Throughput Computation in CSMA Wireless Networks with Collision Effects

Throughput Computation in CSM A Wir eless Networks w ith Collision Effec ts Cai Hong Kai, Soung Chan g Liew Department of I nformation Engineering, The Chine se University of Hong Kong Email: caihong.k ai@gmail.com , soung @ie.cuhk.edu.hk Abstract It is known that link throughputs of CSM A wireless net works can be co mputed from a time-reversible Ma rkov chain arising from an ideal CSMA network model (ICN). I n particula r , this model yields general closed-for m equations of lin k throughputs. H owe ver , an idealized and important assumption made in ICN is tha t the backoff countdow n process is in “contiuo us-time” and carrier sensing is instan taneo us. As a result, t here is no collision in IC N. In pra ctica l CSMA protocols such as IEEE 80 2.1 1, the stations count dow n in “ mini-timeslot” and the pr o cess is therefor e a “discrete-ti me” process. In pa rticular , tw o stations m a y end their backoff pr o cess in the sa me mini-ti meslot and then transmit simultaneously , r es ulting in a packet collision. This paper is a n atte mpt t o study how to co mpute l ink throughputs afte r taking suc h backo ff collision effects into account. We propose a ge neralized ideal CSMA netwo rk model ( GICN) to charact erize the collision states as well as the inter actions and depe ndency among lin ks in t he netw ork. W e show tha t link throughputs and collisio n probability c an be c omputed f rom GICN. Simulation resul ts validate GICN’ s accura cy . Inter e stingly , we also find that the orig inal IC N model y ields fairly a ccurate results despite the fact t hat collisions are not modeled. I Intr oduction W ith the widespread dep loyment of IEEE 802. 1 1 net works, it is co mmon toda y to f ind multiple wireless LANs co -located in the neighb orhood o f each other . The m ul tiple wireles s LANs form an overall large network whose links interact and co mpete for a irtime using the car rier-sense multiple access (CSMA) p rotoco l. When a station hears its neig hbors transmit, it will refrain from tr ansmitting in order to avoid pa cket collisions. For analytical purpo ses, the carrier sensing r elations hips a mong the links are typically captured using a contentio n graph. T he links are modeled b y vertices of the grap h, and an edge j oins two vertice s if the transmitters of the two a ssociated links c an sense e ach other . Since different li nks may se nse different subsets of other links, the links may experience di fferent throughputs. Ref. [1] presented a n anal ytical model, Id eal CSMA Ne twork (I CN), to stud y the behavior o f CSMA networks given their contention graphs. It was shown that the t hroughputs of links can be computed from the stationary pro bability d istribution of the states o f a conti nuous-time Mar kov chai n, even though t he pro cess is not memoryless. T he same mod el has been used in se veral prio r works [2] [3], a ssuming that the bac koff and transmissio n time are exponentiall y distributed. A n important contribution o f [1] is the re moval o f t his ass umption, making t he I CN model applicable to a practical CSMA wireless networ k. Rec ently , an e legant distrib uted adaptive CSM A algorit hm is propo sed b ased on the ICN mod el to achieve the o ptimal throu ghput in CSM A wireless net works [4] [5]. Although the ICN model ha s captured the m ai n feature s o f the CSMA pr otocol, an importa nt simplifying as sumption made in o rder to maintain anal ytical tractabilit y is that there is no collision in the network. In particular, it is assumed that carr ier sensi ng is perfect and instantaneous. T he goal of ICN is to c apture the “first-ord er” intera ctions a mong the li nks due to their car rier-sensing relationships as modeled b y the contentio n graph. Coll isions are tr eated a s a “second-ord er” effect and m odeled away in ICN. An outstandin g issue is ho w to make ICN mo re accurate by incorporatin g the effects of collisions. In prac tical CSMA pr otocol such as IEEE 802 .1 1 , time is divided into discrete minislots, and collisions happen if multiple conflicting links count d own to zero in the same timeslot in their backoff process and then transmit si multaneously 1 . When a co llision o ccurs, all t he involved links lose their packets and will try to retrans mit later . In this paper, we first pr opose a genera lized ideal CSMA network mod el (GICN) to an alyze the effects of collis ions d ue to si multaneous trans missions. GICN ca n be vie wed as a perturbation-a nalytical model of ICN. B ased on the GIC N model, the link throughp uts and co llision probabilit y can be computed. Si mulation result s sho w t hat the GI CN model has high acc uracy . Interestingl y , we find that the effect o f co llisions is not signi ficant as far a s the link throughput s are concerned. That is, t he o riginal ICN model yields good app roximations e ven t hough it do es not capt ure the collision e ffect. II ICN Model and Its Equilibriu m Analysis T o build up the background f or the later s ections, w e briefly revie w the ICN model and its equilibrium anal ysis here. A. ICN Model In ICN, the car rier sensi ng relatio nship amon g li nks is de scribed by a contention graph ( ) , G V E = . Each link is mod eled as a vertex i V ∈ . Edges, on the other ha nd, model the car rier-sensing relationships among links. There is an edge e E ∈ between two vertices i f the tran smitters of t he two associated links can sense eac h other . In this paper we will use th e terms “links” a nd “ vertices” interchangeabl y . At an y time, a link i s i n o ne of t wo possib le state s, ac tive or idle. A li nk is active if ther e is a data transmission between its two end node s. Thanks to carrier sensing, any two links that can hear each other will refrain fro m b eing active at the same t ime. A link sees the c hannel as idle if and only i f none of its neighbor s is active. 1 The “hidden-node” phenomenon can also cause packet co llisions. That is, the transmitters of two links c annot hear the activities of each other , but a c ollision occurs if t he two t ransmissions overl a p. The throughput analysis with col lisi ons incurred by “hidden-node” is left as future work in this pap er. Alternatively , we c an design protocols, such as the algorithms proposed in [6], to remove the hidden-node pheno menon. In ICN, eac h li nk maintains a ba ckoff timer , C , the initial val ue of which i s a rando m variable with an arbitrary d istribution ( ) cd f t . The timer value of the lin k decr eases in a continuous manner with 1 dC dt = − as long as the link se nses the chan nel as idle. If the channel is se nsed busy (due to a neighbor transmitting), the countdown process is frozen and 0 dC dt = . When t he channel beco mes idle again, the co untdo wn continues and 1 dC dt = − with C initialized to the pr evious frozen value. When C reaches 0, the link tra nsmits a packet. The transmission duration is a rando m variable with arbitrary distribution ( ) tr g t . After the transmission, the lin k resets C to a new ra ndom value according to the distribution ( ) cd f t , and the pr ocess repe ats. W e define the a ccess intensity o f a link as the ratio of its mean trans mission duration to its mean ba ckoff time: [ ] [ ] tr cd E t E t ρ = . Let { 0,1 } i s ∈ denote the state o f link i , where 1 i s = if link i is activ e (transmittin g) and 0 i s = if link i is idle (activel y counting down or f r ozen). The overall system state of ICN is 1 2 ... N s s s s = , where N is the nu mber of lin ks i n the net work. Note that i s and j s cannot both be 1 at the same time if lin ks i and j are neighbo rs because ( i) they ca n sense eac h other; and ( ii) the probabilit y o f them counti ng down to zero and tran smitting together is 0 u nder ICN ( because the backoff time is a continuou s rando m variable). The collec tion of feasible states corr esponds to the collectio n o f independent sets of the c ontention graph. An i ndependen t set (I S) of a graph i s a subset of vertices such t hat no edge joins any two of them [7]. As an example, Fi g. 1( a) shows t he conte ntion graph of a network co nsisting of four links. In this network, link 1 only senses link 2 while links 2, 3 and 4 can hear each other . Fig. 1( b) shows the associated state-transitio n diagram under the ICN model. T o avoid clutters, we have m erged the two directional transitions bet ween t wo state s into one li ne i n Fig. 1(b). Each tra nsition fro m le ft to right correspo nds to the begin ning of t he transmission of one particular link, while the reverse transition correspo nds to the endi ng o f t he trans mission o f that link. Fo r exa mple, the transitio n 1000 1010 → is due to link 3 ’ s begin ning to tra nsmit; the reverse transitio n 1010 1000 → is due to link 3 ’ s completin g its transmission. 4 3 1 2 (a) (b) Fig. 1. (a) An example con tention graph and (b) its state-transition d iagram. B. Equilibrium Analysis This part is a quick revie w of the r esult in [1], and the reader is referred to [1 ] for de tails. If we assume that t he backoff ti me and transmission time ar e exponentially distributed, then ( ) s t is a time-r eve rsible Mar kov process. For any pair of neighbo r states in the c ontinuous-t ime Markov chai n, the transitio n fro m t he left s tate to the right state occ urs at r ate [ ] 1 / cd E t λ = , and the tran sition from the right state to the left state occurs at rate [ ] 1 / tr E t µ = . Let S denote the set o f all feasible states, and s n be the number of transmitting links when the system is in state 1 2 ... N s s s s = . The stationary distributio n of state s can be sho wn to be: s n s P s Z ρ = ∀ ∈ S , where s n s Z ρ ∈ = ∑ S (1) The fraction o f time d uring which l ink i transmits is : 1 i i s s s th P = = ∑ , whic h co rresponds to the normalized thro ughput of li nk i . Ref. [1] sho wed that (1) is in fact quite general and does not require the s ystem state ( ) s t to be a Markov pr ocess. I n p articular, (1 ) is insensitive to the distri bution o f t he tra nsmission dur ation ( ) tr g t , and the distributio n of the bac koff duration ( ) cd f t , given the ratio of their mean [ ] [ ] tr cd E t E t ρ = . Note that Z in (1) is a w eighted su m o f indep endent sets of G . In statistica l p hysics, Z is referred to as the partitio n functio n and the co mputation o f Z is the crux of many pro blems, which is known to b e NP-hard [7]. For the case where dif ferent links have differe nce access intensities, (1) can be generalized by replacing s n ρ with the : 1 in i i i s s ρ = ∏ , where i ρ is the access i ntensity of link i . III GICN Mo del and Assu mptions This section describe s o ur generalized ideal CSMA model (GICN) and the ass umptions made in the analysis o f Section IV . Different fro m ICN where both the ba ckoff time a nd tr ansmission time are con tinuous variables, in GICN time is d ivided into mi ni-timeslot s. The b ackoff cou ntdown timer is an integ er var iable which is uniforml y distrib uted b etwee n [0 , CW ], where CW is the contention windo w . T he timer value of the link de creases by one for each timeslot the link senses the c hannel as idle. Further more, we assume that the transmi ssion d uration is a fixed value eq ual to [ ] tr E t (As can be seen later , this assump tion is important i n our analys is.). That is, the acce ss inten sity of a l ink is 2 [ ] tr E t CW ρ = . The feasible state of ICN is the ind ependent set o f the co ntention graph. In GICN, multiple links can countdo wn to zer o in the same timeslot a nd the n trans mit together . That i s, the states o f t wo neighbor links, i s and j s can b oth b e 1 at the same ti me. In this case, we say that a co llision happ ens and the two packets c ollide ( in this pa per we do not consid er the “signal capture ” effect [8]). W e summar ize the assu mption s made in GI CN as follows: 1) The co ntention windo w , CW , is not do ubled upo n collisions. 2) The tra nsmission time is of fix ed length o f [ ] tr E t . 3) The remaining countdo wn ti me of ea ch link is indepe ndent of each other when co llision events are taken i nto account. 4) Only coll isions in which two neighb or links tr ansmit togeth er are considere d. T hat is, w e do not consider the co llisions i n whic h more than t wo neig hbor links tr ansmit at t he same ti me. Ref. [ 2] made the same assu mption as 1 ) and studied the thr oughput co mputatio n and fairnes s issues in CSMA wireless network s. Note that [2] did not co nsider collision eve nts. This paper studies the throughput co mputation o f lin ks i n CSM A wireless networks with co llision s. T o justify t his assumptio n, we will s how the i mpact of contention window doublin g is no t significa nt thro ugh simula tion results in Section V. Note from 2) that i n GIC N, the links involved in a collisi on ar e “sync hronized ”. That is, the li nks involved in a collisio n must be gin transmitti ng in the same timeslo t and will end tra nsmitting in the same timeslo t after [ ] tr E t timeslots. It can be shown that, under this a ssumption, the s ystem pro cess is still time-re versible. Ref. [9] made the sa me assu mption as 1) , 2) and 3 ) to design CSMA-ba sed scheduli ng a lgorithms. Different fro m [9] in which ne w distrib uted sched uling al gorithm i s desi gned, thi s pape r studies the existing CSM A proto col such a s IEEE 8 02.11 netw o rks. IV Throughput Distribution Computation under G I CN W e conduct perturb ation on I CN’ s computatio n of link thro ughputs. The basic ide a is as follo ws: in the state-transition diagra m o f IC N, we ca n figure o ut t he states star ting from which a co llision may occur . Fo r ea ch such state, we could carefully identify the transitio n rates betwee n these collision-relate d states. Then we can co mpute the statio nar y probab ility of each stat e, o btain the normalized li nk throu ghputs and the collisio n prob ability of each link. In P art A b elow , we calculate the co nditional c ollision probabilit y beforeha nd, which will be used in the analysis o f Part B . A. Conditional Collision Pr oba bility For each c ollision c aused b y simulta neous tran smission s, the multiple neig hbor lin ks involved in the collisio n must be actively counting d own befor e the colli sion occurs. Consider the co llision prob abilit y o f a particular li nk co nditioned o n that the link has n actively-cou ntdo wn neighbo rs. In our m o del the b ackoff ti me o f ea ch li nk is unifor mly distributed between [0 , CW]. If we trea t trans mission slo t as one co untdo wn backo ff slot, then the a verage distance between two transmi ssion time slots is 2 CW , and the tra nsmissio n p roba bility of an activel y c ountin g do wn station o ver eac h timeslo t is 1 2 / 2 1 2 CW CW = + + . T ake the typical val ue o f cont ention windo w in I EEE 802.11b network, 31 CW = , as an e xample. The average distance bet ween t wo transmi ssion time slots is 15.5 , and the tr ansmission pr obab ility of an actively co unting d own st ation is o ver each ti meslot 1 /16.5 . The probab ility of collisio n is then 1 ( 15.5 / 16.5) n − . Conditioned on tha t an activel y-countdo wn l ink has n actively-co untdo wn neighbor s, the collision pro babilit y of this lin k, n q , is 2 1 1 1 2 2 n n n CW q CW CW     = − − = −     + +     (2) Remark: From (2) w e ha ve 1 2 2 q CW = + . Filling in the typical value o f 31 CW = , 1 0.0607 q = . When 1 q is small, we ca n app roximate (2) as ( ) 1 1 1 1 n n q q nq = − − ≈ (3 ) W e note that the appr oximation made in ( 3) is im p ortant in the c onstructio n of state -transit ion diagram in P art B . B. Perturbation on ICN’ s Computation Next we illustrate our p ertur bation on ICN’ s computation . W e use the t wo-link network as the illustrating example. The state transition is sho wn in Fig. 2. State “ 00 * * ” is the collision state (i.e ., it is the state 11 in this example) . Since the trans mission ti me is fi xed, starting fro m state “ 00 * * ”, the s ystem will evolve b ack to state “00” d irectly . Proced ure of Conduct ing Pert urbation on I CN’s C omputation S tep 1: From the state-transit ion o f ICN, we id entify t he s tates in which collisions may occ ur . Denote this subset of states by M . For each s M ∈ , the co llisio n states connected to s are represented by a single state, d enoted b y * * S . State * * S includes al l possible states in which two- link collision occur s. The pro bability of a co llision state * * S is the su m of the prob abilities of t hese states. In Fig.2, states 0 1 and 1 0 cann ot lead to a “co llisio n” state. Starting fro m 00, there co uld b e a stat e 1 1 , in which the t wo link s collide. That is, { 00 } M = and { } 00 * * 11 = . S tep2: For each state s in M , identify the tra nsition rate s to/from its neighbor states. Note that the tran sition rate fro m “1” to “0” is fixed to µ no matter whet her the tra nsmission collides with a neighbor ’ s p acket. Also , the tr ansition rate from state * * S to s is µ . T o identify the tr ansitio n rate s of the c ollision -related states, we need to co mpute the c onditional collision p rob ability fir st. Fo r e ach state s in M , w e id entify the co llision probab ility of each link conditioned on that the c urrent state is s , using the re sults in P art A. In our e xample o f Fig. 2 , give n sta te 00 s = , the coll ision prob ability o f bo th l inks 1 and 2 i s 1 q . The transition rate from 00 to ** is then 1 q λ ; the transitio n rate fro m 00 to 01 and 10 becomes ( ) 1 1 q λ − . W e note that the tra nsition r ate leaving state 00 is a bit s maller than 2 λ due to co llision effects. In general, we can count the nu mber of neighbor links o f link i which are activel y counting do wn in state s . Invoking (2) o r (3), w e can obtai n the collision pro babilit y o f link i . The transition rate fro m state s to the collision state * * S is eq ual to n q , wh e re n is the number o f edges joini ng the activel y-countdo wn links in state s . 0 0 * * 1 q λ µ µ µ ( ) 1 1 q λ − ( ) 1 1 q λ − Fig. 2 T wo-link network and the assoc i ated state-transitio n diagram with col lisions S tep 3: Assuming that t he st ationar y probab ility of G ICN has t he pr oduct form as in IC N (No te that loca l ba lance still hold s in the gener alized state-tra nsition diagra m because it can b e sh own that the system pr ocess is time-re versible) , we ca n co mpute the statio nary p robab ility distrib ution of states in GICN. Based on this stationar y distrib ution, we can comp ute: i) the nor malized throu ghputs of links : 1 i i s s s Th P = = ∑ and ii) the collision pro babilit y of each link: Pr{ collision states in which link is co ll ided } Pr{ all states in which link is active } i i p i = . In our e xample of Fig. 2, we have 1 1 2 1 ( 1 ) 1 2 q Th Th q ρ ρ ρ − = = + − and 1 2 1 p p q = = . Given a typical value o f 83 / 15.5 ρ = , the normalized throughp ut o f bo th links i s 0 .441 8, co llision pro bability o f both links is 1 0.0607 q = . MA TL AB simulatio n validated this res ult. W e use two more exa mples to demonstrate t he pro cedure: 1) A three-link networ k For three link c ase, the state transitio n is sho wn in Fig.3. “ 000 * * * ” represents all possible collision states. In Fig. 2 , 1 2 , q q is calculated from Eq. 2. 000 * * * 2 q λ µ µ ( ) 1 1 q λ − ( ) 1 1 q λ − µ µ µ µ ( ) 2 1 q λ − λ λ Fig. 3 Three -link network and the ass ociated state-transitio n diagram with c ollisions Recall that in I CN the throughput is calculated as 2 1 3 2 2 2 =0.7440, = 0.1171 1 3 1 3 Th Th Th ρ ρ ρ ρ ρ ρ ρ + = = = + + + + where 83 / 15.5 ρ = . After taking collision ef fects into account , 2 1 1 1 3 2 1 1 2 2 2 1 1 ( 1 ) ( 1 ) 1 3 2 ( 1 ) ( 1 ) 1 3 2 ( 1 ) q q Th Th q q q Th q q ρ ρ ρ ρ ρ ρ ρ ρ ρ − + − = = + − + − − = + − + − and collisi on probability of both links are 1 1 3 2 2 2 1 , ( 1 ) q p p p q q ρ ρ ρ = = = + − Hence, 1 2 3 0.7374 0.1090 0.7374 Th Th Th = = = , 1 3 2 0.01 0.1174 p p P = = = 2) A four- link network 1 0010 1010 0000 0001 0100 2 3 4 1000 1001 0000 * * * * 1000 * * * * 1 4 q λ µ µ µ µ µ µ µ ( ) 2 1 q λ − ( ) 1 1 q λ − ( ) 3 1 q λ − µ ( ) 2 1 q λ − ( ) 1 1 q λ − ( ) 1 1 q λ − λ µ λ 1 q λ Fig. 4 Four -link n etwo rk and the associated state- t ransition diagram w ith collisions Recall that in ICN the throughput is calculated as 2 1 2 2 2 2 3 4 2 2 =0.7861, = 0.0671 1 4 2 1 4 2 0.4266 1 4 2 Th Th Th Th ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ + = = + + + + + = = = + + where 83 / 15.5 ρ = . After ta king collisio n effects into account, 2 2 1 2 1 1 1 2 2 1 2 1 1 3 2 2 2 1 2 1 1 2 2 2 3 4 2 2 1 2 1 1 ( 1 ) 2( 1 ) ( 1 ) 1 4 4 2 ( 1 ) ( 1 ) ( 1 ) 1 4 4 2( 1 ) ( 1 ) ( 1 ) ( 1 ) 1 4 4 2 ( 1 ) ( 1 ) q q q q Th q q q q q Th q q q q q q Th Th q q q q ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ − + − + − = + − + − + − − = + − + − + − − + − = = + − + − + − and collision pro bability are 1 1 2 2 2 1 1 2 3 2 2 1 1 3 4 2 2 2 1 1 2( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) q p q q q p q q q q p p q q q ρ ρ ρ ρ ρ ρ ρ ρ ρ = + − + − = + − = = + − + − Hence, 1 2 3 4 0.7807 0.0606 0.4093 0.4093 Th Th Th Th = = = = 1 2 3 4 0.0056 0.1709 0.07 p P P P = = = = V Simulation Results W e conduct si mulations to ver ify the accura cy of GICN a s well as I CN. W e use an IC N-si mulator written with MA T LAB to simulate the C SMA proto col with collision s caused b y si multaneous transmission s. The link throu ghputs c omputed by I CN and GICN a re co mpared with that o btained fro m the ICN-si mulator . As can been seen from Fi g. 5, the throughputs and collisio n pr obability c omputed by GICN are very close to simulation results. On the o ther hand, t he li nk thro ughput s do not change largely when collision e ffects are taken i nto account. That is, the origin al ICN m odel yields good appr oximations even thoug h it does not captur e the collisio n effect. T opo logy ICN GICN Simulation Throughput (Mbps) Throughput (Mbps) Collision Prob Throughput (Mbps) Collision Prob (3.3058, 3.3058) (3.20, 3.20) (0.0607, 0.0607) (3.187, 3.19) (0.0603, 0.0604) (2.2684, 2.2684, 2.2684) (2.12, 2.12, 2.12) (0.1 1 74, 0.1 1 74, 0.1 1 74) (2.1208, 2.122, 2.11 9 6) (0.1 1 77, 0.1 1 74, 0.1 1 71) (5.3782, 0.8463, 5.3782) (5.3306, 0.788, 5.3306) (0.01, 0.1 1 74, 0.01) (5.3263, 0.792, 5.3273) (0.0102, 0.1 1 78, 0.0101) (4.1799, 2.2684, 2.2684, 4.1799) (4.1 1 45, 2.1575, 2.1565, 4.1 1 45) (0.033, 0.07, 0.07, 0.033) (4.11 1 4, 2.1603, 2.1555, 4.1 1 92) (0.033, 0.07, 0.0691, 0.0323) 3 4 2 1 (5.6825, 0.4853, 3.0839, 3.0839) (5.6434, 0.4375, 2.9592, 2.9592 (0.0056, 0.1709, 0.07, 0.07) (5.6399, 0.4385, 2.9553, 2.961) (0.0055, 0.1723, 0.0698, 0.0699) (0.1478, 5.9669, 5.9669, 5.9669) (0.1302, 5.9576, 5.9576, 5.9576) (0.1709, 0.0016, 0.0016, 0.0016) (0.1306, 5.9574, 5.9587, 5.9568) (0.1717, 0.0017, 0.0016, 0.0016) Fig. 5 Contention gr aphs of various n etwork topolog i es and the correspondi n g ICN, GI CN compu ted link throughputs an d simulation results. T o justify our Ass umption 1 in Section III , we c onduct a s et o f simulatio ns, co mparing the lin k throughputs and link co llision prob ability with or without co ntentio n-windo w doubli ng up on co llisions. As sho wn in Fi g. 6 , when we disab le the contention- window doubling, the collisio n pr obabilit y is a bit higher . Ho wever , the link t hroughputs r emain al most the sa me. T opo logy W ith windo w doubling W ithou t window doubling Throughput (Mbps) Collision Prob Throughput Collision Prob (3.177, 3.1811) (0.0587, 0.0587) (3.187, 3.19) (0.0603, 0.0604) (2.1 1 37, 2.1 165, 2.1 1 54) (0.1079, 0.1074, 0.1078) (2.1208, 2.122, 2.1 1 96) (0.1 1 77, 0.1 174, 0.1 1 71) (5.4416, 0.6644, 5.4405) (0.0085, 0.1 1 86, 0.0084) (5.3263, 0.792, 5.3273) (0.0102, 0.1 1 78, 0.101) (4.127, 2.1269, 2.1336, 4.1 1 78) (0.0314, 0.0673, 0.0679, 0.0318) (4.11 1 4, 2.1603, 2.1555, 4.1 1 92) (0.033, 0.07, 0.0691, 0.0323) 3 4 2 1 (5.6952, 0.3835, 2.9779, 2.9714) (0.0314, 0.0673, 0.0679, 0.0318) (5.6399, 0.4385, 2.9553, 2.961) (0.0055, 0.1723, 0.0698, 0.0699) (0.0942, 5.9926, 5.9922, 5.9918) (0.1795, 0.0013, 0.0013, 0.001 1 ) (0.1306, 5.9574, 5.9587, 5.9568) (0.1717, 0.0017, 0.0016, 0.0016) Fig. 6 Contention graphs of various network topologies and the corresponding simulation results with/without contention-w i ndow doubling. VI Conclusion This p aper p resented a generalized ICN m od el to characterize the collision eff ects in CSM A wireless networks. Based on the GICN mode l, link t hroughpu ts and co llision p robab ility can be computed. W e found th at the lin k th roug hputs do not change much after taking i nto account the collisions inc urred b y simultaneous tra nsmi ssions. Several assumption s are assumed in the analysis, such as the tra nsmissio n ti me is of the fixed length. The removal o f such a ssumptio ns awaits f urther work. Ref erence [1] S . Liew , C. Kai, J. Leung an d B. W o ng, “Back-of-the-Envelop e Computatio n of Throughput Distrib utions in CSMA Wireless Networks”, IEEE T ransactions on Mobile Computing , Sep. 2010. T echnical rep ort also available at http://arxiv .o rg//pdf/0712.1854. [2] X. W ang an d K. Kar , “Throu ghput Modeling and F airness Issues in CSMA/CA Based Ad hoc n etw orks,” IEEE INFOCOM , Miami, 2005. [3] M. Durvy , O. Dousse, and P . Thiran, “Border Effects, Fairness, and Ph ase Transition in Lar g e W ireless Networks,” IEEE INFOCOM 2008 , P h oenix, USA. [4] L. Jiang and J. W alrand, “A Distribu ted CSMA Algorithm f or Throughput and Uti lity Max imizatio n in W ireless Networks,” IEEE/ACM T ransactions on Networking , V ol. 18, Issue 3, 2010, pp.960-972. [5] M. Chen, S . Liew , Z. Shao and C. Kai, “Markov Approximation for Combinatorial Network Opt imization,” in IEEE INFOCOM , 2010. [6] L. B. Jiang, S. C. Liew , “Hidden-nod e Rem oval an d Its Application in Cellular WiFi Networks,” IEEE Trans . V ehicu lar T ech. , vol. 56, no. 5, Sep. 2007. [7] Independent set, http://en.wikipedia.org/wiki/Independent_set_ (graph_theory). [8] A. Ko chut, A. V asan, A. S hankar , and A. Agrawala, “Sniffing out the correct Ph ysical Layer Captu re model i n 802.1 1 b,” in Pr oceedings of 1 2th IEEE International Confer ence on Netwo rk Pr otocols (ICNP 2004) , Octob er 2004. [9] L. Jiang and J. W alrand , “Approaching Thro ughput-Optimality in Distribu ted CSMA Scheduling Algorithms W ith Co llisions,” IEEE/ACM T ransactions on Networking 2010.