Analysis of Exponential Backoff with Multipacket Reception in Wireless Networks *

Transcription

1 Analysis of Exponential Bacoff with ultipacet Reception in Wieless Netwos * Peng Xuan Zheng, Ying Jun (Angela Zhang and Soung Chang Liew Depatment of Infomation Engineeing The Chinese Univesity of Hong Kong {pxzheng4, yjzhang, soung}@ie.cuh.edu.h Abstact A collision esolution scheme is essential to the pefomance of a andom-access wieless netwo. ost schemes employ exponential bacoff (EB to adjust the tansmission attempt ate accoding to the changing taffic intensity. Pevious wo on exponential bacoff was mostly based on the conventional single-pacet-eception model whee no moe than one pacet can be successfully eceived at any one time. In this pape, we analyze the pefomance of EB based on a uti-pacet Reception (PR model, in which multiple pacets can be eceived successfully at once (i.e., collisions do not occu unless the numbe of pacets tansmitted exceeds a theshold that is moe than. Using a aov chain model, we deive the thoughput expessions fo both caie-sensing and non-caie-sensing netwos with PR capability unde the satuated-taffic condition. We find that the two systems shae a numbe of common pefomance esults. In paticula, the state of both systems can be chaacteized by the same aov-chain model. The binay exponential bacoff (BEB, in which the bacoff facto is set to 2, does not yield the optimum netwo thoughput in both cases. In addition, in both cases, the asymptotic collision pobability goes to / and the maximum asymptotic thoughput inceases oughly linealy with when the population size appoaches infinity. We show how to adjust to achieve the best thoughput pefomance. Ou esults show that the optimal that maximizes the asymptotic thoughput inceases with fo non-caie-sensing systems and BEB is close to optimal fo caie-sensing systems. Simulation esults validate the accuacy of ou theoetical analysis. Index Tems ulti-pacet eception, exponential bacoff, wieless netwos, ALOHA, WLAN, 82., andom access.. Intoduction The collision-esolution scheme used in a andom-access wieless netwo is a significant facto affecting netwo pefomance. Coe to a collision-esolution scheme is a bacoff algoithm which detemines how long a node should wait befoe etansmitting a pacet afte a pacet collision. This is an integal pat that detemines how well the system adjusts to the eve changing taffic intensity in the netwo. Exponential bacoff (EB, in which each collision causes the * This wo was suppoted by the Aeas of Excellence scheme (Poject Numbe AoE/E-/99, the Competitive Eamaed Reseach Gant (Poject Numbe 4856 established unde the Univesity Gant Committee of the Hong Kong Special Administative Region, China, and the Diect Gant (Poject Numbe 2537 of the Chinese Univesity of Hong Kong. bacoff peiod to be multiplied by a constant facto, has been investigated in detail in the last few decades []. ost pevious wo was based on the taditional singlepacet-eception model. With advanced eception techniques at the physical laye, howeve, it is possible fo the eceive to esolve multiple simultaneously tansmitted pacets. With ulti-pacet Reception (PR [2], collisions occu only when the numbe of simultaneously tansmitted pacets exceeds the maximum numbe of simultaneous pacets that the eceive can esolve. To date, thee has been little wo on the behavio and impact of EB unde PR. In this pape, we analyze the pefomance of EB with PR capability unde two common andom-access models, namely models with and without caie-sensing. We fist deive the thoughput expessions fo both systems based on a common aov-chain model assuming a finite population. Then, the asymptotic collision pobability and tansmission pobability fo the infinite-population case is studied. We find that the commonly deployed binay exponential bacoff (BEB is not optimal in tems of the oveall netwo thoughput. We show how to adjust the bacoff facto to achieve the best pefomance. In addition, we popose an infinite population model to investigate the asymptotic thoughput of EB. Results show that the optimal which maximizes the asymptotic thoughput inceases with fo non-caie-sensing systems and BEB is close to optimal fo caie-sensing systems. Besides, the maximum asymptotic thoughput inceases oughly linealy with fo both caie-sensing and noncaie-sensing systems. The accuacy of ou analytical model is validated by extensive simulations. The est of the pape is oganized as follows. Section 2 intoduces the concepts of PR and EB. Section 3 fist descibes the analytical model, and then deives the thoughput of EB with PR fo the non-caie-sensing model. Based on the thoughput expession, we show how to adjust the bacoff facto fo optimal thoughput pefomance. An infinite population model is also pesented to analyze the asymptotic behavio of EB when the numbe of uses in the netwo is lage. Section 4 analyzes the pefomance of an PR AC potocol with caie-sensing. Finally, Section 5 concludes the pape. 2. Bacgound and Related Wo 2.. utipacet Reception With the advent of CDA [3] and multiple-antenna [4] techniques, it is no longe a physical constaint fo the channel to accommodate only one ongoing tansmission. Fo example, we peviously poposed an 82.-lie potocol to suppot

2 PR in wieless local aea netwos, whee the AP has multiple antennas and is capable of eceiving multiple pacets simultaneously [5]. Ghez, Vedu and Schwatz [6] wee the fist to analyze the stability popeties of slotted Aloha with PR capability in the late 98s. By means of a dift analysis, they showed that the channel baclog aov chain is egodic if the pacet aival ate is less than the expected numbe of pacets successfully eceived in a collision of n pacets as n goes to infinity. Not until vey ecently did PR esuface in eseach wo again. Reseaches fom Conell eopened the topic and have pefomed extensive studies on vaious aspects of PR. In [2], L. Tong et al studied the impact of PR-enabling signal-pocessing techniques on the thoughput and design of andom access potocols. In [7], Zhao and Tong poposed a centalized multiqueue sevice oom AC potocol (QSR which was designed especially fo PR. Late in [8], Zhao and Tong futhe poposed anothe centalized AC potocol fo PR, the dynamic queue potocol, which offes a much simple implementation and only maginal pefomance degadation. These medium access schemes all equie a cental contolle to coodinate the tansmissions of the client stations. Chan and Bege poposed a coss-laye designed CSA potocol fo PR (XL-CSA in [9]. XL-CSA is a decentalized andom access scheme and may be applied in vaious situations when a cental contolle is not available. Wo in [5] is the fist attempt to implement PR unde the IEEE 82. DCF setting in wieless LAN. In this pape, we assume that the channel has the capability to accommodate up to simultaneous tansmissions. Heeafte, we efe to this as PR capability. oe specifically, this means the pacets can be eceived coectly wheneve the numbe of simultaneous tansmissions is no moe than. When moe than stations contend fo the channel at the same time, no pacet can be decoded. To focus on the effect of PR on EB, we assume that the channel is eo fee in the sense that all pacet losses ae due to collisions (i.e., moe than pacets ae tansmitted at the same time. This assumption is widely used in the liteatue to simplify analysis and at the same time povide easonable esults. Nonetheless, ou wo can be easily extended to include the effect of andom channel eo by adding a tem of pacet eo ate in the thoughput expession Exponential Bacoff As pointed out in the pevious subsection, wheneve the numbe of simultaneous tansmissions exceeds the channel s PR capability, collision happens and the pacets involved ae gabled. Theefoe, once a collision occus, a collision esolution scheme is needed fo the stations to optimally schedule the etansmission of the colliding pacets. One of the most widely used collision esolution potocols is the binay exponential bacoff (BEB scheme, which is being included as pat of the AC specifications in Ethenet [] and IEEE 82. [] standads. The basic idea of BEB is as follows. In andom access netwos, a collision implies that the channel is liely to be oveloaded. Hence, once a collision occus, the colliding stations double thei contention window size to educe thei tansmission pobability. In this way, the oveall load on the channel is lightened. ost pevious wo in this aea focuses on the stability issues of EB. Since stability is not the focus of ou pape, inteested eades ae efeed to [2] fo a moe detailed liteatue suvey. In this pape, we assume that EB is stable in the sense which will become clea late, and based on that, we pefom a caeful study of the impact of PR on the pefomance of EB. In this pape, we conside the geneal fom of exponential bacoff (EB in which the bacoff facto can be any value lage than one. The BEB is a special case of the geneal EB with = 2. We also investigate the pefomance of EB unde both caie-sensing and non-caie-sensing models. ost of ou geneal conclusions hold fo both systems, except that the thoughput expessions ae diffeent. Although ou focus is on the non-caie-sensing model in Section 3, the aov chain model fo the analysis of the collision pobability and tansmission pobability also applies to the caie-sensing case. The diffeence in the thoughputs of the two systems is manifested though the diffeent time inteval paametes being used fo collision event, successful-tansmission event, etc. in the thoughput expessions. 3. Analysis of EB with ultipacet Reception 3.. odel In the following analysis, we use the so-called time-slotted model, which assumes that the time axis is divided into slots of equal duation. All pacet tansmissions ae of the same length, equal to exactly one slot, and ae synchonized. The model assumed hee is without caie-sensing, while the model assumed in Section 4 is with caie-sensing. Ou analysis of the non-caie-sensing case in this section can be egaded as a genealization of [2] to the PR famewo. Thee ae N stations, each opeating in a satuated mode, which means the stations always have pacets waiting fo tansmission. Afte each tansmission, the tansmitting stations will be infomed of the outcome of the tansmission, i.e., success o failue. We also assume that the feedbac time is extemely shot compaed with the length of pacet tansmissions, theefoe is negligible in ou analysis. This assumption is to mae sue that ou model does not depend on any paticula implementation. The thoughput expession deived in the next subsection only need to be modified a little bit to account fo the effect of any paticula feedbac mechanism. The EB algoithm wos as follows. Initially, any station having pacets to tansmit andomly selects a bacoff time fom to W slots, whee the intege W denotes the minimum contention window size. Afte waiting fo this amount of bacoff time, the station tansmits a pacet in its queue. Evey time the tansmission is unsuccessful, the contention window fo the station will be multiplied by the bacoff facto. Afte i successive failed etansmissions, the numbe of slots fo which the station will wait befoe the i + etansmission attempt, D i, is geneated accoding to the following distibution []:

3 Wi + Fi, =,,, Wi Wi( Wi + P{ Di = } = ( Fi, = Wi Wi + i i whee Wi = W and Fi = W Wi. Once a tansmission is successful the contention window of that station is eset to W. We fist assume that EB is used in each of the stations in the netwo, and that with the use of the EB, the stations will finally each a steady state in which the distibution of the bacoff stages of the stations becomes stationay. So we ae infomally defining the stability as the capability of the EB to adapt to the vaying taffic condition and finally bing the system into a steady state. We use an infinite-state aov chain, as in Fig., to model the opeation of EB with no ety limit at a station. The eason why no ety limit is intoduced is that it is theoetically moe inteesting to loo at the limiting case when m is infinitely lage. Besides, by stiing out m, we ae at an advantage of having fewe vaiables so that cleae elationships between the moe inteesting paametes can be manifested. Having said that, the analysis in ou pape can be easily extended to the case whee thee is a ety limit m. The state in the aov chain in Fig. is the bacoff stage, which is also equal to the numbe of etansmissions expeienced by the station. Theefoe, the contention window size is i = W when the station is in state i. In ou model, we Wi assume that the collision pobability of each tansmission attempt is equal to a constant, no matte which state the station is cuently at. This assumption is accuate as long as W and N is lage enough [3]. - i = i = i = 2 i = Fig. aov chain model fo the bacoff stage. - Let p t be the tansmission pobability of a station in an abitay slot. Fom this aov chain, we can deive the expession fo p t, 2( pc pt =. (2 W ( pc + pc whee p c < is a necessay condition fo the steady state to be eachable [2]. In the steady state, the pobability that a tansmitted pacet is collided is equal to the pobability that the numbe of simultaneous tansmissions fom the othe N stations in the same slot is no less than the channel s PR capability. Thus, we have the following elation: N N p c = pt ( pt. (3 =... p t W =6 (2 (3 with = (3 with = 2 N= N= Fig. 2 Plots of p t as a function of when = 2; dashed lines: p t in (2, dotted lines: p t in (3 with =, = 2, solid lines: p t in (3 with = 2. Fom Eqns (2 and (3, we can solve fo and p t, given N,, and W. The cuves detemined by (2 and (3 ae plotted in Fig. 2. As can be seen, the cuves have a unique intesection which can always be calculated numeically. This unique intesection epesents the oots and p t to Eqns (2 and (3. Fom the figue, we can also see that as N inceases, p t and convege to and.5 (i.e., / when = 2 espectively, egadless of. This obsevation will be poved analytically in the next subsection Pefomance Analysis Let P t be the pobability that thee is at least one tansmission in the slot time. Then P ( N t = pt. (4 Let P s denote the pobability that pacets ae tansmitted simultaneously and the tansmission is successful, conditioning on the fact that thee is at least one tansmission. Since each station tansmits its pacet independently in a given slot, the pobability P s is calculated as N N Ps = pt ( pt Pt. (5 Thus, the nomalized thoughput of the netwo is given by N N T = Ps Pt = pt ( pt = =. (6 Note that the thoughput can be intepeted as the atio of the expected numbe of pacets successfully tansmitted within one time slot to the maximum numbe of pacets that can be tansmitted in that time inteval without PR. Theefoe, it has no unit. This definition confoms to the one used in [2] fo =, but slightly diffeent fom the thoughput definition we use late in the caie-sensing case. In the following discussion, we analyze the asymptotic behavio of and T in ou PR model. Fist we show that conveges to / egadless of the PR capability. It is not difficult to see that lim p t = is necessay fo the system to each steady state which is ou initial assumption.

4 Suppose p t conveges to some nonzeo value, then Np t, the numbe of aveage tansmission attempts in a slot, appoaches infinity as N goes to infinity. This implies that the collision pobability p c =, which does not satisfy the necessay condition < / fo the system to be able to each the steady state. Hence, p t conveges to zeo as N goes to infinity. Fom (2 we now 2( pc pt =. W ( pc + pc Taing the limit, we should have 2( pc lim p t = lim = N W ( pc + pc which implies lim 2( p c =. Theefoe, we have lim = fo all. (7 Fom (7, we conclude that conveges to / egadless of as N goes to infinity. We plot the analytical esults of, obtained by calculating (2 and (3 numeically, as lines and the simulation esults as maes in Fig. 3. To veify the analytical esults, we developed a simulato to ealize the EB algoithm descibed in the pevious subsection along with those assumptions. The simulato is witten in C++ and developed unde Visual C++ 6. IDE. The data wee collected by unning 5,, slots afte,, slots of waming up, as in [2]. The bacoff facto we use in the simulation is equal to 2, so the geneal EB educes to BEB and the minimum contention window sizes we choose ae W = 6, 32, and 64 which confom to the IEEE 82. specification fo diffeent PHY layes. We only show the simulation esults when = and 2 as examples and the cuves fo othe can be infeed fom these two special cases. In Fig. 3, both the cuves fo = and = 2 convege to.5 (i.e., / since = 2. We can also see fom Fig. 3 that the simulation esults match well with the analytical esults, which suppots ou conclusion in (7. It is obvious in the figue that as N inceases, inceases to / moe slowly fo = 2 than fo =. This is self-evident since when the PR capability is inceased it is less liely fo a station to collide with othes = =2 analysis with W =6 analysis with W =32 analysis with W =64 simulation with W =6 simulation with W =32 simulation with W = N Fig. 3 Plots of vesus N when = 2; lines ae analytical esults calculated fom (2 and (3, maes ae simulation esults. We now loo at the thoughput esults based on solving (2, (3 and (6. Fom Fig. 4, we can see that the thoughputs with the same convege to the same constant as N inceases, egadless of W. This phenomenon implies that no matte how cowded the netwo is, EB can guaantee a nonzeo limiting thoughput. This limiting thoughput only depends on the PR capability of the channel and is insensitive to the settings of the initial minimum contention window size. In the next subsection, we will show how to pedict this limiting thoughput fom the infinite-population model. As shown in Fig. 4, howeve, this limiting thoughput inceases as gets lage. This fact confoms to ou intuition, since we would expect highe thoughput with inceased PR capability. Thoughput = = 2 analysis with W =6 analysis with W =32 analysis with W =64 simulation with W =6 simulation with W =32 simulation with W = N Fig. 4 Plots of thoughput vesus N when = 2; lines ae analytical esults obtained numeically, maes ae simulation esults. As a special case of ou geneal PR model when =, Kwa et al [2] poved that, unde the taditional collision model, the collision pobability and netwo thoughput T convege as the numbe of stations N goes to infinity as follows: lim =, (8

5 lim T = ln. (9 Anothe dimension to investigate the thoughput is to see how the thoughput behaves as inceases given a fixed numbe of stations N. The analytical esults obtained when N = 5 ae plotted in Fig. 5. In this figue, the thee cuves go up to thei maximum value as inceases. The maximum thoughput deceases as W inceases. The eason is as follows. Fist, it is obvious that the maximum thoughput is eached when = N. In this case, thee is no collision and all tansmissions ae successful. Theefoe, the contention window size is always equal to W. It can be easily figued out that the maximum thoughput is equal to 2 N ( W +. This implies that the maximum thoughput deceases monotonically with W when N is fixed. Anothe obsevation is that, when is much smalle than N, thoughput inceases oughly linealy with ; when appoaches N, the thoughput eaches the maximum. We will show in the next subsection that the asymptotic thoughput (thoughput when inceases almost linealy with iespective of W. Thoughput W = 6 W = 32 W = Fig. 5 Thoughput vesus when = 2 and N = 5. In pactice, PR capability is achieved by inceasing both the hadwae and computational costs. It is theefoe wothwhile investigating the achievable thoughput pe unit cost in ou system. As shown in Fig. 5, the thoughputs become almost flat as becomes lage than some value. Theefoe, the nomalized thoughput (nomalized by actually inceases as inceases at fist and then goes down. The optimal point should be somewhee aound the cone of the cuve. While this is tue fo finite N, fo infinite N the nomalized thoughput inceases monotonically with. Similaly, the elation between thoughput and given can also be obtained fom (2, (3 and (6. Fig. 6 shows the thoughput vaiation with diffeent settings of. Fom this figue, we can see that, given, thee exists an optimal value of which maximizes the netwo thoughput. This fact can be intepeted as follows. When we incease, the collision pobability is educed. Howeve, we may tend to bacoff too much, thus also educing the tansmission pobability. Thus, to maximize the thoughput by calibating the bacoff facto, we ae indeed balancing between letting the stations be moe aggessive at the is of highe collision pobability and letting them be consevative at the is of wasting pecious ai time when nobody tansmits at all. athematically, the optimal value of which maximizes the thoughput can be obtained by solving the equation dt d =. ( As shown in (9, when =, the asymptotic thoughput is expessed as a function of. The optimal which maximizes the asymptotic thoughput is given by [2] opt = ( e. ( Howeve, when is lage than, it is difficult to expess T as a closed-fom function of, so numeical methods ae needed to find the optimal. Anothe obsevation fom this figue is that the commonly deployed binay exponential bacoff is not optimal in the sense that = 2 does not achieve the maximum aggegate thoughput. Thoughput = 2 = W = 6 W = 32 W = Fig. 6 Thoughput vesus bacoff facto when N = Infinite Population odel In the pevious subsection, we studied EB and achievable thoughput of wieless netwos with PR when the numbe of stations is finite and equal to N. The asymptotic pefomance when N appoaches infinity can be obtained by setting N to be a vey lage numbe in ou peviously deived equations. Howeve, this would mae the numeical esults difficult to obtain. In this subsection, we theefoe adopt an altenative infinite-population model to analyze the asymptotic pefomance of PR. In this model, it is assumed that the numbe of stations N in the netwo is infinitely lage and each station independently tansmits in a slot with pobability p t. Recall that lim p t =, while lim Np t is a constant. Theefoe, the oiginally binomially distibuted numbe of tansmission attempts in a slot can be appoximated by Poisson distibution: n λ λ P{ X = n} = e (2 n! whee the andom vaiable X denotes the numbe of attempts in a slot and λ is the mean which can be expessed as λ = lim Npt. (3 In (7, we have shown that lim =. eanwhile, fom ou infinite population model, we have

6 lim = P{ X } = P{ X }. (4 Thus fom (7 and (4, we get the following equation λ λ λ λ P{ X } = e = e = (5 =! =! which elates λ with the values of and. Hence, we can always calculate λ numeically fom (5 given and. Finally the asymptotic thoughput is given by lim T = P{ X = } = λ P{ X = } = =. (6 = λp{ X } = λ( Note that the asymptotic thoughput is expessed by λ and. Since λ is detemined by and, the asymptotic thoughput is indeed only a function of and. Given and, we can calculate λ fom (5. The asymptotic thoughput is then obtained by plugging the value of λ into (6. Fig. 7 is a plot of the asymptotic thoughput vesus fo vaious. Fom this figue, we can see that the optimal which maximizes the asymptotic thoughput inceases with. This esult sounds counteintuitive, because we usually expect that with lage we should decease to encouage the stations to be moe aggessive. It is tue that inceasing will educe the numbe of attempts in a slot, but on the othe hand this will incease the success pobability of an attempt. It is the weighting of these two effects that decides the optimal opeating point of. Fom the esults, we can conclude that as inceases, the second effect dominates and in the end moves to the ight to achieve the maximum asymptotic thoughput. Anothe diect conclusion fom this figue is that BEB ( = 2 is fa fom optimal. Fo example, when = BEB only achieves about 8 pecent of the maximum asymptotic thoughput. Asymptotic thoughput = = 5 = Fig. 7 Asymptotic thoughput vesus By tuning to the optimal value fo each, we plot the maximum asymptotic thoughput against in Fig. 8. Unlie Fig. 5 in which N is finite, Fig. 8 shows that the asymptotic thoughput inceases almost linealy with. In fact, the esults ae consistent with ou obsevations fom Fig. 5 in the sense that thoughput inceases oughly linealy with when N (infinitely lage hee is much lage than. A close obsevation of Fig. 8 indicates that the slope of the cuve inceases slowly with. Consequently, thoughput nomalized by also inceases with. This implies that as inceases, the achievable thoughput pe unit cost (i.e., bandwidth in CDA systems o antenna in multi-antenna systems also inceases. aximum asymptotic thoughput Fig. 8 aximum asymptotic thoughput by tuning vesus. 4. ultipacet Reception in WLAN In this section, we analyze the pefomance of an AC potocol fo PR with caie-sensing. The eason why we ae inteested in this potocol is that this PR AC potocol closely follows the 82. DCF which employs the BEB as the bacoff algoithm. Details of the PR AC potocol and the PHY implementation can be found in [5]. 4.. Potocol Desciption The poposed potocol follows the 82. DCF RTS/CTS (Request To Send/Clea To Send access mechanism closely, with extension to suppot PR. We biefly descibe the potocol in this subsection. Fo simplicity, we conside a single isolated BSS with an AP and N associated client stations. We assume that the AP is the only station in the BSS with the capability to eceive up to ( pacets simultaneously. Figue 9 illustates the potocol opeation. A station with a pacet to tansmit fist sends an RTS fame to the AP. In ou PR AC model, when multiple stations tansmit RTS fames at the same time, the AP can successfully detect all the RTS fames if and only if the numbe of RTSs is no moe than. When the numbe of tansmitting stations exceeds, collisions occu and the AP cannot decode any of the RTSs. The stations will etansmit thei RTS fames afte a bacoff time peiod as the oiginal 82. potocol.

7 Fig. 9 Time line example fo the PR AC. When the AP detects the RTSs successfully, it esponds, afte a SIFS peiod, with a CTS fame that gants tansmission pemissions to all the equesting stations. Then the tansmitting stations will stat tansmitting DATA fames afte a SIFS, and the AP will acnowledge the eception of the DATA fames by an ACK fame Pefomance Analysis In the following analysis, we assume the numbe of contending stations is fixed and that all stations opeate in satuated conditions (i.e., they always have pacets available fo tansmission. We also assume that all stations can hea each othe (i.e., thee is no hidden teminals and the channel is pefect in that thee is no pacet loss due to fading. We define slot time as the time inteval between two consecutive bacoff time counte decements [3]. Note that the slot time we define hee is not necessaily the constant slot time size. In 82. DCF, the bacoff counte will be fozen when the channel is sensed busy. We define ou slot time to include the fozen time and it is vaiable. Although the PR AC is a caie-sensing potocol, the analysis esults we deive in Section 3 pat still apply. This is because the tansmission pobability, which is defined as the pobability to tansmit in an abitay slot, does not depend on the length of the slot. So (2 and (3 ae tue even in this caie-sensing model. We define the thoughput T to be the atio of payload infomation bits being tansmitted to the total amount of time spent to successfully tansmit the payload. Theefoe, E T = [ payload infomation bits tansmitted in a slot time] E [ length of a slot time] P P E[ L] = ( P + P P T + P ( P T s t = t σ s t s t s c = (7 whee P t and P s ae as defined in Section 3, E[L] is the aveage payload length in bits, T s is the aveage slot time spent when thee ae successful -pacet tansmissions, T c is the aveage slot time when thee ae collisions, and P s = P s = is the conditional pobability of successful tansmissions in a busy time slot. The geneal thoughput expession (7 deived fo PR also incopoates the 82. case. In paticula, when =, the PR thoughput educes to the 82. thoughput. In ou potocol, the RTS/CTS access scheme is employed. Theefoe Ts = RTS+ SIFS+ δ + CTS+ SIFS+ δ + H * + EL [ ]/ R+ SIFS+ δ + ACK+ DIFS+ δ (8 Tc = RTS+ DIFS + δ whee δ is the popagation delay, H = PHY hd + AChd is the total ovehead time to tansmit the pacet heades, E [ L * ] is the aveage length (in bits of the longest payload involved in a -pacet simultaneous tansmission, and R is the data ate fo payload tansmission. In the following numeical investigations, we assume all pacets have the same fixed length, i.e., the aveage length E[ L * ] = E[ L] = L ( and T s = Ts, whee L and T s ae constants. To have a clea pictue of the asymptotic behavio of the PR AC, again we esot to the infinite population model poposed in Section 3. We give the expession fo the asymptotic thoughput of the PR AC diectly Ps Pt E[ L] = lim T = ( P σ + PPT + P( PT t s t s t s c = (9 whee all the symbols have aleady been defined peviously in (7 except that P t and P s should be changed to Pt = P{ X = } = e λ (2 and λ λ Ps = P{ X = } Pt = e Pt, (2! espectively. Paallel to the discussion in the non-caie-sensing case, we plot the asymptotic thoughput against fo vaious in Fig.. The esults shown hee ae fom theoetical analysis and the paamete settings we used hee ae basically those fom 82.g with some modifications as stated in [5] except that thee is no ety limit hee. Fom this figue, we can see that thee still exists an optimal fo each cuve. The appoach to achieve the best thoughput pefomance by adjusting in PR AC is analogous to the one we mentioned in Section 3. Howeve, fom the engineeing s point of view, we ague that BEB (i.e., = 2 aleady achieves a close-to-optimal thoughput fo a lage vaiety of. Results show that about 45% incease in the maximum asymptotic thoughput can be achieved when = 2 compaed with 82.g. Simila to the conclusions we made fo the non-caie-sensing system, Fig. shows that the maximum asymptotic thoughput inceases oughly linealy with.

8 Asymptotic thoughput (bps = = 2 = 5 = Fig. Asymptotic thoughput vesus bacoff facto fo the PR AC. aximum asymptotic thoughput (bps Fig. aximum asymptotic thoughput vesus fo the PR AC (ignoing the addess incease in CTS and ACK. 5. Conclusion This pape has investigated the pefomance of EB in andom-access netwos with PR capability that allows pacets to be simultaneously tansmitted without collision. Both caie-sensing and non-caie-sensing systems have been studied. Insights gained fom ou analysis ae useful fo system design in tems of setting the coect opeating paametes. Extensive simulations have also validated the accuacy of ou theoetical analysis. We have deived the thoughput expessions fo both caie-sensing and non-caie-sensing systems unde satuated-taffic condition, and have analyzed the asymptotic behavio of EB with PR unde infinite-population assumption. In both systems, the collision pobability conveges to / egadless of and the thoughput given conveges to a constant as N inceases (albeit diffeent constants fo the two systems. With the help of ou thoughput expession, we have also analyzed the vaiation of nomalized thoughput with espect to and the vaiation of thoughput with espect to given. Based on the infinite population model, we obseved that the optimal inceases with fo non-caie-sensing systems and BEB is close to optimal fo caie-sensing systems. We have pesented numeical esults showing the possible impovements with the PR capability. Ou esults show that fo caie-sensing systems, thoughput impovement of 45% can be achieved with ou poposed PR AC potocol when = 2 compaed with the conventional 82.g potocol. Fo both caie-sensing and non-caie-sensing systems, netwo thoughput inceases almost linealy with fo the infiniteuse-population case. 6. Refeences [] J. Goodman, A. G. Geenbeg, N. adas, and P. ach, "Stability of binay exponential bacoff," J. AC, vol. 35, no. 3, pp , Jul [2] L. Tong, Q. Zhao, and G. egen, ultipacet eception in andom access wieless netwos: Fom signal pocessing to optimal medium access contol, IEEE Commun. ag., vol. 39, no., pp. 8-2, Nov. 2. [3] P. B. Rapajic and D. K. Boah, "Adaptive SE maximum lielihood CDA multiuse detection," IEEE J. Select. Aeas Commun., vol. 7, no. 2, pp , Dec [4] S. Talwa,. Vibeg, and A. Paulaj, Blind sepaation of synchonous co-channel digital signals using an antenna aay, Pat I: Algoithms, IEEE Tans. Signal Pocessing, vol. 44, no. 5, pp , ay 996. [5] P. X. Zheng, Y. J. Zhang and S. C. Liew, "ultipacet eception in wieless local aea netwos," Poc. IEEE ICC'6, Istanbul, Tuey, June, 26. [6] S. Ghez, S. Vedú, and S. Schwatz, Stability popeties of slotted Aloha with multipacet eception capability, IEEE Tans. Automat. Cont., vol. 33, no. 7, pp , Jul [7] Q. Zhao and L. Tong, A multiqueue sevice oom AC potocol fo wieless netwos with multipacet eception, IEEE/AC Tans. Netwoing, vol., pp , Feb. 23. [8] Q. Zhao and L. Tong, "A dynamic queue potocol fo multiaccess wieless netwos with multipacet eception," IEEE Tans. Wieless Comm., vol. 3, no. 6, pp , Nov. 24. [9] D. S. Chan and T. Bege, Pefomance and Coss-Laye Design of CSA fo Wieless Netwos with ultipacet Reception, Poc. of Asiloma Conf. on Signals, Systems and Computes, vol. 2, pp , Nov. 24. [] ANSI/IEEE Std , IEEE standads fo local aea netwos: caie sense multiple access with collision detection (CSA/CD access method and physical laye specifications, 985. [] IEEE Std , IEEE Std Infomation Technology- telecommunications And Infomation exchange Between Systems-Local And etopolitan Aea Netwosspecific Requiements-pat : Wieless Lan edium Access Contol (AC And Physical Laye (PHY Specifications, Nov [2] B. J. Kwa; N. O. Song, and L. E. ille, "Pefomance analysis of exponential bacoff," IEEE/AC Tans. Netwoing, vol. 3, no. 2, pp , Ap. 25. [3] G. Bianchi, Pefomance analysis of the IEEE 82. distibuted coodination function, IEEE J.. Select. Aeas Commun., vol. 8, no. 3, pp , a. 2.