Acknowledgement-Aware MPR MAC Protocol for Distributed WLANs: Design and Analysis

Transcription

1 Globecom Wireless Commuicatios Symposium Ackowledgemet-Aware MPR MAC Protocol for Distributed WLANs: Desig ad Aalysis Arpa Mukhopadhyay, Neelesh B. Mehta, Seior Member, IEEE, Vikram Sriivasa, Member, IEEE Abstract Multi-packet receptio (MPR, i which a receiver ca decode multiple simultaeous trasmissios, sigificatly improves the uplik throughput of wireless local area etworks (WLANs. However, the medium access cotrol (MAC layer must be redesiged to ecourage, ad ot avoid, simultaeous trasmissios. Asychroous MPR MAC protocols, i which odes idepedetly access the chael so log as the umber of ogoig trasmissios is less tha a threshold, are promisig solutios for eablig MPR i IEEE based WLANs. I this paper, we highlight the problem of ackowledgmet (ACK delays that arises i asychroous MPR whe multiple odes trasmit i successio without the chael becomig idle. We propose a ovel asychroous MAC protocol that reduces the ACK delays, icreases throughput, ad retais the distributed ature of the distributed coordiatio fuctio (DCF. A accurate reewal theoretic fixed-poit aalysis that leads to geeral aalytical expressios for the saturatio throughput is also developed. I. INTRODUCTION IEEE wireless local area etworks (WLANs are very popular, but are facig icreasig demads for higher data rates ad greater spectral efficiecy. These WLANs use the distributed coordiatio fuctio (DCF ad its variats such as ehaced DCF (EDCF for medium access cotrol (MAC [1] [3]. The desig of the DCF is based o the premise that whe multiple statios trasmit simultaeously i a time slot, a collisio occurs ad the receiver caot decode ay trasmissio. Therefore, the DCF uses carrier sese multiple access with collisio avoidace (CSMA/CA to discourage trasmissios by more tha oe user at ay time. This is accomplished by freezig the backoff timer aytime a ode seses a busy chael. However, wireless receivers today are capable of employig sophisticated sigal processig ad multi-user detectio techologies such as successive iterferece cacellatio (SIC, space-time codig, ad multiple ateas [4], ad ca receive multiple packets simultaeously. This capability has bee referred to as multi-packet receptio (MPR i the literature [5] [9]. MPR promises sigificat throughput gais i the uplik, i.e., the lik from the statios to the access poit (AP [10]. However, it also egeders a fudametal redesig of the MAC protocol, which must ow facilitate ad ot discourage simultaeous trasmissios by multiple users. At the same A. Mukhopadhyay is with the Dept. of Electrical ad Computer Eg. at the Uiv. of Waterloo, Caada. N. B. Mehta is with the Dept. of Electrical Commuicatio Eg. at the Idia Istitute of Sciece (IISc, Bagalore, Idia. V. Sriivasa is with Bell Labs Research, Alcatel-Lucet, Bagalore, Idia. A. Mukhopadhyay was at IISc durig the course of this work. s: arpamu@gmail.com, bmehta@ece.iisc.eret.i time, it is desirable that the MAC retais its distributed ature, which is a key reaso for the success of IEEE DCF. While MPR has bee cosidered i the literature, e.g., [7], [10], [11], a sychroous access sceario is ofte assumed. I it, multiple odes ca start trasmittig simultaeously, but o ode ca start a trasmissio if it seses the chael to be busy. This leads to the MPR capability ot beig fully haressed. This problem was firstaddressedbybabichad Comisso i [12], who aalyzed a asychroous protocol usig a Markov chai model. I it, a ode decremets its backoff timer as log as the umber of ogoig trasmissios sesed by it is below a threshold. The umber of receive ateas i the ode determies the umber of trasmissios that it ca sese. However, ackowledgemets (ACKs were ot modeled. It was implicitly assumed that a ode kows whether its trasmissio was successful or ot as soo as its packet trasmissio eds. I fact, i the asychroous MPR setup, ACKs ca get sigificatly delayed. This is because multiple trasmissios to the AP ca ow occur i successio without ay idle period i betwee. Thus, the AP, which is a half-duplex ode, must cotiue to receive these time overlappig trasmissios eve after a particular ode completes its trasmissio. It ca ackowledge all the packets it successfully received oly after the chael becomes idle. However, a ode chooses its backoff timer parameters ad schedules its ext trasmissio oly after it receives its ACK or whe it times out waitig for the ACK. Therefore, this delay ca degrade the protocol s performace. Furthermore, i [12], a memoryless distributio of packet legths was assumed i order to facilitate a Markovia aalysis. This assumptio does ot hold uder heavy traffic situatios whe there are may retrasmissios. I this paper, we first highlight a problem related to ACK delays that is uique to a asychroous MPR MAC protocol. To show that the ACK delays ca degrade the performace of such protocols we compare the performaces of two protocols. The first oe is similar to the oe aalyzed i [12] except that ACKs are ow icorporated i the protocol. The secod protocol, which we propose i this paper, reduces ACK delays by modifyig the rules that gover whe the timer should be decremeted ad froze. We show that the modificatios reduce the ACK delays ad icrease the saturatio throughput, while retaiig the distributed ature of covetioal DCF. Aother importat cotributio of this paper is a fixed-poit aalysis of the proposed asychroous MPR MAC protocol based o reewal theory [13]. Ulike [12], it captures the effect of ACK delays o the protocol s performace ad leads /12/$ IEEE 5087

2 to geeral aalytical expressios for the throughput ad packet collisio probability. It also geeralizes the aalysis i [14], which is applicable oly to covetioal DCF. The packet legths are o loger required to be geometrically distributed i our model. Further, the effect of packet droppig after a fiite umber of retrasmissios is ow explicitly icluded. Thus, our approach provides a more geeral aalysis of the asychroous MPR MAC. The paper is orgaized as follows. Sectio II develops the system model. The proposed protocol is aalyzed i Sectio III. Simulatio results are preseted i Sectio IV, ad are followed by our coclusios i Sectio V. Mathematical derivatios are relegated to the Appedix. II. SYSTEM MODEL AND PROTOCOL DESCRIPTION Cosider a IEEE etwork that cosists of cotedig odes. We assume that the AP ca decode up to two simultaeous trasmissios, ad each ode ca correctly estimate whether there is zero, oe, or more tha oe ogoig trasmissio i the chael, as was also assumed i [12]. 1 This is easily achieved whe each ode is equipped with at least two ateas by meas of directio of arrival (DOA techiques based o eige-decompositio of the received sigal s array correlatio matrix [16] [12, Sec. VI]. All odes follow the biary expoetial backoff scheme, as specified i DCF. Before each packet trasmissio, a ode selects the backoff period i multiples of a slot duratio δ. The multiple is chose uiformly from {0, 1,..., w 1}, where w is called the cotetio widow. It depeds o the umber of failed trasmissios of a packet. I the first attempt, w is set equal to the miimum cotetio widow CW mi. After each usuccessful trasmissio, w is doubled, up to a maximum value of CW max =2 m CW mi. A packet is dropped by a ode after K +1failed trasmissio attempts. A. Protocol 1: Asychroous MPR Protocol i [12] with ACKs I the followig we describe a protocol similar to that i [12] but with ACKs icorporated i it. As metioed earlier, the AP is assumed to be able to decode up to two simultaeous trasmissios. I this protocol, a ode havig a packet to trasmit samples a backoff timer value ad starts decremetig it oce it seses the chael to be idle for a distributed iterframe space (DIFS of duratio T DIFS. The backoff timer is decremeted as log as the ode seses 0 or 1 trasmissios i the chael. Whe the sesed umber of trasmissios becomes greater tha or equal to 2, the ode freezes its backoff timer. It is resumed from the last stored value as soo as the umber of sesed trasmissios agai falls to 1. The ode trasmits its packet whe the backoff timer value becomes zero. 1 We focus o the case with two simultaeous trasmissios sice it is both aalytically rich ad isightful ad practically relevat. As we show, it leads to close to 100% gais i saturatio throughput ad delay over covetioal DCF. Further, it requires odes to have just two ateas, which is feasible today [4], [15]. The aalysis ca be geeralized to hadle more tha two simultaeous trasmissios, though the expressios become more ivolved. Fig. 1. Timig diagram for the proposed scheme showig a system-wide reewal iterval i which two simultaeous asychroous trasmissios to the AP occur. Whe the chael becomes idle after the ed of the overlappig trasmissios, the AP seds a cumulative ACK of duratio T ACK, which ackowledges all the successful trasmissios together [7]. As i covetioal DCF, it waits for the chael to remai idle for a short iter-frame space (SIFS of duratio T SIFS before commecig its trasmissio of the ACK. By settig T DIFS >T SIFS, the ACK gets priority over other trasmissios whe the chael is idle. However, ulike covetioal DCF, a ode expects its ACK to arrive withi a duratio T OUT = T DIFS after the chael becomes idle, ad ot after it has completed its packet trasmissio. I case the ACK does ot arrive, the ode times out, updates its cotetio widow, chooses a ew backoff timer value, ad starts decremetig it. B. Protocol 2: Proposed Protocol We ow propose a asychroous MPR MAC protocol that reduces the ACK delays. As i the previous protocol, a ode with a packet samples a backoff timer ad starts decremetig it whe it fids the chael to be idle for T DIFS duratio. It trasmits its packet whe the timer reaches zero. However, the proposed protocol differs with respect to the coditios uder which the backoff timer is froze ad remais froze. The rule is as follows: Durig the backoff phase, a ode freezes its timer oce the umber of ogoig trasmissios exceeds two or oce it decreases. Thereafter, the ode resumes decremetig its timer oly whe the chael has remaied idle for a duratio T DIFS. The operatio of the AP remais the same as before. The above protocol is illustrated i Figure 1 for three odes A, B, adc. I it, A trasmits first. While A is trasmittig, B ad C cotiue decremetig their timers sice there is oly oe ogoig trasmissio i the chael. Oce B starts trasmittig, C freezes its timer. The timer of C remais froze eve after A s trasmissio eds. Oly after a idle period of duratio T DIFS, do all the odes resume decremetig their timers. I Protocol 1 istead, C will start decremetig its timer as soo as A s trasmissio eds. As a result, its timer may become zero while B is trasmittig. 5088

3 III. ANALYSIS We ow aalyze the proposed MPR scheme i saturated traffic coditios i which trasmissio queue of each ode is assumed to be always o-empty. This gives a limit o the system throughput i heavy traffic loads, ad has bee extesively aalyzed for covetioal DCF; see, for example, [1], [14]. To simplify the aalysis, the odes are assumed to be statistically idetical ad the trasmissio rate is fixed at Ω [1], [14]. A trasmitted packet is received successfully uless it is ivolved i a collisio. To get compact aalytical results, we assume that a data packet trasmissio lasts for a duratio of λ slots. This also illustrates that a o-memoryless packet legth distributio ca be aalyzed, ulike the Markov chai approach of [1], [12]. We also assume that ACKs are successfully received by all the odes. As all the odes use the same backoff parameters, we make the followig two classical decouplig approximatios, which eable a fixed-poit aalysis [1], [14]: 1 Each trasmitted packet suffers a collisio with a probability γ, regardless of the umber of its retrasmissios ad idepedet of all other odes. We shall call γ as the coditioal packet collisio probability. 2 Each ode attempts a trasmissio i a slot where it is allowed to trasmit with a probability β that is idepedet of all other odes. The parameter β will be referred to as the attempt rate. Note that over a sufficietly log time, γ ca be iterpreted as the ratio of the total umber of collisios occurrig i the etwork to the total umber of trasmissio attempts by all odes uder saturated traffic coditios. Similarly, β, ithe log ru, is the average (take over all odes of the ratio of the total umber of trasmissio attempts by a ode to the total umber of slots i which the ode is i its backoff phase. Based o these decouplig approximatios, the followig two reewal processes ca be costructed. Node-specific reewal process: Cosider a give ode, which we heceforth call the tagged ode. Let A j ad B j respectively deote the umber of attempts ad total backoff duratio (i slots eeded by the tagged ode to trasmit its j th packet. From the first approximatio it follows that the backoff process of a tagged ode is a reewal process with reewal lifetimes B j, j 1, ad with the time istats at which the ode starts the fial trasmissio of its j th packet as reewal epochs. If we treat A j, j 1, as the reward gaied i each reewal cycle, the from the reewal reward theorem [13], we have β = E[Aj] E[B j ],wheree[ ] deotes expectatio. It ca be show that the attempt rate β as a fuctio of γ is give by 1+γ + + γ K β G(γ = b 0 + γb γ K, (1 b K ( where b k = k CW mi 1,for0 k K, is the mea backoff duratio (i slots before the (k +1 th trasmissio attempt of a packet. The proof is similar to that i [14], ad is ot show here to coserve space. System-wide reewal process: Cosider the aggregate attempt process by all the odes o the chael. Due to the decouplig approximatios, the aggregate process is aother reewal process. As show i Figure 1, a reewal iterval starts whe all the odes start decremetig their backoff timers ad eds with either a ACK timeout (i case of a collisio or the trasmissio of a cumulative ACK by the AP (i case of a success followed by a idle of duratio T DIFS. Ulike covetioal DCF, more tha oe packet ca get trasmitted i a reewal iterval. We, therefore, defie the first packet ad secod packet of a reewal iterval as follows. A packet is called the first packet i a reewal iterval if the chael is idle whe its trasmissio commeces. A packet is called the secod packet if there is already oe ogoig trasmissio i the chael whe its trasmissio commeces. Give that a tagged ode has trasmitted a packet, let α deote the probability that the packet is the first packet i the reewal iterval. Lemma 1: Give that a tagged ode trasmits i a reewal iterval, the probability α that it is the first packet i the iterval is give by K 1 (β α = K 1 (β+k 2 (β, (2 where K i (β is the probability that the i th packet i a reewal iterval is the tagged ode s packet. Further, K 1 (β = β ad K 2 (β = ( 1β2 (1 β 1 ( (λ 1( 1 ( ( 1. Proof: The proof is relegated to Appedix A. Theorem 1: The coditioal packet collisio probability, γ, as a fuctio of β is give by γ Γ(β =αp 1 (β+(1 αp 2 (β, (3 where P i (β, fori =1, 2, deotes the probability of collisio of the i th trasmitted packet i a reewal iterval. Further, ( 1 (1 β 1 ( 1β(1 β 2 P 1 (β = 1 (1 β ( 1 1 (1 β λ( 1, (4 P 2 (β =1 (1 β 2. (5 Proof: The proof is relegated to Appedix B. Hece, by combiig (1, (2, ad (3, we get the followig fixed-poit equatio: γ = Γ(G(γ. SiceΓ(G(γ is a cotiuous mappig from the closed set [0, 1] to itself, Brouwer s fixed-poit theorem implies that there exists a fixed poit i [0, 1] [14]. Solvig this equatio umerically yields γ. The, (1 directly yields β. 2 A. Saturatio Throughput Let ζ deote the amout of successfully trasmitted data i a reewal iterval of duratio T. From the reewal reward theorem [13], the saturatio throughput, S, isgiveby S = E [ζ] E [T ], (6 2 We have observed that the fixed poit is uique for the parameters of iterest. However, provig uiqueess remais a challegig problem. 5089

4 We ow develop expressios for E [ζ] ad E [T ]. Asshow i Figure 1, a reewal iterval of legth T starts with a idle period of duratio T idle. It is followed by a busy period of duratio T busy, which icludes packet trasmissio(s, a cumulative ACK (if success occurs, ad a idle duratio of legth T DIFS. Depedig o whether a success or collisio has occurred, we refer to the busy period that occurs after the idle period as a success period (of duratio T suc or a collisio period (of duratio T col. Further, let Tcol mi ad Tsuc mi deote the miimum values of T col ad T suc, respectively. It ca be see that Tcol mi = λδ +T DIFS ad Tsuc mi = λδ +T SIFS +T ACK +T DIFS. Lemma 2: The expected legth of the reewal iterval is E [T ]=E [T idle ]+D col + D suc, (7 where D col ad D suc are the cotributios to the average busy period duratio from the collisio ad success evets, respectively. Further, E [T idle ]= 1, [ 1 (1 β β(1 β 1 D col = 1 (1 β ( 1β2 (1 β 2 2(1 (1 β ] T mi col + β (1 ( β 1 1 (1 β 1 ( 1β(1 β 2 (1 (1 β (1 (1 β 1 [ ( 1 (1 β (λ 1( 1 Tcol mi + 1 (1 ( β(λ 1( 1 λ (λ 1(1 β 1 ] (1 (1 β 1 δ, (8 ad D suc = ( 1β2 (1 β 2 +2β(1 β λ( 1 2(1 (1 β Tsuc mi + ( 1β2 (1 β 1 (1 β 2 (1 (1 β (1 (1 β 1 Fig. 2. Proposed protocol: Fiite state machie for the access poit If the umber of sesed trasmissios becomes greater tha or equal to 2 or decreases [ ( 1 (1 β (λ 1( 1 T mi suc + 1 (1 ( β(λ 1( 1 λ (λ 1(1 β 1 (1 (1 β 1 δ. (9 Proof: The proof is relegated to Appedix C. Lemma 3: The expected amout of data trasmitted i a reewal iterval, with a trasmissio rate of Ω, isgiveby ( ( 1β 2 (1 β 2 E [ζ] =2λδΩ 2(1 (1 β + ( 1β2 (1 β [ 2 (1 β 1 (1 β λ( 1] (1 (1 β 1 (1 (1 β β(1 βλ( 1 + λδω 1 (1 β. (10 Proof: The proof is relegated to Appedix D. The expressio for the ormalized saturatio throughput the follows directly from Lemmas 2 ad 3. ] Fig. 3. Proposed protocol: Fiite state machie for a ode IV. NUMERICAL RESULTS We ow preset the results obtaied from Mote Carlo simulatios that use 50,000 samples for differet MAC protocols, ad compare them with aalytical results. A evet-drive simulatio platform writte i C programmig laguage was used to implemet the MPR protocols described i Sectio II. This provides idepedet verificatio of the aalytical results. I it, each ode is modeled as a fiite state machie (FSM havig several states as per the protocols. The FSMs that model the AP ad the odes for the proposed protocol are show i Figures 2 ad 3, respectively. The parameter values used i the simulatios are: δ =20μs, T DIFS = T OUT =50μs, T SIFS =10μs, T ACK = 304 μs, CW mi =32, CW max = 1024, K =7,adλ = 400 slots. Figure 4 plots the ormalized saturatio throughput, S/Ω, 5090

5 Normalized saturatio throughput, S/Ω % 28.3% 18.1% 14.2% 0.5 Proposed protocol Protocol DCF basic access Sychroous MPR MAC Number of odes, Coditioal collisio probability, γ Covetioal o MPR DCF Proposed protocol Aalysis (Proposed protocol Simulatio (Proposed protocol Aalysis ( DCF basic access Simulatio ( DCF basic access Number of odes, Fig. 4. Saturatio throughput as a fuctio of the umber of odes Fig. 6. Coditioal packet collisio probability as a fuctio of the umber of odes Average head of lie packet delay (ms Proposed protocol Protocol DCF basic access Number of odes, Fig. 5. Saturatio delay as a fuctio of the umber of odes as a fuctio of the umber of cotedig odes,, for covetioal DCF, Protocol 1, ad the proposed Protocol 2. Also plotted is the throughput of a sychroous MPR protocol i which the odes freeze their backoff timers if they sese the chael to be busy regardless of the umber of ogoig trasmissios i the chael. However, if oe or two odes trasmit simultaeously, their packets will be received successfully by the AP. The ACKs are icorporated i all the protocols i order to esure a fair compariso. We see that the throughput of the proposed protocol is twice that of covetioal DCF, 10-15% more tha Protocol 1, ad 16-38% more tha the sychroous MPR versio. 3 Figure 5 plots the average head-of lie packet delay, i.e., the average time spet by a packet at the head of a ode s queue util the ode receives a ACK cofirmig successful trasmissio or the ode drops the packet. Agai, the proposed protocol outperforms the two bechmark protocols. The headof-lie delay is a importat performace metric that affects the performace of higher layers of the protocol stack [17]. The coditioal packet collisio probability as a fuctio of 3 For the geeral case where the AP ca decode up to L 2 overlappig trasmissios, Protocol 2 ca be geeralized as follows. A ode freezes its timer whe the umber of trasmissios i the chael exceeds L 1 or whe the umber of trasmissios starts decreasig. For L =3, 4, ad 5, the throughput of Protocol 2 is 2.96, 3.95, ad 4.98 times, respectively, more tha the throughput of covetioal DCF. It is at least 10% more tha the that of Protocol 1. is show i Figure 6. Notice that the aalysis ad simulatio results match each other well. Further, as icreases, the percetage error decreases. This is i cosoace with the results i mea field iteractio theory, which provides a mathematical justificatio for this behavior [18]. V. CONCLUSIONS We saw that makig the odes freeze their backoff timers oce the umber of ogoig trasmissios i the chael has crossed a threshold value or decreased from its previous value reduces the ACK delays i the asychroous MPR setup. It restricts the umber of ovelappig trasmissios i a reewal iterval, ad leads to saturatio throughput gais over the asychroous MPR MAC protocol cosidered i the literature ad over covetioal DCF. We saw that our reewal-theoretic fixed-poit aalysis is accurate ad geeral. It eables the modelig of packet droppig after K retrasmissios. Also, ulike the Markovia aalysis, it does ot eed to assume a memoryless packet legth distributio. Future work icludes evaluatig the effect of adaptive modulatio ad codig ad imperfect estimatio of the umber of ogoig trasmissios i the chael, ad determiig the o-saturated throughput. Correspodig throughput improvemets i the dowlik, which ca be achieved usig trasmissios techiques such as superpositio codig or spatial multiplexig at the AP, are also worth ivestigatig. APPENDIX A. Proof of Lemma 1 Expressio for K 1 (β: Let the tagged ode trasmit i slot t (t 1 of the reewal iterval. This occurs with probability (1 β (t 1 β, sice oe of the odes should have trasmitted i the slots 1,...,t 1 ad the tagged ode must trasmit i slot t. Hece, K 1 (β = β(1 β (t 1 β β = 1 (1 β. t=1 Expressio for K 2 (β: Let the first trasmissio from exactly oe ode other tha the tagged ode begi i slot t 1 of the reewal iterval, where t 1 1, ad let the tagged ode 5091

6 trasmit i slot t 1 + t Clearly, 0 t 2 λ 2, sice the proposed protocol does ot permit the tagged ode to trasmit oce the chael becomes idle. The probability of this evet is (1 β (t1 1 ( 1β(1 β 2 (1 β(1 β ( 1t2 β. Summig the probabilities over t 1 ad t 2 yields the desired expressio for K 2 (β. The expressio i (2 for α i terms of K 1 (β ad K 2 (β follows directly from Baye s rule. B. Proof of Theorem 1 A packet trasmitted by the tagged ode suffers a collisio uder the followig two scearios. 1 It is the first packet i the iterval: I this case, the packet suffers a collisio oly if, i ay of its λ slots, at least two amog the other 1 odes trasmit. The probability that the first i slots (0 i λ 1 of the trasmitted packet are free from collisios is (1 β i( 1. Ad, the probability that two or more odes trasmit i the (i+1 th slot is 1 (1 β 1 ( ( 1β(1 β 2. Thus, we have P 1 = λ 1 i=0 (1 βi( 1 1 (1 β 1 ( 1β(1 β 2, which simplifies to (5. 2 It is the secod packet i the iterval: I this case, a collisio ca occur oly i the first slot of the packet. This happes whe at least oe amog the remaiig 2 odes trasmits i the slot, which occurs with probability 1 (1 β 2. C. Proof of Lemma 2 Let P [ ] deote probability. Expressio for E [T idle ]: Clearly, P [T idle >x]=(1 β x, for x =0, 1,..., because o ode must trasmit for at least x slots. Hece, we get E [T idle ] = x=0 P [T idle > x] = x=0 (1 βx 1 =. Expressio for D col : Whe two or more odes start trasmittig simultaeously just after the idle period eds, we have T col = Tcol mi. This occurs with probability β(1 β 1 ( 1 2 β 2 (1 β 2. The deomiator term of 1 (1 β arises due to coditioig o the evet that the idle period has eded. Now, T col = Tcol mi + xδ, for 1 x λ 1, whe exactly oe amog the odes starts trasmittig after the idle period is over, oe of the remaiig 1 odes trasmit i the ext x 1 slots, ad at least two of the remaiig 1 odes trasmit i the followig slot. This happes with probability β(1 β 1 (1 β (x 1( 1 ( 1 ( 1β(1 β 2.Summig over x results i (8. Expressio for D suc : If just after the idle period, oly oe ode trasmits i the whole reewal iterval or exactly two odes start trasmittig simultaeously the a success occurs ad T suc = Tsuc mi. Its probability of occurrece is ( 1 2 β 2 (1 β 2 +β(1 β 1 (1 β (λ 1( 1.Similarly, it ca be show that P [T suc = Tsuc mi + xδ] = β(1 β 1 (1 β (x 1( 1 ( 1β(1 β 2,for1 x λ 1. Summig over x yields the desired expressio i (9. D. Brief Proof of Lemma 3 If a collisio occurs i a reewal iterval, the the AP caot decode ay of the trasmitted packets. Hece, i this case ζ =0.Otherwise,ζ equals iλδω if i {1, 2} odes trasmit i the reewal iterval. The probabilities of these evets have bee derived i Appedix C. Therefore, (( β(1 βλ( 1 E [ζ] = 1 (1 β λδω+ 2 β 2 (1 β 2 1 (1 β λ 1 ( 1β 2 (1 β 2 (1 β x( (1 β 2λδΩ. (11 x=1 The expressio above simplifies to (10. REFERENCES [1] G. Biachi, Performace aalysis of the IEEE distributed coordiatio fuctio, IEEE J. Sel. Areas Commu., vol. 18, pp , Mar [2] K. Medepalli ad F. A. Tobagi, Towards performace modelig of IEEE based wireless etworks: a uified framework ad its applicatios, i Proc. INFOCOM, pp. 1 12, Apr [3] F. Cal, M. Coti, ad E. Gregori, IEEE protocol: desig ad performace evaluatio of a adaptive backoff mechaism, IEEE J. Sel. Areas Commu., vol. 18, pp , Sep [4] M. Z. Siam ad M. Kruz, A overview of MIMO-orieted chael access i wireless etworks, IEEE Wireless Commu., vol. 15, pp , Feb [5] L. Tog, Q. Zhao, ad G. Merge, Multipacket receptio i radom access wireless etworks: from sigal processig to optimal medium access cotrol, IEEE Commu. Mag., vol. 39, pp , Nov [6] G. D. Celik, G. 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