Outage and Local Throughput and Capacity of Random Wireless Networks

1 Outage and Local Through put and Capacity of Random W ireless Networks Martin Haenggi, Senior Member , IEEE Abstract Outage probab ilities and sing le-hop throu ghpu t are two important perfo rmance m etrics that have been ev aluated for certain specific types of wireless networks. Ho wever , ther e is a lack of comprehen si ve results for larger classes of networks, and there is n o systematic appro ach that permits the convenient compariso n of the perfo rmance of ne tworks with different g eometries a nd lev els of rand omness. The uncertainty cube is introd uced to categorize the uncertainty present in a network. The three axes of the cube represent th e three main p otential sources of uncertainty in in terferen ce-limited networks: the node distribution, the chann el g ains ( fading), and the c hannel access ( set of transmitting n odes). For the perfor mance an alysis, a new parameter, the so-called sp a tial contentio n , is defined. It m easures the slope of the outage prob ability in an ALOHA network as a fu nction of the transmit pro bability p at p = 0 . Outage is defin ed as th e event that the signal- to-interfe rence ratio (SIR) is below a certain threshold in a giv en time slot. It is shown that the spatial contentio n is sufficient t o char acterize outage and throughp ut in large classes o f wireless networks, co rrespond ing to different position s on the un certainty cub e. Ex isting results ar e placed in this framework, and new o nes a re d erived. Further, interp reting the outage pr obability as the SI R distribution, the ergod ic capacity of un it- distance links is d etermined and compared to the through put achievable for fixed (yet optimized) tr ans- mission rates. I . I N T RO D U C T I O N A. Backgr ound In many large wireless n etworks, the achievable performanc e is limited by the interference. Sinc e the seminal pap er [1] the scaling beh avior of the ne twork through put or transport capacity has been the subject o f intense in vestigations, se e, e.g. , [2] and referenc es the rein. S uch “ order-of” resu lts are certainly important but do not p rovide design insight when different protocols lead to the sa me sca ling behavior . On the o ther ha nd, relatively few qua ntitative resu lts on outage and loca l (per-link) throu ghput are avail able. Nov ember 26, 2024 DRAFT 2 While su ch results provide o nly a microscopic view o f the network, we can expect con crete performanc e measures that pe rmit, for example, the fine-tuning o f chann el access probabilities or transmiss ion rates. Using a new p arameter termed spatial conten tion , we classify a nd extend the results in [3]–[6] to general s tochastic wireless ne tworks with u p to three d imensions of u ncertainty: node place ment, ch annel characteristics, and channel access. B. The uncertainty cube The le vel of uncertainty of a network is determined by its position in the uncer tainty cube . The three coordinates ( u l , u f , u a ) , 0 6 u l , u f , u a 6 1 , den ote the degree of uncertainty in the node p lacement, the c hannels, and the chann el a ccess sche me, respectively . V alues of 1 indicate comple te uncertainty (and indepen dence ), as spe cified in T able I. T he value of the u f -coordinate correspond s to the fading Node location u l = 0 Deterministic node placement u l = 1 Poisson point process Channel (fading) u f = 0 No f ading u f = 1 Rayleigh (block) f ading Channel acce ss u a = 0 TDMA u a = 1 slotted ALOHA T ABLE I S P E C I FI C A T I O N O F T H E U N C E RTA I N T Y C U B E . figure (amou nt of fading). For the Nakagami- m fading mode l, for exa mple, we may d efine u f , 1 /m . A network with ( u l , u f , u a ) = (1 , 1 , 1) has its no des distrib uted according to a Po isson point proce ss (PPP), all ch annels a re Rayleigh (block) fading, an d the chan nel ac cess sc heme is slotted ALOHA. The other extreme would be the (0 , 0 , 0) network where the node’ s pos itions are de terministic, the re is no fading, a nd there is a deterministic s cheduling mechanism. Any po int in the unit cube corresp onds to a meaningful practical ne twork—the three a xes are indepe ndent. Our objec ti ve is to cha racterize o utage and throughput for the rele vant corners of this uncertainty cube. W e focus on the interferenc e-limited case , so we d o not conside r n oise 1 . It is ass umed that all nodes 1 In the Rayleigh fading case, the outage expressions factorize into a noise part and an interference part, see (5). So, the noise term is simply a multiplicative factor to p s . Nov ember 26, 2024 DRAFT 3 transmit at the sa me power level that ca n be set to 1 since only relativ e powers matter . The p erformance results are also independen t of the absolute scale of the ne twork since only relative distanc es ma tter . C. Models, notation, and definitions Channel model. For the large-scale path loss (deterministic c hannel component), we assume the stan dard power law whe re the receiv ed power decays with r − α for a path loss exponen t α . If all cha nnels are Rayleigh, this is some times referred to as a “Ray leigh/Rayleigh” model; we denote this c ase as “1/1” fading. If e ither only the desired transmitter or the interferers are sub ject to f ading, we speak of partial fading , denoted as “1/0” or “ 0/1” fading, res pectiv ely . Network model. W e consider a single link of distance 1, with a (desired) transmitter and recei ver in a lar ge ne twork with n ∈ { 1 , 2 , . . . , ∞} other no des as pote ntial interferers. Th e signal power (deterministic channe l) or average sign al power (fading cha nnel) at the rec eiv er is 1. The distances to the interferers are denoted by r i . In the case of a PPP as the node distrib ution, the intens ity is 1 . For regular line networks, the inter -node distanc e is 1 . T ransmit pr obability p . In s lotted ALO HA, every node transmits indepen dently with p robability p in each timeslot. Hence if the nodes form a PPP o f unit intensity , the se t of transmitting n odes in eac h timeslot forms a P PP of inten sity p . The interference from no de i is I i = B i G i r − α i , wh ere B i is iid Bernoulli with parame ter p and G i = 1 (no fading) or G i is iid expone ntial with mea n 1 (Rayleigh fading). Succes s pr obability p s . A transmission is success ful if the ch annel is n ot in an outag e, i.e. , if the (instantaneous ) SIR S/I excee ds a certain thresh old θ : p s = P [SIR > θ ] , whe re I = P n i =1 I i . This is the reception probability gi ven that the de sired transmit-recei ver pair trans mits and listens , respectively . Ef fective distan ces ξ i . The ef fective distance ξ i of a node to the rece i ver is de fined as ξ i , r α i /θ . Spatial con ten tion γ an d spatial efficiency σ . For a network using AL OHA with transmit probability p , define γ , − d p s ( p ) d p    p =0 , (1) i.e. , the slope of the outa ge probability 1 − p s at p = 0 , as the s p atial conten tion me asuring how concurrent transmissions (interference) affect the succ ess probability . γ de pends on the SIR thres hold θ , the geome try of the network, a nd the path los s expo nent α . Its inv erse σ , 1 /γ is the s patial efficiency wh ich qu antifies how efficiently a network uses space as a resource. (Local) pr obabilistic throughput p T . The probabilistic throughp ut is defined to be the success proba bility multiplied b y the probab ility that the transmitter ac tually transmits (in full-duplex operation) and, in Nov ember 26, 2024 DRAFT 4 addition in half-duplex operation, the receiver ac tually listens. S o it is the uncon ditioned rec eption probability . This is the throughput achiev able with a simple ARQ scheme (with error-f ree feedbac k) [7]. For the ALOHA scheme, the half-duplex probabilistic throughput is p h T , p (1 − p ) p s and for full- dup lex it is p f T = p p s . For a TDMA line network where nodes transmit in every m -th timeslot, p T , p s /m . Thr oughp u t T . Th e throughput is defined as the produ ct of the probabilistic throughpu t and the rate of transmission, assuming that c apacity-ach ievi ng codes a re used, i.e. , T , p T log(1 + θ ) . Er godic capac ity C . Finally , interpreting 1 − p s ( θ ) as the d istrib ution of the SIR , we c alculate C , E log(1 + SIR) . I I . R E L AT E D W O R K The study of outag e and throughput performance is related to the prob lem of interference cha racteri- zation. Important results on the interference in lar ge wireless systems have be en deriv ed by [5], [8]–[11]. In [4], ou tage proba bilities for cellular networks a re ca lculated for ch annels with Rayleigh fading and shadowing while [3] determines outage proba bilities to determine the op timum transmiss ion range in a Poisson network. [12] co mbined the two approac hes to determine the optimum transmission ran ge under Rayleigh fading and shadowing. [6] provides a detailed analysis on outage probabilities and routing progress in Poisson networks with ALOHA. For our study o f (1 , 0 , 1) , (0 , 1 , 1 ) , and (1 , 1 , 1) networks, we will draw o n results from [3], [5], [6], [12], as discusse d in the rest of this s ection. A. (1 , 0 , 1) : Infi nite non-fading random networ k s with α = 4 and slotted ALOHA This case is studied in [3]. The characteristic function of the interference is determined to be 2 E e j ωI = exp  − π p Γ(1 − 2 /α ) e − j π/α ω 2 /α  (2) and, for α = 4 , = exp  − π p π / 2(1 − j ) p √ ω  . (3) 2 Note that their notation is adapted to ours. Also, a small mistake in [3, Eqn. (18)] is corrected here. Nov ember 26, 2024 DRAFT 5 B. (0 , 1 , 1) : R e gular fad ing networks with α = 2 an d slotted ALOHA In [5], the autho rs deri ve the distrib ution o f the interference power for o ne- and two-dimensional Rayleigh fading ne tworks with slotted ALOHA and α = 2 . Closed-form express ions are deri ved for infinite regular line networks with r i = i , i ∈ N . Th e Lap lace trans form of the interference is [5, E qn. (8)] L I ( s ) = sinh  π p s (1 − p )  √ 1 − p sinh  π √ s  . (4) The Lap lace transforms of the interference a re particularly co n venient for the determination of outage probabilities in Rayleigh fading. As was noted in [4], [6], [12], the suc cess probability p s can be expressed as the product of the L aplace transforms of the interferenc e and noise: p s = Z ∞ 0 e − sθ d P [ N + I 6 s ] = L I ( θ ) · L N ( θ ) . (5) In the interference -limited regime, the Laplace trans form of the interferenc e itself is suf ficient. Other- wise an exponential factor for the noise term (as suming noise with fixed variance) ne eds to be added. C. (1 , 1 , 1) : R a ndom fading ne tworks with slotted ALOHA In [6], [12], (5) was calculated for a two-dimensional rando m network with Ra yleigh fading and ALOHA. Ignoring the noise, they obtained (see [6, Eqn. (3.4)], [12, (Eqn. (A.1 1)]) p s = e − pθ 2 /α C 2 ( α ) (6) with C 2 ( α ) = 2 π Γ(2 /α )Γ(1 − 2 /α ) α = 2 π 2 α csc  2 π α  . (7) The subs cript 2 in C 2 indicates that this is a c onstant for the two-dimensional c ase. Us eful values inc lude C 2 (3) = 4 π 2 / 3 √ 3 ≈ 7 . 6 a nd C 2 (4) = π 2 / 2 ≈ 4 . 9 . C 2 (2) = ∞ , so p s → 0 as α → 2 for any θ . The spatial contention is γ = θ 2 /α C 2 ( α ) . I I I . T H E C A S E O F A S I N G L E I N T E R F E R E R T o start, we c onsider the cas e of a single interferer at eff ective distance ξ = r α /θ transmitting with probability p , which is the simples t case of a (0 , u f , 1) -network. For the fading, we allow the d esired channe l and the interferer’ s c hannel to be f ading or s tatic. If both are Ray leigh fading (this is called the 1 / 1 case), the success probability is p 1 / 1 s = P [SIR > θ ] = 1 − p 1 + ξ . (8) Nov ember 26, 2024 DRAFT 6 Case Spatial contention γ 1/1 1 1+ ξ 1/0 1 − exp( − 1 /ξ ) 0/1 exp( − ξ ) 0/0 1 ξ 6 1 T ABLE II S P A T I A L C O N T E N T I O N γ I N T H E S I N G L E - I N T E R F E R E R C A S E . For a fading des ired link an d non -fading interferers (deno ted as 1 / 0 fading), I = B r − α with B Bernoulli with parameter p and thus p 1 / 0 s = P [ S > B /ξ ] = 1 − p (1 − e − 1 /ξ ) . (9) In the case of 0 / 1 fading (non-fading desired link, fading interferer), p 0 / 1 s = P [ I < θ − 1 ] = 1 − pe − ξ . (10) For comparison, transmission succes s in the non -fading ( 0 / 0 ) case is guarantee d if ξ > 1 or the interferer does not transmit, i.e. , p 0 / 0 s = 1 − p 1 ξ 6 1 . Hence in all cases the outag e proba bility 1 − p s ( p ) is increasing linearly in p with slope γ . The values of γ a re summarized in T able II. The ordering is γ 1 / 0 > γ 1 / 1 > γ 0 / 1 , with equality only if ξ = 0 , correspon ding to an interferer a t distance 0 that caus es an outage whenever it trans mits, in which c ase a ll γ ’ s are one. T he statemen t that 1 − exp( − 1 /ξ ) > (1 + ξ ) − 1 , ξ > 0 is the sa me as log (1 + ξ ) − log ξ < 1 /ξ , which is evident from interpreting the left s ide as the integral of 1 /x from ξ to 1 + ξ and the right s ide its Riemann u pper approximation 1 /x times 1 . The ordering can also be sh own us ing Jens en’ s inequ ality: γ 1 / 0 > γ 1 / 1 since E (exp( − I θ )) > exp( − θ E I ) d ue to the co n vexity of the exponential. And γ 1 / 1 > γ 0 / 1 since E (1 − exp( − S ξ )) < 1 − exp( − ξ E S ) d ue to the c oncavity of 1 − exp x . T o summarize: Proposition 1 In the sing le-interfer er case, fading in the desired link is h a rmful whereas fading in the channel from the interferer is helpful. W e also o bserve that for small ξ , γ 1 , 1 / γ 0 , 1 , whereas for larger ξ , γ 1 , 1 ' γ 1 , 0 . So if the interferer is relati vely close, it doe s no t ma tter whe ther the desired link is f ading or not. On the other han d, if the interferer is relati vely lar ge, it hardly matters whether the interferer’ s ch annel is fading. Nov ember 26, 2024 DRAFT 7 The res ults can b e generalized to Nakaga mi- m fading in a straightforward manner . If the interferer’ s channe l is Nakag ami- m fading, while the desired link is Ra yleigh fading, we ob tain p 1 /m − 1 s = 1 − p  1 − m m ( ξ − 1 + m ) m  . (11) As a function of m , this is decrea sing for all θ > 0 , an d in the limit con ver ges to p 1 / 0 s as m → ∞ (see (9)). On the other h and, if the desired link is Nakagami- m , the succ ess prob ability is p m − 1 / 1 s = 1 − p  mξ − 1 1 + mξ − 1  m (12) which incr eas e s as m increases for fixed θ > 0 and app roaches (10) as m → ∞ . The three s ucces s probabilities p s ( θ ) are the c omplemetary c umulati ve distrib utions (cc df) of the SIR. I V . N E T W O R K S W I T H R A N D O M N O D E D I S T R I B U T I O N A. (1 , 1 , 1) : On e-dimensiona l fading random networ k s with slotted ALOHA Evaluating (5) in the one-dimens ional (and noise -free) c ase yields p s = exp  − Z ∞ 0 2 p 1 + r α /θ d r  = exp( − pθ 1 /α C 1 ( α )) , (13) where C 1 ( α ) = 2 π csc( π /α ) /α . For finite C 1 , α > 1 is need ed. C 1 (2) = π , C 1 (4) = π / √ 2 = p C 2 (4) . So the spa tial c ontention is γ = θ 1 /α C 1 ( α ) . For a gen eral d -dimensional network, we ma y co njecture that γ = θ d/α C d ( α ) , with C d = c d ( dπ /α ) csc( dπ /α ) and c d , π d/ 2 / Γ(1 + d/ 2) the volume of the d -dim. unit ball. α > d is neces sary for fi nite γ . This g eneralization is co nsistent with [13] wh ere it is shown that for Poiss on point proc esses , a ll conn ectivit y p roperties are a function of θ ′ = θ d/α and do no depend on θ in any other way . B. (1 , 1 , 1 ) : P artially fading random networks with slotted AL OHA If only the desired link is subject to fading (1/0 fading) a nd α = 4 , we can exploit (2), replacing j ω by − θ , to get p 1 / 0 s = L I ( θ ) = e − pπ Γ(1 − 2 /α ) θ 2 /α . (14) For α = 4 , p 1 / 0 s = L I ( θ ) = e − p √ θπ 3 / 2 . ( 15) So γ = π Γ(1 − 2 /α ) θ 2 /α which is larger than for the case with no fading a t all. So, as in the single- interferer case, it hurts the desired link if interferers do not fade. Nov ember 26, 2024 DRAFT 8 C. (1 , 0 , 1) : No n-fading random network s with α = 4 and slotted ALOHA From [3, Eqn. (21)], I − 1 has the cdf F I − 1 ( θ ) = P [1 /I < θ ] = 1 − p s = erf π 3 / 2 p √ θ 2 ! , (16) which is the outage probability for n on-fading c hannels for a trans mitter -receiv er distance 1 . For the spatial contention we obtain γ = π √ θ , a nd it can be verified ( e.g. , by comp aring T a ylor expansions ) tha t 1 − γ p < p s ( p ) < exp( − γ p ) holds. D. (1 , 1 , 1) : Fu lly ran d om ne tworks with e xpon ential path loss In [14] the autho rs mad e a cas e for exponential path loss laws. T o determine their e f fect on the spatial contention, c onsider the expo nential path loss law exp( − δ r ) instead o f r − α . Following the deri vation in [6], we find p s = exp  − 2 π p Z ∞ 0 r 1 + exp( δ r ) /θ d r  = exp  − 2 π p − dilog( θ + 1) δ 2  , (17) where dilog is the dilogarithm function de fined a s dilog( x ) = R x 1 log t/ (1 − t )d t . So γ = − 2 π dilog ( θ + 1) /δ 2 . The (negati ve) dilog function is bounde d by − dilog ( x ) < log( x ) 2 / 2 + π 2 / 6 [15], so γ < π δ 2  log 2 (1 + θ ) + π 2 3  , (18) indicating tha t the spatial contention grows more s lo wly (wit h log θ instead of θ 2 /α ) for lar ge θ than for the power path loss law . In the exponen tial case, finitenes s of the integral is guaranteed for any δ > 0 , in con trast to the power law where α needs to exceed the n umber of network d imensions. Practical pa th loss laws ma y include b oth a n exponential a nd a power law p art, e.g. , r − 2 exp( − δ r ) . Th ere are, howe ver , no closed-form solutions for such p ath loss la ws, and one has to resort to numerical studies. V . N E T W O R K S W I T H D E T E R M I N I S T I C N O D E P L A C E M E N T In this section, we a ssume that n interferers are place d at fixed distanc es r i from the intended receiver . A. (0 , 1 , 1) : F ading network s with slotted ALOHA In this c ase, p s = P [ S > θ I ] for I = P n i =1 S i r − α i and S i iid exponential with mean 1. For general r i and α , we obtain from p s = E [ e − θ I ] = L I ( θ ) p s = n Y i =1  1 − p 1 + ξ i  (19) Nov ember 26, 2024 DRAFT 9 where ξ i = r α i /θ is the ef fectiv e distance. W e find for the s patial contention γ , − d p s ( p ) d p    p =0 = n X i =1 1 1 + ξ i . (20) Since d p s / d p is decreasing, p s ( p ) is co n vex, so 1 − pγ is a lo wer bound on the succes s p robability . On the other hand, e − pγ is an upper bound, since log p s = n X i =1 log  1 − p 1 + ξ i  / n X i =1 − p 1 + ξ i . (21) The upper bound is tight for s mall p o r ξ i lar ge for mo st i , i.e. , if most interferers are far . B. (0 , 1 , 1) : Infi nite r e gular line ne tworks with fading and ALOHA Here we specialize to the case of regular one-dimension al (line) networks, where r i = i , i ∈ N . For α = 2 , we obtain from (4) (or by direct calculation of (20 )) γ = 1 2  π √ θ co th ( π √ θ ) − 1  . (22) Since x coth x − 1 < x < x coth x , this is bounde d by ( π √ θ − 1) / 2 < γ < π √ θ / 2 , with the lower bound being very tight as soon as θ > 1 . Ag ain the success p robability is bou nded b y 1 − γ p < p s ( p ) < exp( − pγ ) , and both the se bounds become tight as θ → 0 , and the upp er bound become s tight a lso as θ → ∞ . For α = 4 , we first establish the analogous result to (4). Proposition 2 F or on e-sided infinite re gular line ne tworks ( r i = i , i ∈ N ) with slotted AL OH A and α = 4 , p s = cosh 2  y (1 − p ) 1 / 4  − cos 2  y (1 − p ) 1 / 4  √ 1 − p (cosh 2 y − cos 2 y ) (23) where y , π θ 1 / 4 / √ 2 . Pr oof: Re write (19) as p s = Q n i =1 (1 + (1 − p ) θ /i 4 ) Q n i =1 (1 + θ /i 4 ) . (24) The factorization o f both nume rator and denominator according to (1 − z 4 /i 4 ) = (1 − z 2 /i 2 )(1 + z 2 /i 2 ) permits the use of Euler’ s produ ct formula sin( π z ) ≡ π z Q ∞ i =1 (1 − z 2 /i 2 ) with z = √ ± j ((1 − p ) θ ) 1 / 4 (numerator) an d z = √ ± j θ 1 / 4 (denominator). The two resu lting express ions are comp lex co njugates, and | sin( √ j x ) | 2 = cosh 2 ( x/ √ 2) − cos 2 ( x/ √ 2) . Nov ember 26, 2024 DRAFT 10 The spatial contention is γ = 1 8 ( y − 1) e 2 y + 4 cos 2 y + 4 y cos y sin y − 2 − ( y + 1) e − 2 y cosh 2 y − cos 2 y . (25) For y ' 2 , the e 2 y (numerator) and cosh 2 y (denominator) terms dominate, so γ ≈ ( y − 1) / 2 for y > 2 . In terms of θ , this implies that γ ≈ π θ 1 / 4 / (2 √ 2) − 1 / 2 , (26) which is quite accurate as soon as θ > 1 . The corresp onding ap proximation p s ≈ e − p ( π θ 1 / 4 / (2 √ 2) − 1 / 2 ) . (27) can b e de ri ved from (23) noting that for y not too small and p not too close to 1 , the cosh terms dominate the cos terms and cosh 2 ( x ) ≈ e 2 x / 4 , 1 − (1 − p ) 1 / 4 ≈ p/ 4 , and (1 − p ) − 1 / 2 ≈ e p/ 2 . For general α , the T a ylor expansion of (20) yields γ ( θ ) = − ∞ X i =1 ( − 1) i ζ ( αi ) θ i . (28) In particular , γ < ζ ( α ) θ . Since ζ ( x ) ' 1 for x > 3 , the series conv erges quickly for θ < 1 / 2 . For θ > 1 , it is unsuitable. C. (0 , 1 , 1) : P artially fading r e gular networks If only the desired link is subject to fading, the succe ss proba bility is g i ven by p s = e − pθ P n i =1 r − α i , (29) thus γ = P n i =1 1 /ξ i . Compared with (20), 1 + ξ is replaced by ξ . S o the spatial contention is la r ger than in the case of full f ading , i.e. , fading in the interferer’ s chann els h elps, as in the s ingle-interferer case. For regular line networks ξ i = i α /θ , so γ = θ ζ ( α ) and p s = e − pθ ζ ( α ) . D. (0 , 1 , 0) : Regular line network s with fading an d TDMA If in a T DMA s cheme, only every m -th node transmits, the relative distances of the interferers are increased b y a factor of m . Fig. 1 s hows a two-sided regular line network with m = 2 . Sinc e ( mr ) α /θ = r α / ( θ m − α ) , having every m -th node transmit is equ i valent to reduc ing the thresh old θ by a factor m α and setting p = 1 . Proposition 3 The succe s s pr obab ility for one-sided infinite re gular line networks with Ra yleigh fading and m -phase TDMA is: F or α = 2 : p s = y sinh y , where y , π √ θ m , (30) Nov ember 26, 2024 DRAFT 11 ... −4 −3 −2 −1 1 2 3 4 ... T R Fig. 1. T wo-sided regular line network with T DMA with m = 2 , i.e. , ev ery second node tr ansmits. The filled circles indicate the transmitters. The tr ansmitter denoted by T is the i ntended transmitter , the others are interferers. T he receiv er at t he origin, denoted by R , is the intended receiv er . In t he one-sided case, the nodes at positions x < 0 do not exist. and for α = 4 : p s = 2 y 2 cosh 2 y − cos 2 y , where y , π θ 1 / 4 √ 2 m . (31) Pr oof: Ap ply L ’H ˆ opital’ s rule for p = 1 in (4) and (23) (for α = 2 , 4 , respectively) and replace θ by θ m − α . The follo wing propos ition es tablishes sharp bounds for a rbitrary α . Proposition 4 The su c cess probability for one -side d infinite r e gular line networks , Rayle igh fading, and m -phase TDMA is bounded by e − ζ ( α ) θ/m α / p s / 1 1 + ζ ( α ) θ m α . (32) A tighter upper bound is p s / 1 1 + ζ ( α ) θ m α + ( ζ ( α ) − 1) θ 2 m 2 α . (33) Pr oof: Up per b ound: W e only nee d to proof the tighter b ound. Let θ ′ , θ /m α . The expansion o f the prod uct (19), p − 1 s = Q ∞ i =1 1 + θ ′ /i α , ordere d according to powers o f θ ′ , has only positi ve terms an d starts with 1 + θ ′ ζ ( α ) + θ ′ 2 ( ζ ( α ) − 1) . There a re more terms with θ ′ 2 , but their coe f ficients are relativ ely small, so the bound is tight. T he lo wer bound is a special case of (21 ). Note that all b ounds approa ch the exact p s as θ /m α decreas es. Interestingly , for α = 2 , 4 , the upper bound (32) co rresponds exactly to the expressions ob tained wh en the den ominators in (30) and (31) are replaced by their T aylor expan sions of order 2 α . Higher -order T aylor expa nsions, howev er , deviate from the tighter bound (33). The s uccess probabilities p ′ s for two-sided regular ne tworks a re obtained s imply by s quaring the probabilities for the one-side d networks, i.e. , p ′ s = p 2 s . This follows from the fact that the distances are related as follo ws: r ′ i = r ⌈ i/ 2 ⌉ . Nov ember 26, 2024 DRAFT 12 E. Spatial contention in TDMA networks In order to us e the spatial contention framewor k for TDMA networks, Let ˜ p , 1 /m be the fraction of time a node transmits. Now d p s / d ˜ p | ˜ p =0 = 0 s ince p s depend s on m α rather than m itself. So for TDMA, we define γ , − d p s d( ˜ p α )    ˜ p =0 (34) and find γ = ζ ( α ) θ , which is identical to the spatial conten tion of the ALOHA line network with non-fading interferers. T able III summarizes the results on the s patial contention established in this s ection. Uncertainty Spatial contention γ Eqn. #dim. Remark (1 , 1 , 1) 2 π θ 1 /α csc( π /α ) /α (13) 1 T wo-sided n etwork 2 π 2 θ 2 /α csc(2 π /α ) /α (6) 2 Fr om [6]. π 2 √ θ / 2 (6) 2 Special ca se for α = 4 π Γ(1 − 2 /α ) θ 2 /α (14) 2 Non-fading interferers π 3 / 2 √ θ (15) 2 For α = 4 and non-fading interferers (1 , 0 , 1) π √ θ (16) 2 No f ading , for α = 4 (0 , 1 , 1) P n i =1 1 / (1 + ξ i ) (20) d Determi nistic nod e plac ement, n nodes π √ θ co th ( π √ θ ) / 2 − 1 / 2 (22) 1 One-sided regular network, α = 2 ≈ π θ 1 / 4 / (2 √ 2) − 1 / 2 (26) 1 One-sided regular network, α = 4 P n i =1 1 /ξ i (29) d Det. node place ment, no n-fading interf. θ ζ ( α ) (29) 1 Regular network, non-fading interferers (0 , 1 , 0) p s ' e − ζ ( α ) θ/m α (32) 1 TDMA in o ne-sided regular n etworks. T ABLE III S P A T I A L C O N T E N T I O N PA R A M E T E R S F O R D I FF E R E N T T Y P E S O F S L OT T E D A L O H A N E T W O R K S . F O R C O M PA R I S O N , T H E T D M A C A S E I S A D D E D . “ R E G U L A R N E T W O R K ” R E F E R S T O A N I N FI N I T E L I N E N E T W O R K W I T H U N I T N O D E S PA C I N G . Nov ember 26, 2024 DRAFT 13 V I . T H RO U G H P U T A N D C A P A C I T Y A. ( u l , u f , 1) : Networks with s lotted ALOHA For networks with slotted ALOHA, define the pr obabilistic throughput as full-duplex: p f T , p p s ( p ) ; half-duplex: p h T , p (1 − p ) p s ( p ) . (35) This is the unconditional prob ability of s uccess , taking into acc ount the probab ilities that the des ired transmitters actually transmits and, in the ha lf-duplex ca se, the de sired recei ver actually listens. Proposition 5 (Maximum probabilistic thr oughp ut in ALOHA networks with fading) Conside r a net- work with ALOHA and Rayleigh fading with spatial co n tention γ such that p s = e − pγ . The n in the full-duple x ca se p opt = 1 /γ ; p f T max = 1 eγ (36) and in the half-duplex cas e p opt = 1 γ + 1 2  1 − r 1 + 4 γ 2  . (37) and p h T max ' 1 + γ (2 + γ ) 2 exp  − γ 2 + γ  , (38) Pr oof: Full-duplex: p opt = 1 /γ maximizes p exp( − pγ ) . Half-duplex: Maximizing log p h T ( p ) yields the qua dratic equation p 2 opt − p opt (1 + 2 σ ) + σ = 0 wh ose solution is (37). Any approximation of p opt yields a lower bound on p h T . Sinc e p opt (0) = 1 / 2 , and p opt = Θ( γ − 1 ) for γ → ∞ , a simple yet ac curate choice is p opt ' 1 / (2 + γ ) which results in the bound in the propos ition. Numerical calculations show that the lower bound (38) is within 1 . 4% of the true maximum over the whole range γ ∈ R + . B. (0 , 1 , 0) : T wo-sided re gular line network s with TDMA Here we cons ider a two-sided infin ite regular line network with m -pha se TDMA (see Fig. 1). T o maximize the throughput p T , p s /m , w e use the b ounds (32 ) for p s . Since the ne twork is now two- sided, the expressions need t o be squared . Le t ˜ m opt ∈ R an d ˆ m opt ∈ N be estimates for the tr ue m opt ∈ N . W e find  θ ζ ( α )(2 α − 1)  1 /α < ˜ m opt <  θ ζ ( α )2 α )  1 /α , (39) Nov ember 26, 2024 DRAFT 14 0 5 10 15 20 0 5 10 15 20 25 Θ [dB] m opt α =2 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Θ [dB] p s α =2 Fig. 2. Left: Optimum TDMA parameter m as a function of θ [dB] for α = 2 . The dashed lines sho w the bounds (39), the circles indicate t he true optimum m opt , the crosses the estimate ˆ m opt in (40). Right: p s for the optimum m as a function of θ [dB] for α = 2 . T he dashed lines show the approximations (41), the solid l ine the actual value obtained numerically . where the lower and up per bounds stem from maximizing the upper and lower bound s in (32), respectiv ely . The factor 2 in 2 α ind icates that the network is two-sided. Rounding the average of the two b ounds to the nearest integer yields a good estimate for m opt : ˆ m opt = ⌈  θ ζ ( α )(2 α − 1 / 2 )  1 /α ⌋ (40) Fig. 2 (left) shows the bo unds (39), ˆ m opt , and the true m opt (found numerically) for α = 2 as a func tion of θ . For most values of θ , ˆ m opt = m opt . T he resu lting difference in the maximum a chiev able throug hput p T max is negligibly small. W e can ob tain estimates on the su ccess probability p s by inserting (39) into (32):  1 − 1 2 α  2 ≈ p s ≈ e − 1 /α . (41) In Fig. 2 (right) , the actual p s ( θ ) is shown with the two approximations f or α = 2 . Since m opt is increasing with θ , the relative error ˜ m opt /m opt → 0 , so we expect lim θ → ∞ p s ( θ ) to lie between the approx imations (41). C. Rate optimization So far we have assumed that the SIR threshold θ is fixed and giv en. Here we add ress the problem of finding the o ptimum rate of transmission for ne tworks where γ ∝ θ d/α , where d = 1 , 2 indicates the Nov ember 26, 2024 DRAFT 15 number of network dimensions. W e defin e the thr oughput as the produc t of the proba bilistic throughput p T and the (normalized) rate of transmission log (1 + θ ) (in nats/s/Hz). As before, we distinguis h the c ases o f half-duplex and full-duplex ope ration, i.e. , we max imize p f T ( θ ) log (1 + θ ) (full-duplex) or p h T ( θ ) log (1 + θ ) (half-duplex), resp ectiv ely . Proposition 6 (Optimum SI R threshold f or full-duplex o peration) The thr oughput T = p exp( − pγ ) log(1 + θ ) is max imized a t the SIR thr eshold θ opt = exp  W  − α d e − α/d  + α d  − 1 , (42) where W is the principa l branch of the Lamber t W function and d = 1 , 2 is the number of network dimensions. Pr oof: Gi ven γ , the optimum p is 1 /γ . W ith γ = cθ d/α , we need to maximize T ( α, θ ) = 1 ecθ d/α log(1 + θ ) , (43) where d = 1 , 2 is the number o f dimensions. Solving ∂ T /∂ θ = 0 yields (42). Remark. θ opt in the two-dimensional ca se for a path los s exponen t α equa ls θ opt in the one -dimensional case for a path loss exponen t α/ 2 . In the two-dimensional case, the optimum thres hold is smaller than one for α < 4 log 2 ≈ 2 . 77 . The optimum (normalized) transmission rate (in nats/s/Hz) is R opt ( α ) = log(1 + θ opt ) = W  − α d e − α/d  + α d , d = 1 , 2 . (44) R opt ( α ) is c oncave for α > d , and the de ri vati ve a t α = d is 2 for d = 1 and 1 for d = 2 . So we have R opt ( α ) < α − 2 for d = 2 and R opt ( α ) < 2( α − 1) for d = 1 . In the half-duplex ca se, clos ed-form solutions are no t av ailable. Th e results of the numerical throug hput maximization a re shown in Fig. 3 , together with the res ults for the full- duplex case. As can b e s een, the maximum throug hput sca les almost linearly with α − d . The optimum transmit proba bilities do no t depe nd strongly on α and are arou nd 0 . 105 for full-duplex operation and 0 . 08 for h alf-duplex opera tion. The achiev able throughput for full- dup lex operation is quite exactly 10% higher than for half-duplex operation, over the entire practical range of α . D. (1 , 1 , 1) : Ergodic cap a city Based on our definitions, the e r godic c apacity can be generally exp ressed as C = E log(1 + SIR) = Z ∞ 0 − log(1 + θ )d p s , (45) Nov ember 26, 2024 DRAFT 16 2.5 3 3.5 4 4.5 5 0 2 4 6 8 10 12 α θ opt Half−duplex Full−duplex 2.5 3 3.5 4 4.5 5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 α Throughput T Half−duplex Full−duplex Fig. 3. Left: Optimum threshold θ opt for full- and half-duplex operation as a function of α for a two-dimensional network. Right: Maximum throughp ut. where p s ( θ ) is the ccdf of the S IR. Proposition 7 (Ergodic capac ity for (1 , 1 , 1) networks ) Let C be the er godic c a pacity of a link in a two-dimensional (1 , 1 , 1) network with transmit probabilit y p . F or α = 4 , C = 2 ℜ{ q } cos ( c p ) − 2 ℑ{ q } sin ( c p ) , q , Ei(1 , j c p ) , (46) where c p = pC 2 ( α ) and Ei(1 , z ) = R ∞ 1 exp( − xz ) x − 1 d x is the e xpo nential inte gral. F or general α > 2 , C is lower bounde d a s C > log 2 ·  c − α/ 2 p γ (1 + α/ 2 , c p ) +  α 4 − 1  exp( − √ 2 c p ) + exp( − c p )  + α 2 Ei( √ 2 c p ) , (47) where γ ( a, x ) = R x 0 t a − 1 exp( − t )d t is the lo wer incomplete gamma function. The one-dimensional network with path loss exponent α (and c p = pC 1 ( α ) ) h a s the same ca pacity as the two-dimensional network with path los s e xpo nent 2 α . Pr oof: Let c p , pγ θ − 2 /α = pC 2 ( α ) . W e have C = 2 c p α Z ∞ 0 log(1 + θ ) θ 2 /α − 1 exp( − c p θ 2 /α )d θ (48) = c p Z ∞ 0 log  1 + t α/ 2  exp( − c p t )d t . (49) So, the 2 /α -th moment of the SIR is expone ntially distrib uted with me an 1 /c p . As a co nseque nce, the capac ity of the ALOHA channe l is the cap acity of a Rayleigh fading cha nnel with mean S IR c − 1 p with Nov ember 26, 2024 DRAFT 17 an “SIR bo ost” exponent of α/ 2 > 1 . Note that s ince a significa nt part of the proba bility mass may b e located in the interval 0 6 θ < 1 , this does not mean that the capa city is larger than for the s tandard Rayleigh case. This is only true if the SIR is high on av erage . For g eneral p an d α , the integral do es not have a c losed-form expression . For α = 4 , d irect calculation of (49) yields C = exp( − j c p ) Ei(1 , j c p ) + exp( − j c p ) Ei(1 , − j c p ) , (50 ) which equals (46). T o find an analytical lower bound, re write (49) as (by substituting t ← t − 1 ) C = c p Z ∞ 0 log(1 + t − α/ 2 ) exp( − c p /t ) t 2 d t (51) and lo wer bound log(1 + t − α/ 2 ) by L ( t ) gi ven b y L ( t ) =              − α 2 log t for 0 6 t < √ 2 / 2 log 2 for √ 2 / 2 6 t < 1 log(2) t − α/ 2 for 1 6 t . (52) This yields the lo wer bound (47). For rational values o f α , pseudo -closed-form expressions are available us ing the Meijer G func tion. Fig. 4 displays the cap acities a nd lower bo unds for α = 2 . 5 , 3 , 4 , 5 . For sma ll c p (high SIR on average), a simpler bound is C > Z ∞ 1 − log( θ )d p s = α 2 Ei(1 , pC ( α )) , (53) T o obtain the sp atial cap acity , the e r god ic capac ity needs to be multiplied by the prob ability (de nsity) of trans mission. It is expe cted that there exists an optimum p maximizing the product pC in the cas e of full-duplex o peration or p (1 − p ) C in the cas e of half-duplex operation. The c orresponding curves a re shown in Fig. 5. Interestingly , in the full-duplex cas e, the optimum p is dec reasing with increa sing α . In the half-duplex ca se, p opt ≈ 1 / 9 quite exactly — independe nt of α . E. TDMA line networks Proposition 8 (Ergodic capac ity bounds f or TDMA line networks) F or α = 2 , 2 log  2 m π  < C < log  1 + 7 ζ (3) π 2 m 2  (54) and E √ SIR = π 4 m ; E S IR = 7 ζ (3) π 2 m 2 . (55) Nov ember 26, 2024 DRAFT 18 0 0.2 0.4 0.6 0.8 1 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 p log C α =2.5 α =5 Fig. 4. Ergodic capacity for a two-dimensional fading network with AL OHA for α = 2 . 5 , 3 , 4 , 5 as a function of p . The solid lines are t he actual capacities (49), the dashed lines t he l o wer bounds (47). 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Fullduplex Operation p pC α =2.5 α =5 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Halfduplex Operation p p(1−p)C α =2.5 α =5 Fig. 5. Spatial capacities for α = 2 . 5 , 3 , 4 , 5 as a function of p . Left plot: Full-duplex operation. Ri ght plot: Half-duple x operation. T he star marks the optimum p . Nov ember 26, 2024 DRAFT 19 F or general α > 1 , C > e ζ ( α ) /m α Ei(1 , ζ ( α ) /m α ) (56) and E SIR > 1 ζ ( α ) m α . (57) Pr oof: α = 2 : Using (45) and (30) and substituting t ← π √ θ /m yields C = Z ∞ 0 log 1 +  mt π  2 ! t cosh t − sinh t sinh 2 t d t (58) Replacing log (1 + x ) by log x results in the lower bo und which ge ts tighter a s m increase s. It a lso follows that π √ SIR /m is distrib uted as P ( π √ SIR /m < t ) = e 2 t − 2 te t − 1 e 2 t − 1 (59) from whic h the mome nts of the S IR follow . The upper bound in (54) stems from Jense n’ s inequa lity . General α : Use the lo wer bound (32) on p s and calculate directly . Fig. 6 shows the ergodic c apacity for the TDMA line ne twork for α = 2 , together with the lower bound s (54) and (56 ) and the upp er boun d from (54). As can be se en, the lower bo und s pecific to α = 2 gets tighter for larger m . U sing the lower bound (57) on the SIR together with Je nsen’ s inequ ality would result in a good approximation C ≈ log(1 + m α /ζ ( α )) . From the slope of C ( m ) it can be see n that the optimum spatial reuse factor m = 2 maximizes the spatial capac ity C /m for α = 2 . For α = 4 , m = 3 yields a slightly h igher C /m . Th is is in ag reement with the observation made in Fig. 5 (left) that in ALOHA p opt slightly decrease s as α increases . V I I . D I S C U S S I O N A N D C O N C L U D I N G R E M A R K S W e have introduced the unc ertainty c ube to class ify wireless networks acco rding to their underlying stochas tic proce sses. For lar ge cla sses of n etworks, the outage probability P (SIR < θ ) of a unit-distance link is determined by the spatial c ontention γ . Summa rizing the outage results: • For (1 , u f , 1) networks (PPP networks with ALOHA), γ ∝ θ d/α . W ith Rayleigh fading, p s = exp( − pγ ) , otherwise p s 6 exp( − pγ ) . • For regular line ne tworks with ALOHA (a class of (0 , 1 , 1) networks), γ ≈ cθ d/α − 1 / 2 . So, the regularity is refl ected in the shift in γ by 1 / 2 , i.e. , γ be comes a f fine in θ d/α rather than linear . • Quite generally , with the exception of deterministic networks without fading interferers, γ is a function of θ only through θ d/α (see T able III ). Nov ember 26, 2024 DRAFT 20 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 3.5 4 m C Capacity Lower bound 1 Lower bound 2 Upper bound Fig. 6. Ergodic capacity for TDMA line netwo rk for α = 2 as a function of the reuse parameter m . The solid line is the actual capacity ( 49), lower bound 1 and the upper bound are from (54), and lower bound 2 is (56). • For regular line n etworks with m -ph ase TDMA (a class o f (0 , 1 , 0) n etworks), p s ≈ exp( − ˜ p α ζ ( α ) θ ) , where ˜ p = 1 /m . So the incre ased ef ficien cy of TDMA sched uling in line n etworks is refle cted in the exponent α of ˜ p . The follo wing interpretations of γ = σ − 1 demonstrate the fundamental nature of this parameter: • γ determines how fast p s ( p ) decays a s p increases from 0: ∂ p s /∂ p | p =0 = − γ . • For any ALOHA network with Ra yleigh fading, the re exists a unique parameter γ such that 1 − pγ 6 p s 6 exp( − pγ ) . Th is parameter is wha t we call the sp atial contention. From a ll the networks studied, we conjecture that this is true for ge neral ALOHA networks. • In a PPP network, the suc cess probability equals the proba bility that a disk of area γ around the receiv er is free from con current transmitters. So an e q uivalent dis k model could be devised where the interference radius is p γ /π . For a trans mission over distance R , the disk rad ius would scale to R p γ /π . • In full-duplex operation, the probabilistic throughput is p f T = pe − pγ , and p opt = m in { σ , 1 } . So the spatial efficiency equ als the optimum tr ans mit probability in ALOHA, and p f T = σ /e . The throug hput is proportional to σ . • The transmission c apacity , introduced in [16], is defined as the maximum s patial density of concurrent Nov ember 26, 2024 DRAFT 21 transmission allowed gi ven an outage cons traint ǫ . In our framework, for small ǫ , p s = 1 − pγ = 1 − ǫ , so p = ǫσ . So the transmission capacity is proportional to the s patial efficiency . • Even if the cha nnel access protoco l used is different from ALOHA, the spatial contention offers a single-parameter characterization of the network’ s capa bilities to use spa ce. Using the express ions for the succ ess probab ilities p s , we have determined the optimum ALOHA trans- mission prob abilities p and the optimum T DMA p arameter m that maximize the probabilistic through put. Further , p s ( θ ) e nables determining b oth the o ptimum θ (rate of transmission) and the ergodic capac ity . For the case s where γ ∝ θ d/α , S IR d/α is expo nentially distributed. The optimum rates and the throughpu t are roughly linea r in α − d , the spatial ca pacity i s abou t 2 . 5 × larger than t he throughp ut, an d the penalty for half-duplex op eration is 10-20%. Th e o ptimum transmit probab ility p opt is around 1/9 for b oth o ptimum throughput (Fig. 3, right) an d max imum spa tial c apacity (Fig. 4, right). T he mean distance to the nea rest interferer is 1 / (2 √ p opt ) = 3 / 2 , s o for optimum performance the nea rest interferer is , o n average, 50% further away from the rec eiv er than the des ired trans mitter . In line n etworks with m -ph ase TDMA, E SIR grows with m α . The res ults ob tained ca n be generalized for (desired) link distan ces other than o ne in a straightforward manner . 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Nov ember 26, 2024 DRAFT p~ 0 0.2 0.4 0.6 0.8 1.0 C 0 2 4 6 8 10 Ergodic Capacity and Lower Bound for α=4 0 5 10 15 20 0 2 4 6 8 10 12 14 16 18 20 Θ [dB] m α =2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p p s Θ =20dB Θ =10dB Θ =0dB Exact expression Approximation 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p p s ξ =1/4 ξ =2 Exact (fading) Approx. (fading) Exact (part. fading) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 transmit probability p success probability p s ξ =1/3 ξ =2 1/1 1/0 0/1 0/0 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Θ [dB] Throughput g( Θ ) α =2 Theoretical optimum For integer m