assume Q L ′ 1 = Q R ′ 1 = 0. W e ev a luate the c hange in T ( k ) from one CRP to the next, i.e., w e ev aluate the diﬀerence in left endp oin ts of initial allo cation in terv als o f successiv e CRPs. F or the assumed initial in terv al of size φ 0 , this change is at most φ 0 . Ho w ev er, if left allo cation in terv als hav e collisions or captures (e.g., R L in Figure 7 .1), then the corresp onding righ t allo catio n in terv als (e.g., R R in Figure 7.1) are returned to the waiting in terv al, and the change is less t ha n φ 0 . Let rando m v ariable F denote the fraction of φ 0 returned in this manner ov er a CRP , so that φ 0 (1 − F ) is the c hange in T ( k ). W e distinguish b et wee n tw o cases: 1. If a left allo catio n interv al of type ( L, i ) has a collision or a capture, then the corresp onding righ t allo catio n in terv al ( R, i ) is returned to the w a iting interv al. Let U L i denote the probability that ( L, i ) has a collision or a capture. Hence, 7.4. Throughpu t Analysis 139 U L i denotes the probability that ( L, i ) has tw o or more pack ets. Th us, U L i = P L i ,L i +1 + P L i ,C i +1 . Using (7.1 4) and (7.15), w e obtain U L i = 1 − (1 + G i ) e − G i 1 −  1 + G i − 1 + G 2 i − 1 4  e − G i − 1 ∀ i > 1 . (7.51) 2. If a left allo cation in terv al o f t yp e ( L ′ , i ) has a collision o r a capture, t hen the corresp onding rig h t allo catio n in t erv al ( R ′ , i ) is returned to the w aiting in t erv al. Let U L ′ i denote the pro babilit y t ha t ( L ′ , i ) has a collision o r a capture. Hence, U L ′ i denotes the probability that ( L ′ , i ) has tw o o r mo r e pack ets. Thus , U L ′ i = P L ′ i ,L i +1 + P L ′ i ,C i +1 . Using (7.2 7) and (7.28), w e obtain U L ′ i = 1 −  1 + G i  e − G i 1 − (1 + G i − 1 ) e − G i − 1 ∀ i > 2 . (7 .52) In either case, the fraction of the original allo cation in terv al returned o n suc h a collision o r a capture is 2 − i . Therefore, the exp ected v alue of F is E [ F ] = ∞ X i =1 ( Q L i U L i + Q L ′ i U L ′ i )2 − i , (7.53) where w e assume U L ′ 1 = 0. F rom (7.6), (7 .5 0) a nd (7 .5 3), w e observ e that E [ K ] and E [ F ] are functions only of the pro duct λφ 0 . Note t hat as i → ∞ , G i = 2 − i λφ 0 → 0. Using the T a ylor series expansion for e x or L’Hˆ opital’s Rule, w e can easily prov e that: 1. lim i →∞ P L ′ i ,R ′ i = 1 2 , (7.54) lim i →∞ P L ′ i ,L ′ i +1 = 1 4 , (7.55) lim i →∞ P L ′ i ,C i +1 = 1 8 , (7.56) lim i →∞ P L ′ i ,L i +1 = 1 8 , (7.57) 2. lim i →∞ P R ′ i ,R 0 = 1 , (7.58) lim i →∞ P R ′ i ,C i +1 = 0 , (7.59) lim i →∞ P R ′ i ,L i +1 = 0 , (7.60) 140 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks 3. lim i →∞ P L i ,R i = 0 , (7.61) lim i →∞ P L i ,L ′ i +1 = 1 2 , (7.62) lim i →∞ P L i ,C i +1 = 1 4 , (7.63) lim i →∞ P L i ,L i +1 = 1 4 , (7.64) 4. lim i →∞ P R i ,C i +1 = 1 2 , (7.65) lim i →∞ P R i ,L i +1 = 1 2 . (7.66) The pro ofs of these res ults are given in Appendix A. Hence, Q L i , Q L ′ i , Q R ′ i and Q C i tend to zero with increasing i as 2 − i , while Q R i tends to zero with increasing i as 4 − i . Th us, E [ K ] and E [ F ] can b e easily ev aluated num erically as functions of λφ 0 . Deﬁne the time bac klog to b e the diﬀerence b et we en the curren t t ime and the left endp oin t of the allo cation interv al, i.e., k − T ( k ). Note that all pack ets that arriv ed in the in terv al T ( k ) , k ha ve no t yet b een successfully transmitted, i.e., they are back logged. Moreo ver, w e deﬁne the drift D to b e the exp ected change in t ime bac klog, k − T ( k ), o v er a CRP , assuming an initial allo cat io n in terv al of φ 0 . Th us, D is the exp ected n umber of slots in a CRP less the exp ected c hange in T ( k ), and is give n by D = E [ K ] − φ 0 (1 − E [ F ]) . (7.67) The drift is negative if E [ K ] < φ 0 (1 − E [ F ]). Equiv alen tly , the drift is nega t iv e if λ < λφ 0 (1 − E [ F ]) E [ K ] =: ζ . ( 7 .68) The righ t hand side of (7.68), ζ , is a function of λφ 0 and is plotted in Figure 7.7. W e observ e that ζ take s its maxim um v alue at λφ 0 = 1 . 4. More precisely , ζ has a numerically ev a lua ted maxim um of 0 . 551 8 at λφ 0 = 1 . 4. If φ 0 is c hosen to b e 1 . 4 0 . 5518 = 2 . 54, then (7.68) is satisﬁed for all λ < 0 . 5518. Thu s, the exp ected time bac klog decreases whenev er it is initially la rger than φ 0 , a nd we infer that the algo r it hm is stable for λ < 0 . 5518. W e hav e therefore pro v ed the follo wing result. Prop osition 7.4.5. Th e maxi m um stable thr oughput of the PCFCFS algorithm is 0.5518 . 7.5. Numerical Results 141 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 expected number of packets in original allocation interval ζ Power Controlled FCFS Splitting Algorithm Figure 7.7: Plot of ζ ve rsus λφ 0 . 7.5 Numerical Res ults In our n umerical exp erimen ts, w e use v alues of system parameters that are commonly encoun tered in wireless netw orks [42]. W e compare the p erfo r ma nce of the following algorithms: 1. F CFS with uniform p ow er P 1 , 2. PCF CFS. F or eac h algorithm, the v alue of the initial allo cat io n interv al is chosen so a s to a chiev e maxim um stable throughput. F or FCFS, m axim um stable throughput o ccurs when its initial allo cation in terv a l, α 0 = 2 . 6 [22]. F rom Section 7.4, the maxim um throughput of PCF CFS o ccurs a t φ 0 = 2 . 54. Let n suc denote the n umber o f successful pack ets in [0 , τ ) and d i denote the departure time of i th pac ke t. 142 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks F or a given set of system parameters, we compute the follow ing p erformance metrics: Throughput = n suc τ , (7.69) Av erage Dela y = P n suc i =1 ( d i − a i ) n suc , (7.70) Av erage P o we r = P n suc i =1 P d i k = ⌈ a i ⌉ P i ( k ) n suc . (7.71) Keeping all other parameters ﬁxed, we observ e the eﬀect of increasing the arriv al rate on the throughput, av erage delay and av erage p ow er. P arameter Sym b ol V alue comm unication threshold γ c 7 dB noise p o w er sp ectral d ensit y N 0 -90 dBm path loss exp onent β 4 transmitter-receiv er distance D 10 0 m initial allo cation in terv a l of F CFS α 0 2.6 s initial allo cation in terv a l of PCF CFS φ 0 2.54 s algorithm op eration time τ 3 × 10 5 s T able 7.1: System par a meters for perfo rmance ev aluation of P CF CFS and F CFS algo- rithms. The syste m parameters for our n umerical exp erimen t s are sho wn in T able 7.1. F rom (7.4) a nd (7.5), we o btain P 1 = 0 . 50 mW and P 2 = 3 . 01 m W. W e v ary the arriv al rate λ from 0 . 40 to 0 . 60 pac ke ts/s in steps of 0 . 01. Figure 7.8 plots the throughput v ersus arriv al rate for the PCFC FS and F CFS a lgorithms. Figure 7.9 plots the av erage dela y p er successful pack et v ersus arr iv al rate for b o th the algorithms. Finally , Figure 7 .1 0 plots the av erage p ow er p er successful pac k et v ersus arriv al rat e for b oth the algo rithms. F or arriv al rates exce eding 0 . 56, the throughput of PCF CFS is less than the arriv al rate (Figure 7.8) and the a v erage dela y o f PC F CFS increases rapidly (F igure 7 .9 ), whic h leads to a substan tial increase in t he n umber of back logged pac kets and system insta- bilit y . Hence, the maxim um stable thr o ughput of PCFC FS is b et w een 0 . 55 a nd 0 . 56. Th us, Figures 7.8 and 7.9 corrob orate our result that the maxim um stable throughput of PCFC FS is 0 . 551 8 (see Section 7 .4). 7.5. Numerical Results 143 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0 0.1 0.2 0.3 0.4 0.5 packet arrival rate λ (packets/sec) throughput (packets/sec) γ c = 7.0 dB, N 0 = −90 dBm, β = 4.0, D = 100 m, P 1 = 0.5 mW, P 2 = 3.0 mW α 0 = 2.6 s, φ 0 = 2.54 s, τ = 3 × 10 5 s FCFS PCFCFS Figure 7.8: Throughput ve rsus arriv al ra te for PCF CFS and F CFS a lg orithms. 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0 0.5 1 1.5 2 2.5 x 10 7 packet arrival rate λ (packets/sec) average delay per successful packet (s) γ c = 7.0 dB, N 0 = −90 dBm, β = 4.0, D = 100 m, P 1 = 0.5 mW, P 2 = 3.0 mW α 0 = 2.6 s, φ 0 = 2.54 s, τ = 3 × 10 5 s FCFS PCFCFS Figure 7.9: Av erag e dela y ve rsus arriv al rate for PCF CFS and F CFS algo r it hms. 144 Chapter 7. P ow er Con trolled F CFS Splitting Algorithm for Wireless Net wo rks 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0 1 2 3 4 5 6 packet arrival rate λ (packets/sec) average power per successful packet (mW) γ c = 7.0 dB, N 0 = −90 dBm, β = 4.0, D = 100 m, P 1 = 0.5 mW, P 2 = 3.0 mW α 0 = 2.6 s, φ 0 = 2.54 s, τ = 3 × 10 5 s FCFS PCFCFS Figure 7.10: Average p ow er v ersus arriv al rate fo r PCF CFS and FC FS alg o rithms. F or b oth PCF CFS and FCFS, the departure r a te ( t hro ughput) equals the arriv al rate for all arriv al rates up to 0.487 (F igure 7.8). Hence, both these algorithms are stable for arriv al rates b elow 0.487. F or ar r iv al rates exceeding 0.487, the departure rate of F CFS is strictly lo w er than its arriv al rate, leading to pac k et back log and system instability . On the other hand, for PCFC FS, the departure rate still equals its a r r iv al rate f or arriv al rates b et w een 0.487 and 0.551 8. In other w ords, the PCF CFS algorithm is stable for a hig her range of arriv al rates compared to F CFS algorithm. How ev er, the PCFC FS algorithm b ecomes unstable f or arriv al rates exceeding 0.5518. The PCF CFS algorithm achie v es higher throughput and lo we r a v erage dela y than the FC FS algo r it hm, alb eit at the cost of exp ending higher av erage p o wer. F o r example, at λ = 0 . 5 5, PCF CFS achie v es 13 . 3% higher thro ughput and 96 . 7% lo wer a ve rage delay than F CFS, at the cost of 170% hig her p ow er. 7.6 Conclus ions In this chapter, w e ha v e considered r a ndom access in wireless netw orks under the phys - ical in terference mo del. By recognizing that the receiv er can successfully deco de the 7.6. Conclusions 145 strongest pack et in presence of mu ltiple transmissions, w e hav e pr o p osed PCF CFS, a splitting algorithm that mo dulates transmission p ow ers of users based on observ ed chan- nel feedbac k. PCF CFS achiev es higher throughput and substan tially lo w er dela y t ha n those of the well known F CFS algorithm with uniform tra nsmiss ion p ow er. W e sho w that the maximum stable throughput of PCF CFS is 0 .5 518. PCF CFS can b e imple- men ted in those scenarios where users are willing to tra de some p ow er for a substan tial gain in t hro ughput. Moreo v er, if users can estimate the arriv al rate of pac kets , then they can emplo y FCFS algorithm for arriv al rates up to 0.4871 and PCF CFS algor it hm for higher ar r iv al rates, th us leading to further reduction in av erage transmission pow er. Chapter 8 Flo w Con trol: An Information Theory Viewp oin t This thesis has so far explored v arious asp ects of link sc heduling in wireless net works . An equally in teresting problem is to ana lyze ﬂow con trol. W e fo rm ulate the problem o f con tro lling the rate of pac kets at the ingress o f a pa ck et net w ork (p o ssibly a wireless link) so as to maximize the m utual information b et w een a source and a destination. W e discuss v arious nuances of the pro blem and describ e related w o rk. W e then deriv e t he maximum en tropy of a pac k et lev el ﬂo w that conforms to linearly bounded traﬃc constrain ts, by taking in t o account the co v ert information presen t in the randomness of pac k et lengths. Our res ults pro vide insigh ts tow ards the design of ﬂo w con trol mec ha nisms employ ed b y an Inte rnet Service Provider (ISP). The res t of the c hapter is organized as follow s. In Section 8 .1, w e deﬁne the problem of infor mation theoretic analysis of ﬂow con t r ol in a pac ket netw ork. In Section 8.2, w e in tro duce a Generalized T o k en Buck et Regulator (GTBR) as o ur ﬂo w con tro l mec hanism. The concepts of ﬂow en tropy and informat ion utilit y ar e deﬁned in Section 8.3. W e form ulate the problem of determining the G TBR with maxim um info rmation utility in Section 8.4. In Se ction 8.5, we deriv e a necessary condition for the optimal G TBR a nd compute its parameters. W e explain the resu lts f rom a n infor ma t ion theoretic vie wp oin t in Section 8.6 and discuss the implications of our w ork in Section 8.7. 147 148 Chapter 8. Flo w Control: An Information Theory Viewp oin t 8.1 System Mo del Flow Control Mec hanism Destination Source Wireless Netw or k X Y Figure 8.1: Flow control of a source’s pack ets ov er a pack et net work. Our sys tem mo del is sho wn in Figure 8 .1, whe rein a s ource se nds pac k ets to a des- tination ov er a pack et-switc hed net work (p ossibly a wireless netw ork). The pack ets transmitted b y the source are regulated (p oliced) by a ﬂow control mec hanism at the ingress of the net w ork. W e are in terested in the pac ke t probabilit y distribution that maximizes the m utual information b et w een the source a nd the destination. In other w ords, g iven the description of the ﬂo w con t rol mec hanism a nd the sto c hastic c harac- terization of the pac ket netw ork, w e seek t he maxim um amount of information (in the Shannon sense) that c an b e transmitted from the source to the destination. The problem can b e stated as: max p X I ( X ; Y ) , (8.1) where X = random v ariable represen ting randomness in pack et con tents, lengths and timings a t the source , y = random v ariable represen ting randomness in pac k et c on ten ts, le ngths and timings a t the destination , p X = probabilit y dis tribution of X . (8.1) can b e simpliﬁed to: max p X  H ( X ) − H ( X | Y )  . (8.2) Th us, to maximiz e the infor ma t ion tra nsfer from the source to the destination, w e not only hav e to c haracterize the en trop y of the source’s pac k ets H ( X ), but also the con- ditional entrop y of the source’s pa c k ets giv en the pack ets receiv ed at the destination H ( X | Y ). W e state the following remarks ab out our problem formulation: 8.1. System Mod el 149 1. It is w ell-kno wn that, in a pac ket-switc hed net w ork, information can b e transmit- ted not only by the con t ents, but also b y the lengths and timings of pac ke ts. [20] is p erhaps the ﬁrst w o rk to recognize this fact. Info rmation transmitted b y the lengths and timings of pa c k ets is referred to as c ove rt inform ation or side infor- mation . The c hannel that is used to conv ey co v ert information is called c overt channel . Co v ert c hannels hav e b een inv estigated in [20], [21], [90]. 2. By ﬂow control, w e mean a ra te con tro l mec hanism that regulates the pack ets transmitted b y a source (subscriber) at the ingress of a net w ork. Note that w e do not consider end-to-end ﬂo w con trol mec hanisms such as T ransmission Con- trol Proto col (TCP). F or simplicit y , w e consider a ﬂow con trol mechanis m t ha t is described b y a linearly bounded service curv e 1 [91]. 3. In the pac k et net work, pack ets can b e receiv ed incorrectly at the de stination due to ﬂuctuations in the c hannel, lik e that in a wireless channe l. W e assume the existence of link lay er mec hanisms suc h as F orw ard Error Correction (FEC) whic h ensure t ha t all pack ets a r e correctly r eceiv ed at the destination. The pack et netw ork sho wn in Fig ure 8.1 only guara n tees that the conten ts and lengths of the pac k ets tra nsmitted by the source are the same as tho se at the destination. Ho wev er, t he netw ork can arbitrarily v ary the timings b et wee n pa c k ets. Equiv a len tly , the netw ork can highly distort the co vert timing info r ma t ion carr ied by the pac ke ts. T aking a cue from this, w e o nly tak e into accoun t informa t io n that is carried b y the conten ts and lengths of the pac kets . Consequen tly , the probability distribution of pac ke t con ten ts and lengths at the destination is the same a s that at the source. Hence, H ( X | Y ) = 0 and (8.2) simpliﬁes to max p X H ( X ) . (8 .3 ) In other w ords, we seek the probabilit y distribution of pack et conten ts and lengths that maximize t he source entrop y H ( X ). 1 Consider a ﬂow through a sy s tem S with input and output functions A ( t ) and B ( t ) r esp ectively . S oﬀer s to t he ﬂo w a s ervice cur ve ϑ ( t ) if and o nly if ϑ ( t ) is a wide sense incr easing function, with ϑ (0) = 0, a nd B ( t ) > inf s 6 t { A ( s ) + ϑ ( t − s ) } for all t > 0. 150 Chapter 8. Flo w Control: An Information Theory Viewp oin t T ypically , the en tity that owns a netw ork, sa y an Interne t Service Pro vider (ISP), implemen ts certain mec hanisms to ensure that pac k ets transmitted by a subscrib er are not lost in the net w ork. Ho wev er, to allo cate netw ork resources eﬃcien tly and g ua ran tee zero loss of pac k ets, the en tit y also mandates t ha t the aggregate traﬃc of a subscrib er b e upp er b ounded by an env elop e or a service curv e. F or example, the en tit y can mandate that the aggrega te traﬃc of the subscriber b e line arly b ounde d. A linearly b ounded service curv e can b e implemen ted b y a class o f regulator s kno wn as toke n buc k et r egulators. T est Destination tokens arrive p er io dically T oken Buc ket Data Buﬀer arrive bits Source r B Netw ork Pac ket Figure 8.2: T ok en buc ket regulatio n of a source’s pac kets ov er a pac ket netw ork. The system mo del that w e analyze incorp orates a T ok en Buck et Regulator ( TBR) and is sho wn in Figure 8.2. A source transmits pac k ets to a destination ov er a net w ork, where ev ery pack et consists of an inte ger num b er of bits. The pac kets transmitted by the source are regulated by a TBR or leaky buc ket regulator [92]. Intuitiv ely , the regulator collects to k ens in a buc k et of depth B , whic h ﬁlls up at a certain rate r . Each toke n corresp onds to the permission to transmit one bit in to the net w ork. The pac k ets to b e transmitted b y the source acc um ulate in its data buﬀer ov er time. If there is a pac k et of length n bits in the data buﬀer at a giv en time, then it can b e sen t into the net work only if n 6 B + r . If the pac ket is transmitted, then n tok ens are depleted fro m the tok en buc k et. A TBR can be used to smoo t hen the burst y na t ure of a subscrib er’s traﬃc. W e as- sume t hat t he net work is owned by an ISP . F rom a Qualit y of Service (QoS) p ersp ectiv e, a TBR can b e considered to b e a part o f the Service Lev el Agreemen t (SLA) b et w een a 8.2. Generalized T ok en Buck et Regulator 151 subscriber and an ISP . The SLA ma nda t es tha t the ISP should provide end-to- end loss and delay guara ntees to a subscrib er’s pack ets, provided the traﬃc pr o ﬁle of the sub- scrib er adheres to certain TBR constraints. Sp eciﬁcally , the on us of the ISP is t o ensure that ev ery pac k et of a confo r ming source succe ssfully reac hes it s destination within a certain p ermissible dela y . The Sta ndard T ok en Buck et Regulator (STBR), as deﬁned b y the In ternet Engi- neering T ask F orce (IETF) and sho wn in F ig ure 8.2, enforces linear-b oundedness on the ﬂo w. An STBR is c haracterized by its tok en incremen t rate r and buc ket depth B . W e will b e more g eneral and consider a TBR in whic h the tok en incremen t rate and buc ke t depth (maxim um burst size) can v ary from slot to slot. Suc h a TBR, w hic h w e deﬁne as a Generalized T ok en Buc k et Regulator (G TBR), can b e used to regulate V ariable Bit Rate (VBR) traﬃc 2 from a source [93]. The con tin uous-time analogue of a GTBR is the time -v arying leaky buc k et shap er [94] in which the token rate and buc k et depth parameters can change at sp eciﬁed time instan ts. The idea is to dev elop the notion of infor ma t ion utilit y of a GTBR. Sp eciﬁcally , w e derive the maxim um infor ma t ion that a GTBR-confor ming traﬃc ﬂo w can conv ey in a ﬁnite time interv al, by taking into accoun t the additio nal informa t ion presen t in the randomness of pac k et lengths. These asp ects are f urt her elucidated in subsequen t sections. 8.2 Generalize d T ok en Buc k et Regulator In this s ection, w e mathematically describe our system model and deﬁne a GTBR. W e also explain the diﬀerences betw een our sy stem mo del and those considere d in existing literature. Consider a system in whic h time is divided in to slots and a source which has to com- plete its data transmission within S slots. In our discrete-time mo del, w e will ev aluate the system at time instan t s 0 , 1 , . . . , S − 1 , S . Slot k is deﬁned to be the time in terv al [ k , k + 1), i.e., data transmission commences with slot 0 and terminates with slot ( S − 1). 2 F or exa mple, a pre-r ecorded video s tream. 152 Chapter 8. Flo w Control: An Information Theory Viewp oin t time slot k + 1 k u k r k ℓ k B k k Figure 8.3: Relat ive time instan ts of parameters deﬁned in ( 8 .4). The traﬃc from the source is regulated by a GTBR. W e deﬁne: r k = tok en incremen t for slot k , B k = buc ket depth f o r slot ( k + 1) , ℓ k = length of pac ke t (in bits) transmitted in slot k , u k = residual tok ens a t start of slot k . (8.4) r k , B k , ℓ k and u k , whose relativ e time instan ts are sho wn in Figure 8.3, are all no n- negativ e in tegers. Let r := ( r 0 , r 1 , . . . , r S − 1 ) denote the t o k en incremen t sequence a nd B := ( B 0 , B 1 , . . . , B S − 2 ) de note the buc k et depth sequ ence. The system starts with zero tok ens. So, u 0 = 0. A GTBR with the ab ov e parameters is denoted as R g ( S, r , B ). The constrain ts imposed b y R g ( S, r , B ) on the pa c k et lengths is ℓ i 6 u i + r i ∀ i = 0 , 1 , . . . , S − 1 . ( 8 .5) If ( 8 .5) is satisﬁed, then ℓ = ( ℓ 0 , ℓ 1 , . . . , ℓ S − 1 ) is a conforming pack et length vec tor and the num b er of residual toke ns will ev olve a ccording to u 0 = 0 , u i +1 = min( u i + r i − ℓ i , B i ) ∀ i = 0 , 1 , . . . , S − 2 , u S = u S − 1 + r S − 1 − ℓ S − 1 . (8.6) (8.6) is referred t o as the tok en evolution equation. Note that if r i = r ∀ i = 0 , 1 , . . . , S − 1 and B i = B ∀ i = 0 , 1 , . . . , S − 2, then the GTBR R g ( S, r , B ) degenerates t o the STBR R s ( S, r , B ). W e should p oint out that our system mo del is similar to t hat of [95]. Ho wev er, unlike [95], our tr a ﬃc regulator is a deterministic mapping of an input se quence to an output 8.3. Notion of Information Util it y 153 sequence . Also, the rate of our regulato r is deﬁned b y the av erage tok en incremen t rate and not by the p eak rate. The system mo del encompasses that of [90], wherein the authors hav e derive d t he information utility of an STBR a nd suggested a pricing viewp oint for its a pplication. Our in t erest, ho w ev er, is more theoretical. Sp eciﬁcally , w e consider an STBR as a sp ecial case of a GTBR and des crib e a framew ork for their information-theoretic comparison. The main o b jective is to in v estigate whether a G TBR can achie v e hig her ﬂo w en tropy than an STBR and explain the pro p erties of entrop y-maximizing GTBRs. These asp ects are addressed in the following sections. 8.3 Notion of Information Utility In this section, we in tro duce the concept of information utilit y of a GTBR. W e de- riv e the en t r o p y of a ﬂo w tha t is regulated b y a G TBR b y considering the informat io n presen t in the con ten ts and lengths of the pac k ets. W e form ula te the pro blem of com- puting the maxim um ﬂo w en tropy and subsequen tly describ e a tec hnique to compute the info rmation utility of the GTBR. Consider a source whic h has a large amoun t o f data to se nd and whose traﬃc is reg- ulated b y a GTBR. W e seek to maximize the informa t io n that the s ource can con vey to the destination in the give n time interv al or the entrop y presen t in the source traﬃc ﬂow in a n information-t heoretic sense. F or a giv en transmis sion interv al S , token incremen t sequence r and buck et depth sequence B , the maximum en tropy ac hiev able by an y ﬂow whic h is constrained b y the GTBR R g ( S, r , B ) is termed as the inform ation utility of the G TBR R g ( · ). The source can send informat io n to its destination via t w o channe ls: i. Ov ert c hannel: The conten ts of eac h pack et. Let ℓ i denote the length of a pac ket in bits. The v alue of eac h bit is 0 or 1 with equal probability and is indep enden t of the v alues ta ken by the pre ceding and succeeding bits. Th us, this pac k et con tributes ℓ i bits of informatio n. ii. Co v ert c hannel: W e consider the length of a pac k et a s an even t and asso ciate a probabilit y with it. Thus , side information is t r ansmitted b y the randomness in 154 Chapter 8. Flo w Control: An Information Theory Viewp oin t the pack et lengths. The join t en trop y of i. and ii . is the sum of their entropies. During an y slot k , the only method by whic h past transmissions can constrain the rest of the ﬂow is by the residual n umber o f tokens u k . So, u k captures t he state of the sy stem. The k ey observ atio n is that the fut ure en tr o p y dep ends only on the tok en buc k et lev el u k in slot k . Hence, en tro p y is a function of system state u k and is denoted b y H k ( u k ). During slot S , the source signals the termination of the current ﬂo w b y transmitting a special string of bits (ﬂag). The information tra nsmitted by this ﬁxed sequence of bits is zero. Th us H S ( u S ) = 0 ∀ u S . (8.7) F or a giv en state u k of the system, if a pack et of length ℓ k bits is t r a nsmitted with probabilit y p ℓ k ( u k ), then 1. The ov ert information transmitted is ℓ k bits, 2. As the ev en t o ccurs with probability p ℓ k ( u k ), the cov ert information transmitted is ( − log 2 p ℓ k ( u k )) bits, 3. Since ℓ k is random, u k +1 is also random (from (8.6 )). Th us, H k +1 ( u k +1 ) is also a random v ariable. Adding a ll of the a b o ve and a v eraging it ov er all confor ming pac ke t lengths, we o btain the entrop y in the curren t slot (stage) H k ( u k ) = u k + r k X ℓ k =0 p ℓ k ( u k )  ℓ k − log 2  p ℓ k ( u k )  + H k +1  min( u k + r k − ℓ k , B k )   ∀ k = 0 , . . . , S − 1 . (8.8) The equation abov e, which will be referred to as the ﬂow entr opy e quation , intuitiv ely states that the ﬂow en trop y o f the current state is giv en b y the sum of the en trop y of t he pac ke t con t ents, the en trop y of the pac ke t lengths a nd the ﬂow entrop y of p o ssible future states in the next slot. Note that (8.8) is similar to the back w ard recursion equation from dynamic pro g ramming [9 6]. Finally , the pac ket length probabilities must satisfy u k + r k X ℓ k =0 p ℓ k ( u k ) = 1 ∀ k = 0 , . . . , S − 1 . (8.9) 8.3. Notion of Information Util it y 155 Let p k ( u k ) = ( p 0 ( u k ) , p 1 ( u k ) , · · · , p u k + r k ( u k )) de note the vec tor of pack et le ngth proba- bilities for slot k with u k residual tokens . The dep endence of p ℓ k and p k on u k is assumed to b e understo o d a nd is not alw a ys s tated explicitly . So, p k = ( p 0 , p 1 , · · · , p u k + r k ). Our ob jectiv e is to determine t he sequence of probability mass functions ( p ∗ S − 1 , p ∗ S − 2 , · · · , p ∗ 0 ) whic h maximizes t he ﬂow entrop y H 0 (0) for a given GTBR R g ( S, r , B ). F rom (8.7) H ∗ S ( u S ) = 0 . (8.10) F rom (8.8) H k ( u k ) = u k + r k X ℓ k =0 p ℓ k  ℓ k − log 2 ( p ℓ k ) + H ∗ k +1  min( u k + r k − ℓ k , B k )   ∀ k = 0 , 1 , . . . S − 1 . (8.11) Giv en H ∗ k +1 ( u k +1 ) ∀ u k +1 , there exists an optimum probability v ector p ∗ k = ( p ∗ 0 , p ∗ 1 , . . . , p ∗ u k + r k ) whic h maximizes t he ﬂow en tropy H k ( u k ), i.e., H ∗ k ( u k ) = u k + r k X ℓ k =0 p ∗ ℓ k  ℓ k − log 2 ( p ∗ ℓ k ) + H ∗ k +1  min( u k + r k − ℓ k , B k )   ∀ k = 0 , 1 , . . . , S − 1 . (8.12) Th us, the problem of computing the en tire sequence of proba bility ve ctors ( p ∗ S − 1 , p ∗ S − 2 , · · · , p ∗ 0 ) decouples into a sequence of subproblems. The subproblem for slot k is: Given the function H ∗ k +1 ( u k +1 ) ∀ u k +1 , determine the pro ba bilit y v ector p k = ( p 0 , p 1 , . . . , p u k + r k ) so as to maximize u k + r k X ℓ k =0 p ℓ k  ℓ k − log 2 ( p ℓ k ) + H ∗ k +1  min( u k + r k − ℓ k , B k )   , sub ject to u k + r k X ℓ k =0 p ℓ k = 1 . (8.13) (8.13) is an equalit y-constrained optimization problem and can be solv ed using the tec hnique of Lagrange multipliers [97]. Deﬁne the Lagrangia n L ( p k , λ k ) = u k + r k X ℓ k =0 p ℓ k  ℓ k − log 2 ( p ℓ k ) + H ∗ k +1  min( u k + r k − ℓ k , B k )   + λ k  u k + r k X ℓ k =0 p ℓ k − 1  . (8.14) 156 Chapter 8. Flo w Control: An Information Theory Viewp oin t A t the optimal p o in t ( p ∗ k , λ ∗ k ), w e must ha v e ∂ L ∂ p ℓ k     ( p ∗ k , λ ∗ k ) = 0 ∀ 0 6 ℓ k 6 u k + r k , (8.15) ∂ L ∂ λ k     ( p ∗ k , λ ∗ k ) = 0 . (8 .1 6) Solving (8.16) yields u k + r k X ℓ k =0 p ∗ ℓ k ( u k ) = 1 , (8.17) whic h is (8.9) for the case of optimal probabilities. So lving (8.1 5), w e obtain p ∗ ℓ k ( u k ) = 2 ℓ k − log 2 e + H ∗ k +1 (min( u k + r k − ℓ k ,B k ))+ λ ∗ k ( u k ) ∀ 0 6 ℓ k 6 u k + r k . (8.18 ) F rom (8.17) and (8.1 8), the o ptimal L a grange multiplie r is giv en b y λ ∗ k ( u k ) = log 2 e − log 2  u k + r k X ℓ k =0 2 ℓ k + H ∗ k +1 (min( u k + r k − ℓ k ,B k ))  . (8.19) F rom (8.18) and (8.1 9), the o ptim um pa ck et length probability is giv en b y p ∗ ℓ k ( u k ) = 2 ℓ k + H ∗ k +1 (min( u k + r k − ℓ k ,B k )) P u k + r k α k =0 2 α k + H ∗ k +1 (min( u k + r k − α k ,B k )) . (8.20) F rom (8.12) and (8.2 0), w e ﬁ nally obtain H ∗ k ( u k ) = log 2  u k + r k X ℓ k =0 2 ℓ k + H ∗ k +1 (min( u k + r k − ℓ k ,B k ))  . (8 .21) (8.21) will b e referred to as the optimal ﬂow entr opy e quation . The information utility o f the GTBR R g ( S, r , B ) is deﬁned to b e H ∗ 0 (0), the maxi- m um ﬂo w en tropy . H ∗ 0 (0) is computed b y starting with H ∗ S ( u S ) = 0, and using (8.2 1) to compute the o ptimal ﬂo w en t rop y H ∗ k ( u k ) for all u k and then pro ceeding ba c kw ar d recursiv ely for k = S − 1 , S − 2 , . . . , 0. 8.4 Problem F orm u l ation Ha ving dev elop ed a metho d to compute the information utility of a GTBR in Section 8.3, we seek answ ers to the follo wing questions: 8.4. Problem F ormulatio n 157 a. Can a GTBR a c hiev e higher informa t ion utility than that of an STBR? b. If y es, what is the increase in information utility? F or the information-theoretic comparison of a GTBR R g ( S, r , B ) and an STBR R s ( S ′ , r , B ), we imp ose the follow ing conditions: 1. R g ( · ) a nd R s ( · ) must op erat e ov er the same num b er of slots, i.e., S = S ′ . (8.22) 2. The aggregate tok ens of R g ( · ) a nd R s ( · ) must b e equal, i.e., S − 1 X i =0 r i = S r . (8.23) 3. The aggregate buc ket depth o f R g ( · ) must not exceed that of R s ( · ) 3 , i.e., S − 2 X i =0 B i 6 ( S − 1) B . (8.24) 4. The buc k et depth of R s ( · ) cannot b e v ery high compared to its tok en incremen t rate. T o quan tif y this, we mandate 4 2 r 6 B 6 5 r. (8.25) 5. The tok en incremen t rate of R g ( · ) in ev ery slot mus t not exceed the buc k et depth of R s ( · ), i.e., r i 6 B . ( 8 .26) 3 Equality is present in (8.23) b ecause every additiona l token directly tra nslates to the p ermiss ion to transmit o ne more bit, leading to increase in information utility . As this is not nec e s sarily true for buck et depth, w e p ermit inequalit y in (8.24). 4 This assumption is pr a ctically justiﬁable. F or example, in [94], t he autho rs use r = 6 Mbps a nd B = 12 Mbps fo r their simulations. 158 Chapter 8. Flo w Control: An Information Theory Viewp oin t If Conditions 1, 2, 3, 4 and 5 are satisﬁed, t hen GTBR R g ( · ) a nd STBR R s ( · ) a r e said to b e c omp ar able to each other. The optimal GTBR pro blem is formally stated as: Giv en an STBR R s ( S, r , B ), determine the tok en incremen t sequence r and buck et depth sequence B of a comparable GTBR R g ( S, r , B ) so as to maximize H ∗ 0 (0) , sub ject to P S − 1 i =0 r i = S r , (8.27) P S − 2 i =0 B i 6 ( S − 1) B . (8.28) Note that we a re maximizing a real-v alued function o ver tw o ﬁnite sequences of non- negativ e in tegers. 8.5 Results In t his section, we deriv e a neces sary condition for the optimal GTBR in terms of aggregate buck et depth. W e also compute the parameters of the optimal GTBR for some represen ta tiv e cases. 8.5.1 Analytical Result Prop osition 8.5.1. F or an optimal GTBR, e quality m ust hold in ( 8 . 28), exc ept when S is smal l. I n other wor d s , i f B ∗ is the bucket depth se quenc e of an optimal GTBR, it must sa tisfy S − 2 X i =0 B ∗ i = ( S − 1) B . (8.29) Pr o of. W e pro v e b y con tradiction. D eﬁne g k ( u ) = 2 H ∗ k ( u ) . Since H ∗ k ( u ) > 0, g k ( u ) > 1. F rom (8.21) g k ( u ) = u + r k X ℓ =0 2 ℓ g k +1  min( u + r k − ℓ, B k )  . (8.30) 8.5. Results 159 g S − 1 ( u ) = 2 u + r S − 1 +1 − 1 is an increasing s equence in u . Using (8.30), we can show that g k ( u ) is an increasing se quence in u ∀ k = 0 , . . . , S − 1. Let µ i = maxim um n um b er of tok ens p o ssible in slot i . Thus µ 0 = 0 , (8.31) µ i = min( µ i − 1 + r i − 1 , B i − 1 ) ∀ i = 1 , . . . , S − 1 . (8.32) If u i 6 µ i , then we sa y that state u i is r e achable in slot i , otherwise it is unr e achable. Let R g ( S, r , B ) b e an optimal G TBR, for whic h equalit y do es not hold in (8.2 8). Then P S − 2 i =0 B i 6 ( S − 1) B − 1. Consider ano ther GTBR R ′ g ( S, r ′ , B ′ ) with r ′ = r and B ′ = ( B 0 , . . . , B k − 1 , B k + 1 , B k +1 , . . . , B S − 2 ) for some k . Let H ′ k ∗ ( u ) denote the optimal ﬂo w en tro py of R ′ g ( · ) in slot k with u residual tok ens. Deﬁne g ′ k ( u ) = 2 H ′ k ∗ ( u ) . F rom (8.21) g ′ k ( u ) = u + r k X ℓ =0 2 ℓ g ′ k +1  min( u + r ′ k − ℓ, B ′ k )  . (8.33) B ′ satisﬁes (8.28 ) . g ′ i ( u ) = g i ( u ) ∀ i = k + 1 , . . . , S and ∀ u . Since min( u + r k − ℓ, B k + 1) > min( u + r k − ℓ, B k ), it f o llo ws that g k (min( u + r k − ℓ, B k + 1)) > g k (min( u + r k − ℓ, B k )) > 1. If w e determine a reac ha ble state u suc h that g ′ k ( u ) > g k ( u ), then g ′ 0 (0) > g 0 (0), since the ﬂow entrop y in slot 0 is computed slot-b y-slot as a linear sum of future p ossible ﬂow en tropies with p o sitiv e we igh t s. Th us, the problem no w reduces to determining a slot k and a reachable state u suc h that g ′ k ( u ) > g k ( u ). One of t he following m ust hold: 1. There exists an i ∈ { 1 , . . . , S − 1 } suc h that µ i = B i − 1 < µ i − 1 + r i − 1 , or 2. There is no i suc h that µ i = B i − 1 < µ i − 1 + r i − 1 . Case 1: Consider the smallest i suc h that µ i = B i − 1 < µ i − 1 + r i − 1 . Substituting k = i − 1 in (8.30 ), w e obtain g i − 1 ( u ) = u + r i − 1 X ℓ =0 2 ℓ g i  min( B i − 1 , u + r i − 1 − ℓ )  , ∴ g i − 1 ( u ) = u + r i − 1 − B i − 1 − 1 X ℓ =0 2 ℓ g i ( B i − 1 ) + u + r i − 1 X ℓ = u + r i − 1 − B i − 1 2 ℓ g i ( u + r i − 1 − ℓ ) . (8.34) 160 Chapter 8. Flo w Control: An Information Theory Viewp oin t Substituting k = i − 1 in ( 8 .33), we obtain g ′ i − 1 ( u ) = u + r i − 1 X ℓ =0 2 ℓ g i  min( B i − 1 + 1 , u + r i − 1 − ℓ )  , ∴ g ′ i − 1 ( u ) = u + r i − 1 − B i − 1 − 1 X ℓ =0 2 ℓ g i ( B i − 1 + 1 ) + u + r i − 1 X ℓ = u + r i − 1 − B i − 1 2 ℓ g i ( u + r i − 1 − ℓ ) . (8.35) (8.34) and (8.3 5 ) hold only if u + r i − 1 − B i − 1 − 1 > 0 . (8.36) u = µ i − 1 is a state whic h is r eachable in the origina l system as w ell as in the primed system and satisﬁes (8.36 ). Since g i ( u ) is an increasing sequence in u , (8.34) and (8.35) imply g ′ i − 1 ( µ i − 1 ) > g i − 1 ( µ i − 1 ). Consequen tly , g ′ 0 (0) > g 0 (0). Case 2: If no suc h i exists, then w e m ust hav e B i > r 0 + · · · + r i ∀ i = 0 , . . . , S − 2. Adding t hese ( S − 1) inequalities and using r i 6 B (fro m (8.26)) , S − 2 X i =0 B i > ( S r − r S − 1 ) + ( S r − r S − 1 − r S − 2 ) + ( S r − r S − 1 − r S − 2 − r S − 3 ) + · · · , > ( S r − B ) + ( S r − 2 B ) + ( S r − 3 B ) + · · · , (8 .37) = S ( S − 1) r − α B . (8.38) W e cannot ha ve r i = B ∀ i (from (8.25), (8.26) and (8 .2 7)). Th us, α cannot be of the order o f S 2 . Th us, the low er b ound on P S − 2 i =0 B i giv en by (8.37) and (8.38) is a lo ose lo wer b ound. F rom (8.25), (8.28) and (8.3 8 ), P S − 2 i =0 B i gro ws as S 2 and is upp er- b ounded b y 5( S − 1) r , whic h is imp ossible (except when S is small). So, w e discard Case 2. F rom the result of Case 1, H ′ 0 ∗ (0) > H ∗ 0 (0). So, our assumption that R g ( · ) is an optimal GTBR is incorrect. Therefore, equalit y mus t hold in (8 .28) for ev ery opt ima l GTBR.  8.5.2 Numerical Results F or a giv en data tr ansmission time S , tok en incremen t sequence r and buc ke t depth se- quence B , w e determine t he optimal GTBR by exhaustiv e searc h ov er the reduced searc h space o btained from Prop osition 8.5.1. Our computation results are sh o wn in T able 8.1. H s and H ∗ g denote the informa t io n utility o f the STBR R s ( S, r , B ) and the optimal 8.5. Results 161 STBR optimal tok en incremen t optimal buck et depth information utilit y parameters sequence of GTBR sequence of GTBR H s H ∗ g p ercen tage ( S , r , B ) r ∗ B ∗ (bits) (bits) increase (4,3,6) (6 3 3 0) (6 6 6) 20.04 20.9 2 4.4 % (4,3,7) (6 4 2 0) (6 8 7) 20.08 21.1 6 5.4 % (4,3,8) (7 3 2 0) (7 9 8) (8 3 1 0) (8 9 7) 20.10 21.3 2 6.1 % (4,3,9) (8 3 1 0) (8 10 9) (9 2 1 0) (9 10 8) 20.10 21.44 6.7% (4,3,10 ) (9 3 0 0) (9 12 9) 20.10 21.5 1 7.0 % (4,3,11 ) (10 2 0 0) (10 12 11) (11 1 0 0) (11 12 10) 2 0.10 21.54 7.2% (4,3,12 ) (12 0 0 0) (12 12 12) 20.1 0 21.56 7.2% (4,3,13 ) (12 0 0 0) (13 13 13) 20.1 0 21.56 7.2% (4,4,8) (8 4 4 0) (8 8 8) 25.08 26.0 4 3.8 % (4,4,9) (8 5 3 0) (8 10 9) (9 4 3 0) (9 10 8) 25.11 26.24 4.5% (4,4,10 ) (9 5 2 0) (9 12 9) 25.13 26.3 9 5.0 % (4,4,12 ) (11 4 1 0) (11 14 11) 25.1 4 26.59 5.8% (4,4,16 ) (16 0 0 0) (16 16 16) 25.1 4 26.70 6.2% (4,5,10 ) (10 5 5 0) (10 10 10) 29.9 1 30.92 3.4% (4,5,12 ) (11 6 3 0) (11 14 11) 29.9 6 31.24 4.3% (4,6,12 ) (11 7 6 0) (11 13 12) (12 7 5 0) (12 13 11) 3 4.60 35.66 3.1% (5,3,6) (6 3 3 3 0) (6 6 6 6) 25.68 26.5 7 3.5 % (5,3,9) (8 3 3 1 0) (8 10 10 8) 25.88 27.33 5.6% (5,3,12 ) (11 2 2 0 0) (11 13 13 1 1) 25 .90 27.59 6.5% (5,3,15 ) (15 0 0 0 0) (15 15 15 1 5) 25 .90 27.64 6.7% (6,2,4) (4 2 2 2 2 0) (4 4 4 4 4) 23.00 23.77 3.4% (6,3,6) (6 3 3 3 3 0) (6 6 6 6 6) 31.33 32.23 2.9% T able 8.1 : Entrop y-maximizing GTBR for giv en data transmission time, tok en rate and buc k et depth of a comparable STBR. 162 Chapter 8. Flo w Control: An Information Theory Viewp oin t 6 7 8 9 10 11 12 13 14 20.9 21 21.1 21.2 21.3 21.4 21.5 21.6 bucket depth B of comparable STBR information utility of optimal GTBR (bits) N=4, r=3 Figure 8.4: Infor ma t io n utility of GTBR vs. buc k et de pth of comparable STBR. 3 3.5 4 4.5 5 5.5 6 6.5 7 20 25 30 35 40 45 token increment rate r of comparable STBR information utility of optimal GTBR (bits) N=4, B=15 Figure 8.5: Information utilit y of GTBR vs. t o k en increm en t rate of compar a ble STBR. 8.5. Results 163 GTBR R g ( S, r ∗ , B ∗ ) resp ectiv ely . W e also observ e the v ariation in info rmation utility of the optimal GTBR with important parameters of the comparable STBR, namely its buc k et depth B a nd tok en incremen t rate r . F or a data transmission time of 4 slots and tok en incremen t rate of 3 bits, Fig ur e 8.4 sho ws the v ariation of information utilit y of the GTBR v ersus the buc k et depth of the comparable STBR. F or a data transmission time of 4 slots and buck et depth of 15 bits, Figure 8.5 sho ws the v ariation of informa t ion utilit y of the G TBR v ersus the tok en incremen t rate of the compara ble STBR. Based on our computat io ns, w e draw the f o llo wing inferences: 1. A generalized tok en buc ke t regulator can ac hiev e higher information utility than that of a standard tok en buc ket regulator. The increase in information utility is signiﬁcan t (up to 7 .2%), esp ecially for higher v alues o f B . 2. The optimal buc k et de pth sequence B ∗ is uniform 5 or near-uniform (the standard deviation is ve ry small compared to the mean). 3. The o ptima l tok en incremen t sequence r ∗ is a decreasing sequence and is not uniform. 4. F o r a ﬁxed data t r a nsmission time S and t o k en incremen t rate r : (a) If B = 2 r , B ∗ is alwa ys uniform a nd r ∗ is uniform except fo r t he terminal v alues. (b) As B increases from 2 r to min(5 , S ) r , the v ariance of r ∗ increases rapidly with a concen tration of toke ns in ﬁrst few stages, the v ariance of B ∗ increases slo wly , while H ∗ g initially increases and then satura t es at some ﬁnal v alue. H ∗ g is an increasing and conca ve sequence 6 in B (see F igure 8.4). 5. F o r a ﬁxed data tr a nsmission time S and buc k et depth B , H ∗ g an increasing, highly linear and sligh tly concav e sequence in r (se e Figure 8.5). 5 B ∗ 0 = B ∗ 1 = · · · = B ∗ S − 2 . 6 The sequence of ﬁrst-order diﬀerences ( B ∗ 1 − B ∗ 0 , B ∗ 2 − B ∗ 1 , · · · , B ∗ S − 2 − B ∗ S − 3 ) is a decreasing and non-negative sequence. 164 Chapter 8. Flo w Control: An Information Theory Viewp oin t 8.6 Information-Theo retic In terp retation In this s ection, we provide explanations fo r e mpirical results in Section 8.5. The expla- nations are in tuitive and rely on basic results from info rmation theory . Consider a system with n states, where p i denotes t he probability of state i and P n i =1 p i = 1 . F ro m classical inf o rmation t heory , system en tropy H increases with de- creasing Kullbac k-L eibler distance b etw een the g iv en probabilit y mass function (pmf ) and the uniform pmf [98]. H is maximized only if p 1 = · · · = p n = 1 n . Also, H ∗ in- creases with n . Analogously , a GTBR can ac hiev e higher information utilit y than t ha t of an STBR b ecause the pmfs of the pac ket lengths at eac h stag e a r e more uniform and ha ve a larger supp ort. Recall that, for giv en r and B , information utilit y is computed recursiv ely usin g (8.6) and (8.21). W e a rgue that the o ptimal buc k et depth sequence B ∗ m ust b e unifo rm or near- uniform for maxim um information utility . If B ∗ is neither uniform nor near uniform, then B j = min i B i is m uc h smaller than B . This restricts the range of v alues tak en b y u j +1 and ℓ j +1 (from (8.5 ) and (8.6)). The supp ort of pa c k et length pmfs at stag e j + 1 is reduced, leading to low er ﬂow entrop y at stage j + 1 and consequen tly low er information utilit y . Th us, B ∗ m ust b e uniform o r near- uniform to maximize the minimu m supp ort of pac k et length pmfs at e ach stage . In T able 8.1, the o bserv ation tha t min i B ∗ i = B − 1 or min i B ∗ i = B throughout corrob ora tes our claim that B ∗ is near-uniform. W e argue that for maxim um information utilit y , the optimal toke n incremen t se- quence r ∗ m ust b e a decreasing seq uence, sub ject to r i 6 B i for ev ery i . If r i > B i for an y i , then a pac ket of length zero cannot b e transmitted in slot i (fro m (8.6)) a nd will ha ve zero probabilit y . This decreases the suppor t of the pac k et length pmfs in slot i and leads to low er information utility . More imp or tan tly , from (8.8), H ∗ 0 (0) = r 0 X ℓ 0 =0 p ∗ ℓ 0 (0)  ℓ 0 − log 2  p ∗ ℓ 0 (0)  + H ∗ 1  min( r 0 − ℓ 0 , B 0 )   . (8.39) The ma jor con tribution to information utilit y H ∗ 0 (0) is f r om the supp ort of the pac k et lengths [0 , r 0 ] and the pmf of the pac k et lengths ( p ∗ 0 (0)), while the con tribution from H ∗ 1 ( · ) is relatively smaller. So, to maximize H ∗ 0 (0), r 0 should b e a llow ed to tak e its maxim um possible v alue, sub ject to r 0 6 B 0 , and t he pmf of the pack et le ngths should b e close to the uniform pmf. The observ ation tha t r 0 = B 0 consisten tly in T able 8.1 8.7. Discussion 165 corrob orates this. Also, a high v alue of r 0 leads to larger supports of pack et length pmfs at in t ermediate and lat er stages. Similarly , the ﬁrst few elemen ts of r ∗ tend to take large v alues till the aggregate tok ens are exhausted. Ho w eve r, their contribution to H ∗ 0 (0) is not as pronounced and equalit y ma y not hold in r i 6 B i . Th us, r ∗ m ust b e a decre asing sequence and the ﬁrst few elemen ts o f r ∗ tend to tak e t heir maximum p ossible v a lues, sub ject to r i 6 B i , to ac hiev e uniformit y and larger supp orts of pac k et length pmfs at interme diate and later stages . This “greedy” nature of r ∗ is eviden t whe n S and r are k ept constant and B increases (Result 4b). A similar a rgumen t is applicable when S and B are k ept constant and r increases (Result 5) . The only diﬀerence is tha t a unit increase in r will necessarily increase H ∗ g b y a t least S bits ( S bits are con tributed by the pack et conten ts alone, whic h also explains the dominan t linear v ariation in Figure 8.5), while a unit increase in B will increase H ∗ g only by an amount equal to the diﬀerenc e in co ve rt information. The increase in co v ert information is p ositiv e only if the optimal tok en incremen t and buc k et depth sequences ( r ∗ and B ∗ ) result in larger supp ort and more uniformit y for the pack et length pmfs. Indeed, wh en B increases b ey ond the maxim um n umber of tok ens p ossible at an y stage (max i { µ i } ), clamping the residual n um b er of tok ens at ev ery stage b ecomes ineﬀectiv e a nd the sys tem b ehav es as if buc ket depth constrain ts w ere not impo sed a t all (Fig ure 8.4) . 8.7 Discuss ion In this c ha pter, w e ha v e studied linearly b ounded ﬂow s ov er a pac k et net work. W e con- sidered a source whose traﬃc is regulated b y a generalized tok en buc k et regulato r and whic h seeks to maximize the en tropy of the r esulting ﬂo w. Recognizing that the random- ness in pack et lengths acts as a cov ert channel in the net w ork, the source can ac hiev e maxim um en trop y by sizing its pac k ets appropriately . W e ha v e form ulated the pro blem of computing the GTBR with maxim um info rmation utilit y in terms of constrained to k en incremen t and buck et depth sequences. A GTBR can ac hiev e higher inf o rmation utilit y than that of a standard IETF tok en buc k et regula tor. Finally , w e hav e information- theoretically in terpreted the observ atio n that an en trop y-maximizing GTBR alw ay s has a near-unifo rm buck et depth sequence and a decreasing t o k en incremen t seq uence. Chapter 9 Conclusions The recen t revolution in wireless comm unications has motiv ated researc hers and engi- neers alik e to design ev er b etter wireless netw orks that deliv er high data rates to users. The joint design of ph ysical a nd MA C lay ers is the k ey to breaking the “bandwidth b ottlenec k” of wireless net w o rks, whic h has b een the primary inspiration for this thesis . This t hesis has fo cused on link sc heduling in wireless mes h net w orks b y taking into accoun t ph ysical lay er characteristics . The a ssumption made througho ut t his thesis is that a pack et is receiv ed successfully only if the SINR at the receiv er exceeds a cer- tain threshold, termed as comm unication threshold. The thesis has also discussed the complemen tary problem of ﬂow control. The ﬁ rst part of this thes is has considered link s c heduling in STDMA w ireless net- w orks. The net work is mo deled by a ﬁnite set of store-a nd-forw a r d no des that commu- nicate ov er a w ireless c hannel c haracterized by pro pagation path lo ss. W e hav e consid- ered tw o nuances of the sc heduling problem: p oin t to p oint link sc heduling wherein a transmitted pac k et is in tended for a single neigh b or only , and p oin t to m ultip oint link sc heduling w herein a transmitted pac k et is intende d for all neigh b ors in the vicinit y . Sp eciﬁcally , in Chapter 2, we hav e in tro duced the sys tem mo del of an STDMA wireless net work. W e ha v e discussed tw o prev alen t mo dels for sp ecifying the criteria for success ful pac ke t reception: the prot o col in terference mo del whic h mandates a “silence zone” aro und a receiv er and is b etter suited to represen t WLANs, and the physic al in terference mo del whic h mandates that the SINR at a r eceiv er b e no less than the comm unication threshold a nd is more appropriate to represen t mesh netw orks. W e hav e 167 168 Chapter 9. Conclusions described the equiv alence b etw een a link sc hedule and the coloring of edges of a certain graph represen tation of the net w ork, termed as commun ication gra ph. W e hav e argued that STDMA link sc heduling algorithms can be broadly categorized in to three class es: those based en tirely on a comm unication gr aph represen ta tion of the net w ork, those based on communic ation g raph and SINR threshold conditions and those based en tirely on an SINR graph represen tation of the netw ork. W e hav e review ed represe n ta tiv e researc h pap ers from eac h o f these classes. W e hav e described limitations of algorithms that are based only on the comm unicatio n gra ph. Subsequen tly , we hav e intro duced spatial reuse as a p erformance metric that corr esp onds to aggregat e net w ork throughput. Next, in Chapter 3, we ha v e critically examined Arb oricalLinkSche dule, a p oin t to p oin t link sc heduling algorithm pro p osed in [16]. While this is one of the earlier w orks o n link s c heduling with nice theoretical prop erties, it could yield a h igher sc hedule length in practice. Sp eciﬁcally , the metho dology emplo ye d b y Arb oricalLinkSc hedule is to represen t the net work b y a communic ation graph, partition the gr aph in to minim um n umber o f subgraphs and color eac h subgraph in a greedy manner. W e ha ve modiﬁed the a lg orithm to reuse colors while coloring s uccessiv e subgra phs of the comm unicatio n graph. W e hav e sho wn that the mo diﬁed a lgorithm yields lo w er sc hedule length in practice, alb eit a t a cost of sligh tly higher running time complexit y . Subsequen tly , w e ha v e prop osed the ConﬂictF reeLinkSc hedule algorithm that not only utilizes the comm unication graph, but also ve riﬁes SINR threshold conditions a t receiv ers. W e hav e demonstrated that the pro p osed algorithm achiev es higher spatial reuse than existing algorithms, ev en under fading and shado wing c hannel conditions. W e hav e argued that the running time complexity of t he pr o p osed algorithm is only marginally higher than those of existing algo r ithms. T aking a step ahead, in Chapter 4, we hav e provide d a somewhat diﬀerent p er- sp ectiv e on p oint to point link sc heduling. F or an STDMA net w or k, we recognize that in terferences b et we en pairs of links can b e embedded in to edge weigh ts and nor ma lized noise p o w ers at receiv ers of links can b e embedded in to v ertex we igh ts of a ce rtain gr a ph represen tation o f the netw ork, termed as SINR gr aph. W e ha v e then prop osed SINR- GraphLinkSc hedule, a no v el link sc heduling algorithm that is based on the SINR graph. W e hav e pro v ed t he correctness of the algorithm and sho wn that it has p o lynomial run- ning time complexit y . W e hav e demonstrated that the prop o sed algorithm achie v es high 169 spatial reuse compared to algorithms whic h utilize a comm unication graph mo del of the net work, including ConﬂictF reeLinkSc hedule algorithm. In Chapter 5, w e ha v e considered p oin t to m ultip oin t link sch eduling a nd generalized the deﬁnition of spatial reuse for this scenario. W e hav e prop osed a sc heduling algorithm based on a comm unication graph represen tat ion of the netw ork and “neighbor-av erage” SINR threshold conditions. Moreov er, w e ha v e demonstrated that t he prop osed algo- rithm ac hiev es higher spatial reuse than existing algorit hms, without an y increase in running time complexit y . Ov erall, w e hav e observ ed the tra deoﬀ b etw een accuracy of the net work represen ta- tion, spatial reuse a nd algorithm running time complexit y in our successiv e results. F or a more accurate netw ork represen tation, higher spatial reuse is ac hiev ed, but at a cos t of higher running time complexit y . F or example, since the SINR g raph represen tatio n of an STDMA netw ork is more accurate than the comm unicatio n graph represen tation, SINR- GraphLinkSc hedule ac hiev es higher spatial reuse than that of ConﬂictF reeLinkSc hedule, but at a cost of increased running time complexit y . A summary of e xisting and prop osed link sc heduling algorithms in v estigated in the ﬁrst pa r t of the thesis is prov ided in T able 9.1. Type of link Wireless net work Existing Prop osed scheduling mo del algor ithms algor ithm communication graph Arb oric alLinkSchedule [1 6] ALSReuseCo lors (Chapter 3) Poin t communication gr aph GreedyPhysical [27] to and TGSA [32] ConﬂictF r eeLinkSchedule po in t SINR co nditions (Chapter 3) SINR gr aph SINR GraphLinkSchedule (Chapter 4) Poin t communication gr aph Broadca stSc hedule [16] to communication graph m ultipo in t and MaxAverageSINRSchedule SINR conditions (Chapter 5) T able 9.1 : Link sc heduling algorithms in v estigated in Chapters 3, 4 a nd 5. The second part of this thesis has considered link sc heduling in random access wireless net works . Sp eciﬁcally , it has fo cused on random acce ss algorithms for wireless net works 170 Chapter 9. Conclusions that take into accoun t c hannel e ﬀects and SINR conditions at the receiv er. In Chapter 6, w e ha ve review ed represen tativ e researc h pap ers on suc h random access tec hniques. W e ha v e also motiv ated the use of v ariable transmission p ow er in random access wireless netw orks. Subsequen tly , in C hapter 7 , w e ha v e in ves tigated a random acc ess scenario wherein m ultiple transmitters (users) at tempt to commun icate with a single receiv er ov er a wire- less c hannel c haracterized by propagation path loss. W e hav e assumed that the receiv er is capable of p o w er based capture a nd prop osed an in terv al splitting algor it hm that v aries tra nsmission p ow ers of users based o n their arriv al times and quaternary c han- nel feedbac k. W e hav e mo deled the algorithm dynamics b y a Discrete Time Mark ov Chain and conseq uen tly sho wn that its maxim um stable throughput is 0.5518. W e hav e demonstrated that the prop osed a lgorithm has higher throughput and lo wer dela y than the FCFS interv al splitting algorithm with uniform transmission p ow er. The third and ﬁnal part of this thesis has considered info rmation-theoretic analys is of ﬂow con trol in pack et netw orks. W e hav e deﬁned the problem o f maximizing the information carried b y pac k ets from a source to a destination, sub ject to a ﬂow con trol mec hanism at the ing r ess of the netw ork. W e ha v e considered a linearly b ounded ﬂo w and fo cused on the information carried by the randomnes s in pack et conten ts and lengths. Consequen tly , we ha ve form ulated the pro blem of maximizing the en trop y of a pac k et lev el ﬂo w that is shap ed b y a g eneralized tok en buc ket regulator. W e hav e demonstrated that t he optimal regulator has a decreasing token incremen t sequence and a near-uniform buc k et depth se quence. Finally , we hav e pro vided information theoretic in terpretations for these observ ations. T o sum it up, in this thesis, w e hav e inv estigated b oth ﬁxed and random access ﬂa vors of link sch eduling problems in wireless net w orks from a ph ysical lay er viewpoint. Finally , we hav e discussed a ﬂo w control problem in pac ket net works . V a rious av en ues for further researc h hav e e merged from our in ves tigations. W e out- line some p ossible directions for future w ork. 1. It w ould b e in teresting to deriv e a ppro ximation b o unds of ConﬂictF reeLinkSc hed- ule and SINRGraphLinkSc hedule algorithms under reasonable assumptions on no de deplo ymen t and in terference regions. The assumptions and appro ximation 171 tec hniques employ ed in [27] ma y prov ide some pointers in t his direction. 2. Though distributed link sc heduling algorithms fo r STDMA wireless ne t w orks un- der the proto col interferenc e mo del ha ve b een prop osed in [31], [99], the design of distributed link sc heduling algorithms under the ph ysical interfere nce mo del remains a challenging pro blem. 3. V ario us generalizations of the PCF CFS algorithm are w or th inv estigating. F or example: (a) D esign a v ariable p o w er splitting algor it hm under the a ssumption that users are at unequal distances from the receiv er and can adjust their minim um transmission p ow ers accordingly . (b) Design a splitting algo rithm for the case when the rec eiv er is capable o f de- co ding more than o ne pack et correctly (as in wideband systems) and t he users can employ n transmission p ow er lev els, where n > 2 . (c) Analyze the throughput improv emen t in CSMA/CA based WLANs when p o w er control is emplo y ed in conjunction with binar y expo nen tial back oﬀ. The w ork done in [100] can b e a useful s tarting p oint. 4. A ch allenging task w ould be to a nalyze the exp ected dela y of the PCF CFS algo- rithm. A useful starting would b e [101], [102], which ha v e emplo y ed techniq ues to obtain upp er and low ers b ounds on the exp ected dela y of the FC FS algorithm. 5. Our results in Chapter 8 show the existence of upp er b ounds on the en tro p y of regulated ﬂow s. It would b e interes ting to construct source co des whic h come close to this b ound. F urthermore, it would b e insigh tful to dev elop a rate-distortion framew ork for a generalized token buc k et regulator, p erhaps using the t echniq ues emplo ye d in [9 5]. App endix A Pro ofs of Limi ting T ransition Probabili ti es According to L’Hˆ opital’s Rule, if lim x → c f ( x ) and lim x → c g ( x ) are b oth zero or a re b oth ±∞ and, if lim x → c f ( x ) g ( x ) has a ﬁnite v alue or if the limit is ±∞ , then lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) . (A.1) W e will emplo y L’Hˆ opital’s Rule to prov e (7.54) - ( 7 .66) In this app endix, w e will only provide the pro ofs of (7.5 4), (7.55), (7.5 6 ) and (7.57). The pro ofs o f (7.58) - (7.66) a re similar to those of (7.54) - (7.57) and ar e omitt ed. A.1 Pro of of (7.54) Pr o of. In (7.25), substitute G i = x . F ro m (7.7), G i − 1 = 2 G i = 2 x . As i → ∞ , G i = 2 − i λφ 0 → 0. Thu s, using L’Hˆ opital’s Rule successiv ely , w e obta in 173 174 App endix A. Proofs of Limiting T ransition Probabilities lim i →∞ P L ′ i ,R ′ i = lim x → 0 (1 − e − x ) xe − x 1 − (1 + 2 x ) e − 2 x , = lim x → 0 d dx ( xe − x − xe − 2 x ) d dx (1 − e − 2 x − 2 xe − 2 x ) , = lim x → 0 e − x + xe − x − e − 2 x 4 xe − 2 x , = lim x → 0 d dx ( e − x + xe − x − e − 2 x ) d dx (4 xe − 2 x ) , = lim x → 0 2 e − 2 x − xe − x 4 e − 2 x − 8 xe − 2 x , ∴ lim i →∞ P L ′ i ,R ′ i = 1 2 .  A.2 Pro of of (7.55) Pr o of. In (7 .2 6), substitute G i = x . Thus , using G i − 1 = 2 x and applying L’Hˆ opital’s Rule successiv ely , w e obta in lim i →∞ P L ′ i ,L ′ i +1 = lim x → 0 (1 − e − x − xe − x ) e − x 1 − (1 + 2 x ) e − 2 x , = lim x → 0 d dx ( e − x − e − 2 x − xe − 2 x ) d dx (1 − e − 2 x − 2 xe − 2 x ) , = lim x → 0 e − 2 x + 2 xe − 2 x − e − x 4 xe − 2 x , = lim x → 0 d dx ( e − 2 x + 2 xe − 2 x − e − x ) d dx (4 xe − 2 x ) , = lim x → 0 e − x − 4 xe − 2 x 4 e − 2 x − 8 xe − 2 x , ∴ lim i →∞ P L ′ i ,L ′ i +1 = 1 4 .  A.3 Pro of of (7.56) Pr o of. In (7 .2 7), substitute G i = x . Thus , using G i − 1 = 2 x and applying L’Hˆ opital’s Rule successiv ely , w e obta in A.4. Pro of of (7. 57) 175 lim i →∞ P L ′ i ,C i +1 = lim x → 0 x 2 4 e − x 1 − (1 + 2 x ) e − 2 x , = lim x → 0 d dx ( x 2 4 e − x ) d dx (1 − e − 2 x − 2 xe − 2 x ) , = lim x → 0 1 2 e − x − x 4 e − x 4 e − 2 x , ∴ lim i →∞ P L ′ i ,C i +1 = 1 8 .  A.4 Pro of of (7.57) Pr o of. In (7 .2 8), substitute G i = x . 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