Performance Analysis of Contention Based Medium Access Control Protocols

Transcription

1 1 Performance Anayss of Contenton Based Medum Access Contro Protocos Gaurav Sharma, Ayavad Ganesh, and Peter Key, Member, IEEE Abstract Ths paper studes the performance of contenton based medum access contro MAC protocos. In partcuar, a smpe and accurate technque for estmatng the throughput of the IEEE DCF protoco s deveoped. The technque s based on a rgorous anayss of the Markov chan that corresponds to the tme evouton of the back-off processes at the contendng nodes. An extenson of the technque s presented to hande the case where servce dfferentaton s provded wth the use of heterogeneous protoco parameters, as, for exampe, n IEEE 82.11e EDCA protoco. Our resuts provde new nsghts nto the operaton of such protocos. The technques deveoped n the paper are appcabe to a wde varety of contenton based MAC protocos. Index Terms IEEE 82.11, IEEE 82.11e, CSMA/CA, Performance evauaton, Fxed pont anayss, Fud mt, Dffuson Approxmaton, Wreess LANs, Performance of the MAC protocos. I. INTRODUCTION Wreess oca area networks WLANs based on the IEEE standard are one of the fastest growng wreess access technooges n the word today. They provde an effectve means of achevng wreess data connectvty n homes, pubc paces and offces. The ow-cost and hgh-speed WLANs can be ntegrated wthn the ceuar coverage to provde hotspot coverage for hgh-speed data servces, thus becomng an ntegra part of next generaton wreess communcaton networks. The fundamenta access mechansm of IEEE MAC s the Dstrbuted Coordnaton Functon DCF. The DCF s a carrer sense mutpe access protoco wth coson avodance CSMA/CA. In addton to DCF, the IEEE standard aso defnes an optona Pont Coordnaton Functon PCF, whch uses a centra coordnator for assgnng the transmsson rght to statons, thus guaranteeng a coson free access to the shared wreess medum. Whe DCF has ganed enormous popuarty and been wdey depoyed, the use of PCF has been rather mted. Whereas the IEEE standard was targeted at besteffort servce for data transfer, t s expected that n the future WLANs w need to support a mx of QoS-senstve, mutmeda and nteractve traffc, n addton to data traffc whch s ony senstve to the throughput. Future WLANs Gaurav Sharma s wth D. E. Shaw & Co., 12 West 45th St, New York, NY 136. E-ma: sharmag@deshaw.com. Ayavad Ganesh s wth Department of Mathematcs, Unversty of Brsto, Cfton, Brsto BS8 1TW, U.K. E-ma: a.ganesh@brsto.ac.uk. Peter Key s wth Mcrosoft Research, Roger Needham Budng, 7 J J Thomson Avenue, Cambrdge CB3 FB, UK. Ema: peterkey@mcrosoft.com. must therefore provde servce dfferentaton n order to better support the dverse QoS requrements of the appcatons runnng on them. A new standard, namey IEEE 82.11e, has been proposed for ths purpose; t defnes two new access mechansms: EDCA an enhancement to DCF, and HCF an enhancement to PCF. Of the two, EDCA appears to be ganng more eary acceptance. In ths paper we study the performance of contenton based MAC protocos, wth a specfc emphass on DCF and a smpfed verson of EDCA. There have been severa prevous works on the performance of DCF; these ncude smuaton studes [1], [2] as we as anaytca studes based on smpfed modes of DCF [3], [4], [5], [6], [7], [8]. Most of the anaytca work s based on a decoupng approxmaton, frst proposed by Banch n [3]; we henceforth refer to the smpfed mode wth ths decoupng assumpton as Banch s mode. More recenty, severa studes [9], [1], [11], [12], [13] have evauated the performance of EDCF, an earer verson of EDCA see [14]. Wth the excepton of [13], where the authors propose an extenson of the Banch s mode for anayzng EDCF, a these studes are smuaton based. The man contrbuton of ths paper s a nove technque for estmatng the throughput and other parameters of nterest for the contenton based MAC protocos. Our technque s based on a rgorous anayss of the drft of the Markovan mode of the system, and does not requre the decoupng assumpton of Banch. In fact, through the nsghts t yeds nto the system dynamcs, t provdes an ntutve justfcaton of Banch s smpfyng assumptons. The technque s easy to appy, and we use t to anayze DCF as we as a smpfed verson of EDCA. We now brefy sketch the key deas behnd our approach. A common feature of a the contenton based MAC protocos s the concept of back-off stage for a staton. The statons can be n dfferent back-off stages; the back-off stage for a staton depends on the number of cosons that t has encountered snce ts ast successfu transmsson and, possby, other nformaton and can be thought of as ts estmate of the current eve of contenton at a statons. The statons use ths estmate to contro ther access probabtes. The key observaton we make n ths paper s that, when the number of statons s arge, the Markov chan assocated wth the back-off process stays cose to what we ca a typca state, whch can be obtaned as the unque equbrum pont of the drft equatons assocated wth the back-off process. We can obtan qute accurate estmates of the throughput and other parameters of nterest by assumng the system to be n ths typca state at a tmes.

2 2 We fnd that the accuracy of the throughput estmates obtaned usng our technque s about the same as those obtaned usng Banch s anayss. But, n addton, we are abe to provde some key nsghts about the system dynamcs; n fact, our resuts provde a justfcaton for Banch s approxmaton, whch may be of separate nterest. The rest of the paper s organzed as foows. We provde a bref descrpton of DCF and EDCA, and dscuss some reated work, n the next secton. Our technque for performance evauaton s dscussed n the context of DCF n secton III. An extenson of our technque n the context of EDCA s dscussed n Secton V. Some concudng remarks are presented n Secton V. Due to space constrants, a technca detas and proofs are deferred to Appendx A and B. DIFS RTS DATA Source SIFS SIFS SIFS Destnaton CTS Other NAV RTS NAV CTS Defer Access Fg. 2. RTS/CTS Access Method. ACK DIFS Backoff Back off After Defer II. DCF, EDCA, AND RELATED WORK In ths secton, we provde a bref descrpton of DCF and EDCA, and dscuss some reated work n the terature. We start wth a descrpton of DCF. A. IEEE DCF The DCF s a Carrer Sense Mutpe Access wth Coson Avodance CSMA/CA MAC protoco. The coson avodance scheme of DCF s based on the bnary exponenta back-off BEB scheme [15], [16]. The DCF defnes two access mechansms for packet transmssons: basc access mechansm, and RTS/CTS access mechansm. Source Destnaton Other DIFS DATA SIFS ACK NAV DATA DIFS Backoff probem see, for exampe, [17]. A staton that wshes to send a DATA frame frst senses the channe for a DIFS duraton. If the channe s determned to be de, then a RTS frame s sent to the destnaton. Otherwse, the back-off agorthm s trggered after the end of the current transmsson and a further DIFS nterva. Upon successfu transmsson of the RTS frame, the destnaton wats for a SIFS nterva, and then sends a CTS frame back to the source. The source can start sendng the DATA frame a SIFS nterva after the recepton of the CTS frame. As n the basc access mechansm, upon successfu recepton of the DATA frame, the destnaton wats for a SIFS nterva, and then sends an ACK frame back to the source. A staton that hears ether the RTS, CTS, or DATA frame updates ts NAV based on the Duraton/ID fed of the correspondng frame see Fgure 2. The four way handshake prevents any DATA-DATA cosons that mght occur due to the hdden termna probem. Snce the RTS and CTS frames are very sma n sze, the RTS/CTS access scheme sgnfcanty reduces bandwdth oss due to cosons. Defer Access Back off After Defer Fg. 1. Basc Access Method. In the basc access mechansm see Fgure 1, any staton, before transmttng a DATA frame, senses the channe for a duraton of tme equa to the Dstrbuted Interframe Space DIFS to check f t s de. If the channe s determned to be de, the staton starts the transmsson of a DATA frame. A statons whch hear the transmsson of the DATA frame set ther Network Aocaton Vector NAV to the expected ength of the transmsson, as ndcated n the Duraton/ID fed of the DATA frame. Ths s caed the vrtua carrer sensng mechansm. The channe s consdered to be busy f ether the physca carrer sensng or the vrtua carrer sensng ndcates so, and n that case, the staton enters nto a wat perod determned by the back-off procedure to be expaned ater. Upon successfu recepton of the DATA frame, the destnaton staton wats for a SIFS nterva foowng the DATA frame, and then sends an ACK frame back to the source staton, ndcatng successfu recepton of the DATA frame. The RTS/CTS access mechansm uses a four-way handshake n order to reduce bandwdth oss due to the hdden termna The back-off procedure s mpemented by means of the back-off counter and back-off stages. Intay, upon recevng a new frame to be transmtted, the staton starts n backoff stage, wndow CW sze set to CW mn. Foowng an unsuccessfu transmsson attempt coson, the back-off stage s ncremented by 1 and the contenton wndow sze s doubed unt the maxmum sze of the contenton wndow, CW max, s reached, after whch the back-off stage and the contenton wndow sze reman unchanged on subsequent cosons. The back-off wndow sze as we as the back-off stage are set back to ther nta vaues of CW mn and after a successfu transmsson attempt or f the retry count mt for the frame s reached. At the start of each back-off stage, the back-off counter s set to an nteger chosen unformy at random between zero and the vaue CW 1 of the contenton wndow for the current back-off stage. The back-off counter s decremented by 1 n every subsequent sot, as ong as the channe s sensed de n that sot. Here, a sot s an nterva of fxed duraton specfed by the protoco. If a transmsson by some other staton s detected, then the staton freezes ts backoff counter, and resumes ts count from where t eft off after the end of the transmsson pus an addtona DIFS nterva.

3 3 When the back-off counter reaches, the staton transmts. The scheme descrbed above treats a the statons equay. We now brefy descrbe the enhanced dstrbuted channe access EDCA mechansm, whch s an extenson of the DCF mechansm, and ams at provdng servce dfferentaton. B. IEEE 82.11e EDCA The EDCA has been desgned from the perspectve of provdng QoS n WLANs. The EDCA defnes four dfferent ACs, each mantanng ts own channe access functon an enhanced varant of the DCF. The man dfferences between the EDCA and DCF are: 1 The mnmum specfed de duraton tme, caed the arbtraton nter frame space AIFS, s not a constant vaue unke the DIFS n the case of DCF. 2 The contenton wndow mts, CW mn and CW max, are dfferent for dfferent ACs. In secton IV we consder a heterogeneous settng smar to the one as under the above EDCA mechansm. C. Reated Work One of the earest anayses of the throughput of DCF was carred out n [4] usng a greaty smpfed back-off mode, namey that the back-off counter vaue s geometrcay dstrbuted wth constant parameter p, rrespectve of the current back-off stage of the staton. A more reastc mode was proposed n the semna paper of Banch [3]. Here, the evouton of the back-off stage at each node s descrbed by a Markov process; the Markov chans at dfferent nodes evove ndependenty, but n an envronment specfed by the coson probabty p for any transmsson attempt. The parameter p s a constant derved from the mean transmsson probabty n the assocated Markov chans. Ths formuaton eads to a fxed pont equaton for p. Note that the mode s anaogous to mean-fed modes n statstca physcs; the ony nteracton between the Markov processes at dfferent nodes s through the parameter p, whch represents a mean vaue of the envronment. It s not a goa n [3] to provde a rgorous justfcaton of the mean-fed assumpton. The assumpton s justfed through smuatons, whch show that the mode predctons are qute accurate. Severa subsequent studes have but on the work of Banch. In [7], the authors obtan smar fxed pont equatons usng the same decoupng assumpton but wthout the Markovan assumptons of Banch; extensons of ths fxed pont formuaton are studed n [8]. In [6], the authors present As n [3] and majorty of the reated terature, n our anayss, we gnore the facts that the back-off procedure s not nvoked mmedatey after a successfu transmsson or durng the transmsson of the frst data packet, and the back-off counter s not decremented f the channe s sensed to be busy. For a more accurate mode of the back-off procedure, we refer the reader to [18]. The parameters acw mn and acw max depend on the physca ayer. In order to avod confuson arsng from the superfcay smar termnoogy, we emphasze that the fxed ponts we tak of n ths work are dfferent from the fxed ponts n [3], [7], [8]. Ther fxed ponts are for the 1- dmensona coupng parameter p; our fxed ponts are for the n-dmensona state descrptor n a jont Markovan representaton of the back-off stages at a n statons. The detas are provded n the next secton. an approxmate deay anayss based on Banch s mode, and aso extend the mode to account for channe errors. Recenty, Proutere et a. [?] have shown that the mean fed anayss of Banch s asymptotcay n the nfnte staton mt accurate. In partcuar, they have used deas from the theory of propagaton of chaos to show that Banch type decoupng hods aysmptotcay as number of statons s aowed to ncrease to nfnty. Severa works have evauated the performance of EDCF, an earer verson of EDCA see [14]. Most of these have empoyed smuaton [9], [1], [11], [12]. An excepton s [13], where the authors use an extenson of Banch s mode to anayze the performance of IEEE 82.11e MAC protoco. More recenty, the performance of EDCA has been anayzed n [2], [21], usng theoretca modes based on Banch type assumptons. Our approach dffers fundamentay from the work descrbed above n that we do not make the decoupng assumpton ntroduced by Banch, and common to a of them except [?]. Instead, startng from a Markov chan descrpton that expcty takes nto account the nteractons between statons, we show that n a arge system, namey one wth a arge number of statons, the Markov chan converges to a typca state. Thus, one can approxmate the coson probabty seen by any snge staton by that seen n the typca state. Our anayss therefore provdes a rgorous justfcaton for Banch s mode, whch has been the bass of much subsequent work. In addton, t provdes an aternatve approach to performance anayss of MAC protocos; performance measures of nterest can be derved drecty from anayss of the typca state. We vadate ths approach by showng that the performance predctons thus obtaned are cose to those seen n smuatons. Fnay, we pont out that we focus on DCF and EDCA protocos n ths paper because they are key to be the two most wdey depoyed wreess MAC protocos n the near future; however, we do not specfcay advocate ther use. Severa works see, for exampe, [22], [23], [24], and the references theren have dentfed the mtatons of these protocos, and proposed aternatve MAC protcos that can provde better performance. The technques deveoped n ths paper are very genera, and can be apped to evauate the performance of these aternatve MAC protocos as we. III. PERFORMANCE EVALUATION OF IEEE DCF In ths secton, we present a performance anayss of DCF. We start wth a descrpton of our mode. A. The Mode We consder a wreess LAN wth n statons empoyng the IEEE DCF. Every staton can hear every other staton n the network,.e., there are no hdden statons. Our dscusson covers both ad hoc networks, where there s no centra access pont AP through whch a the traffc must pass, as we as ntrastructure based networks, where an AP connects the wreess network wth the wred nfrastructure. In order to smpfy the anayss, we assume, n common wth most

4 4 reated work, that a statons aways have a packet to send. The throughput obtaned under such saturaton condtons s commony referred to as the saturaton throughput. In some cases see, for exampe, [25], t can be shown that the queues at a the nodes are stabe f the arrva rate at each node s ess than the saturaton throughput. s We make the foowng addtona assumptons: A1 The back-off duratons are geometrcay dstrbuted,.e., when a staton s n back-off stage, t makes a transmsson attempt n the next sot wth a probabty p. In order to mantan the same average watng tme as n the IEEE DCF, we set p = 2 W, where W 1 s the contenton wndow sze n back-off stage. A2 The back-off stage s reset to ony after a successfu transmsson,.e., the retry count mt, as defned n Secton II-A, s nfnte. Ths assumpton s not necessary for our anayss, but smpfes the exposton consderaby. A statons use the same back-off parameters. There are M 1 back-off stages, abeed to M. We adopt a dscrete tme mode ndexed by the sot number t. To avod confuson, note that the term sot n our usage refers to a dfferent quantty from the sot n the IEEE protoco descrpton. We use the term to denote the tme perod at the end of whch statons may modfy ther back-off counters. In partcuar, the duraton of a sot s not a fxed physca ayer parameter, but vares dependng on whether t represents an de sot, a successfu transmsson or a coson. The state of the system at tme t can be represented by a vector X n t = X n t,..., X nm t denotng the number of statons n each of the back-off stages through M. It s easy to see that X n t, t =, 1,... forms an rreducbe and aperodc Markov chan on the state space { } M S n x Z M1 : x = n; x for a. = In prncpe, one coud sove for the statonary dstrbuton of X n t and thereby obtan parameters of nterest about the system. However, the number of states, n M1, s too arge to make ths feasbe for systems of practca nterest. The key nsght we provde n ths paper s that, when n s arge and exact computaton expensve, the Markov chan X n t stays cose to what we ca a typca state. Moreover, accurate estmates of varous parameters such as throughput can be obtaned by assumng that X n t s n ths typca state at a tmes. We remark for purposes of comparson that Banch [3] modes the system as a Markov chan wth typcay an even arger state space of sze M 1 n by consderng the back-off stage at each staton. The anayss s smpfed by repacng ths n-dmensona Markov chan by n 1-dmensona Markov chans wth M1 states each whch are assumed to be condtonay ndependent, condtona on the coson probabty p. We do not make any such ndependence assumptons. We now proceed wth the anayss of the Markov chan X n t. Let us ook at the expected change n X n t over one tme sot. For x n S n, et f n x n E{X n t 1 X n t X n t = x n } = x n, :x n S n P n where P n x n s the probabty of makng a transton from x n to x n over one tme sot. We now compute f n x n for {, 1,..., M}. Frst consder =. Let Ix n M = 1 p xn, where p denotes the transmsson probabty for a staton n backoff stage. Note that Ix n s the probabty of an de sot when the system s n state x n. The foowng events can resut n a change n the number of statons n back-off stage : A successfu transmsson by a staton n back-off stage, {1, 2,..., M}, resutng n an ncrease n the number of statons n back-off stage. An unsuccessfu transmsson attempt by one or more statons n back-off stage, resutng n a decrease n the number of statons n back-off stage. For the former event to occur, the staton tsef must transmt and no other staton n the network shoud transmt; ths has probabty p 1 p Ix n. Notng that there are x n statons n the back-off stage to choose from, and summng over, we obtan M x n Ix n p 1 1 p =1 to be the expected ncrease n the number of statons n backoff stage due to successfu transmssons by statons n other back-off stages. Lkewse, a node n the back-off stage transmts unsuccessfuy wth probabty p 1 Ixn 1 p and moves to back-off stage 1. Therefore, x n p 1 Ixn 2 1 p s the expected decrease n the number of statons n the back-off stage due to unsuccessfu transmsson attempts by statons n the back-off stage. Combnng Eqs.1 and 2, we obtan M f n x n = = x n p Ix n 1 p x n p. 3 Next, et {1, 2,..., M 1}. We now need to consder the foowng events: A transmsson attempt by a staton n back-off stage. An unsuccessfu attempt by a staton n back-off stage 1. A staton n back-off stage attempts to transmt wth probabty p, foowng whch, t ether moves to back-off stage successfu transmsson or to back-off stage 1 coson. Thus, the expected decrease n the number of statons n backoff stage at tme t s x n p. 4 A staton n back-off stage 1 transmts wth probabty p 1 and moves to back-off stage f t suffers a coson,

5 5.e., f one or more others staton n the network aso transmt, whch happens wth probabty 1 Ixn 1 p 1. Thus, x n 1 p 1 1 Ixn. 5 1 p 1 s the expected ncrease n the number of statons n back-off stage due to unsuccessfu transmsson attempts by statons n back-off stage 1. Combnng Eqs.4 and 5, we obtan f n x n = x n 1 p 1 1 Ixn 1 p 1 x n p 6 for {1, 2,..., M 1}. Fnay, et = M. In ths case, we need to consder the foowng events: A successfu transmsson attempt by a staton n back-off stage M. An unsuccessfu transmsson attempt by a staton n back-off stage M 1. A staton n back-off stage M transmts wth probabty p M and, f no other staton n the network transmts, an event of probabty Ixn 1 p M, then the staton moves to back-off stage ; otherwse t stays n the back-off stage M. The expected decrease n the number of statons n back-off stage M at tme t due to a successfu transmsson s thus: x n M p M Ix n. 7 1 p M A staton n back-off stage M 1 transmts wth a probabty p M 1 and, f at east one other staton n the network aso transmts, an event of probabty 1 Ixn 1 p M 1, then the staton enters nto back-off stage M. The expected ncrease n the number of statons n back-off stage M at tme t due to cosons s thus «x n M 1 pm 1 1 Ixn. 8 1 p M 1 Combnng Eqs.7 and 8, we obtan «f n M xn = x n M 1 pm 1 1 Ixn x n Ix n M 1 p pm. M 1 1 p m 9 Coectng Eqs.3, 6, and 9, at one pace, we have f n x n = MX x n Ix n p x n p, 1 1 p = «x n p, < < M, f n x n = x n 1 p 1 1 Ixn 1 p 1 11 «f n M xn = x n M 1 pm 1 1 Ixn x n Ix n M 1 p pm. M 1 1 p M 12 Let B n {x R M1 : M x = n; x }, and E B n /n. Let f n : R M1 R M1 be the functon wth components f n specfed by Eqs.3, 6, and 9. It s essentay the one-step drft of the Markov chan X n t. We have so far defned the functon f n x for x S n ony; we = now extend the defnton of f n to x B n by usng the same equatons on the extended doman. In Appendx A, we anayze an appropratey scaed verson, Y n t = X n nt /n, of the process X n t for n = 1, 2,..., and show that for a t, t satsfes: m sup n s t Y n s Y s = a.s., where Y t s a determnstc process gven by the unque souton of the dfferenta equaton dy t dt = FY t for t, wth nta condton y = m n Y n = X/n, where Fx = m n f n nx for x E. In words, we prove a functona aw of arge numbers mt theorem for the process Y n. We aso show that the error nvoved n approxmatng X n t wth ny t/n s amost surey On β for a β > 1/2. In Appendx B, we show that the equaton Fx = has a unque souton If M = 1, we can further show that Y t converges to x from a possbe nta states. We conjecture that such a resut hods for a M as observed n our smuatons. In vew of the resuts n Appendx A and B, we can expect that, for arge t, the process X n t remans cose to the unque pont x n B n satsfyng f n x n =, whch w henceforth be referred to as the equbrum pont of the system. B. Throughput Cacuaton We now estmate the throughput of IEEE DCF, assumng that the system stays cose to ts equbrum pont x n at a tmes. Let T The normazed throughput of the system. P c The condtona coson probabty. I The probabty of an de sot n state x n. P The payoad duraton. T c The average tme the channe s sensed busy durng a coson. T s The average tme the channe s sensed busy because of a successfu transmsson. σ The duraton of an de sot. Note that some of the above defned quanttes may vary wth n. For the sake of brevty, we do not make expct ths dependence. To cacuate the throughput, observe that a staton n backoff stage, transmts wth a probabty p, and the transmsson s successfu f no other staton n the network transmts, an event of probabty I 1 p. Snce there are x n statons n back-off stage, the probabty that a staton n back-off stage transmts successfuy s x n p I 1 p. In ths paper, we consder the payoad duraton to be fxed. Varabe payoad duraton can aso be anayzed as n [3].

6 6 TABLE I IEEE DSSS PHY PARAMETER SET [26] AND OTHER PARAMETERS USED TO OBTAIN NUMERICAL RESULTS PARAMETER VALUE Basc Bt Rate BBR 1Mb/s Bt Rate BR 11Mb/s PHY Header PH 192 bts MAC Header MH 272 bts H PH/BBR MH/BR ACK 112/BR PH/BBR RTS 16/BR PH/BBR CTS 112/BR PH/BBR Propagaton Deay δ 1µs SIFS 1µs Sot Tme σ 2µs DIFS 5µs 1 P c Smuaton Resuts Error bars of ength = 2 x Std. Devaton Banch s Mode Our Technque Number of Nodes Fg. 3. Success probabty 1 P c for M = 5 and W = Banch s Mode Our Method Smuaton.3 Summng over a possbe back-off stages, we obtan the probabty of a successfu transmsson to be M = x n p I 1 p. Snce the probabty that at east one staton transmts n a gven sot s 1 I, we have M = P c = 1 xn I p 1 p 13 1 I The normazed throughput of the system can be expressed as Expected Payoad duraton per sot T =. 14 Sot duraton The expected payoad duraton per sot s 1 I1 P c P. The expected duraton of a sot s ready obtaned consderng that, wth a probabty I a sot s de; wth a probabty of 1 I1 P c t contans a successfu transmsson, and wth a probabty of 1 IP c t contans a coson. And puggng ths s Eq.5,we obtan T = 1 I1 P c P 1 I1 PT s 1 IP c T c Iσ 15 The vaues of T c and T s depend on the access mechansm beng used. Let δ be the propagaton deay, then one can ready obtan for detas, see [3] Ts rts Tc rts Ts bas = RTS CTS H P ACK 3SIFS 4δ DIFS = RTS DIFS δ = H P ACK SIFS 2δ DIFS T bas c = H P DIFS δ 16 where T rts c correspondngy, Tc bas and Ts rts correspond- represent the T c and T s vaues for the RTS/CTS ngy, T bas s based access correspondngy, basc access mechansm, respectvey; the parameters RTS, CTS, H, ACK, DIFS, and SIF S are a physca ayer dependent. We w use the vaues of these parameters as defned n the DSSS PHY see Tabe II. Attempt Probabty 1 I Number of Nodes Fg. 4. Attempt probabty 1 I for M = 5 and W = 128. C. Performance Comparson We have performed extensve smuatons wth dfferent vaues of M and W. The smuaton resuts match extremey we wth the numerca resuts obtaned usng our technque and Banch s mode. The resuts for the RTS/CTS access mechansm wth M = 5 and W = 128 are shown n Fgures 3-5. As s evdent n these fgures error bars are barey vsbe, the varaton of resuts across varous smuaton runs s qute sma, thereby showng the hgh confdence eve of the smuaton resuts. An nterestng thng to note s that athough our technque and Banch s mode are fundamentay dfferent, they both resut n roughy the same fxed pont n terms of P c and I, and correspondngy, the estmates of throughput obtaned usng the two technques are very cose. Smar resuts have been obtaned for the basc access mechansm as we. An nterestng speca case M = 1 under whch t s possbe to cacuate the exact throughput of DCF s dscussed n Appendx C. IV. EXTENSION TO A HETEROGENEOUS SETTING The anayss presented n the prevous secton can easy be extended to a heterogeneous settng, where dfferent nodes can run wth dfferent vaues of the protoco parameters. Indeed, n Appendx D we consder a settng where mutpe access categores, each usng ts own unque set of protoco parameters, are runnng smutaneousy at each node as specfed n the EDCA mechansm. However, unke the EDCA

7 7.464 Banch s Mode Smuaton Resuts Error bars of ength = 2 x Std. Devaton Our Technque nodes can hear each other; accountng for the hdden node probem remans an mportant research chaenge. Throughput Number of Nodes Fg. 5. Throughput T for M = 5 and W = 128. mechansm, our anayss does not aow for the varabe nterframe spacng. We pan to address ths ssue n our future work. V. CONCLUDING REMARKS We studed the performance of contenton based medum access contro MAC protocos. We deveoped a nove technque for estmatng the throughput, and other parameters of nterest, of such protocos. Our technque s based on a rgorous anayss of a Markovan framework deveoped n the paper. The anayss shows that n a mtng regme of arge system szes, the stochastc evouton of the back-off stages at dfferent statons converges to a determnstc evouton; moreover, ths determnstc process has a unque fxed pont. Thus, our anayss provdes nsght nto the dynamcs of the MAC protocos, showng that they gude the system to a typca operatng pont. Ths then aows us to obtan the saturaton throughput and other performance measures of nterest wthout havng to cacuate the statonary dstrbuton of the Markov chan, whch woud be nfeasbe for systems of reastc sze. To the best of our knowedge, our technque for performance anayss of MAC protocos s the frst one of ts knd wth a quantfabe accuracy. Our resuts provde a justfcaton for the decoupng approxmaton of Banch [3]. Fnay, athough we focused on two representatve MAC protcos IEEE DCF and IEEE 82.11e EDCA, the technques deveoped n the paper are qute genera and are appcabe to a wde varety of MAC protocos. Our performance anayss s based on the assumpton that the system remans at ts equbrum pont at a tmes. A natura refnement s to consder fuctuatons around ths pont, whch w typcay be sma. A mathematca framework for studyng such fuctuatons s provded by the dffuson approxmaton a functona centra mt theorem for the Markov process. Ths s a topc for future research. Secondy, our current framework cannot fuy hande protocos ke EDCA that aow the use of dfferent nter-frame spacng as a means of achevng servce dfferentaton. It remans to extend our anayss technques to dea wth ths, and wth other forms of heterogenety. Fnay, we have assumed throughout that a ACKNOWLEDGEMENT The frst author thanks Rav Mazumdar and Ness Shroff for ther encouragement and support. REFERENCES [1] B. P. Crow, Performance evauaton of the IEEE Wreess Loca Area Network Protoco, M. S. Thess, Dept. of ECE, Unv. of Arzona, Tuscon, AZ, [2] J. Wenmer, M. Schager, A. Festag, and A. Wosz, Performance study of access contro n wreess LANs - IEEE DFWMAC and ETSI RES 1 HIPERLAN, Mobe Networks and Appcatons, vo. 2, no. 3, pp , [3] G. 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8 8 [23] J. Deng and Z. J. Haas, Dua Busy Tone Mutpe Access DBTMA: A New Medum Access Contro for Packet Rado Networks, n IEEE ICUPC, Oct [24] F. Ca, M. Cont, and E. Gregor, Dynamc tunng of the IEEE Protoco to acheve a theoretca throughput mt, IEEE/ACM Trans. on Networkng, vo. 8, no. 6, pp , 2. [25] A. Kumar and D. Pat, Stabty and throughput anayss of unsotted CDMA-ALOHA wth fnte number of users and code sharng, Teecommuncaton Systems, vo. 8, pp , [26] Wreess LAN Medum Access Contro MAC and Physca Layer PHY Specfcatons, IEEE Std , Aug [27] S. N. Ether and T. G. Kurtz, Markov Processes: Characterzaton and Convergence, John Wey and Sons, [28] T. G. Kurtz, Approxmaton of Popuaton Processes, Socety for Industra and Apped Mathematcs, [29] P. Bngsey, Convergence of Probabty Measures, Wey, New York, APPENDIX A Reca the settng of Secton III-A. For each n, et E n {k/n : k S n }, and consder the famy of stochastc process {Y n t} defned as foows: Y n t X n nt. 17 n Observe that each for each n, Y n t s just a scaed verson where the scang s both n tme as we as magntude of X n t. We aso defne another famy of stochastc process {Z n t} as foows: Z n t X nnnt, 18 n where Nt s a Posson process wth unt ntensty, ndependent of the sequence of Markov chans {X n t}. Observe that {Z n t} s a sequence of jump Markov process on E n, wth transton rates ntenstes q n k,k/n nk, for k E n. We w need the foowng assumpton: Assumpton 1: There exst postve constants c, c 1,..., c M such that p n = c /n for, 1,..., M and a n. = np n Remark 1: We note that p n and M are kept fxed n the IEEE DCF, ndependent of the number of nodes n to scae wth n to avod trvates; for exampe, f M and p n were kept fxed for a n, then as n the throughput woud drop to zero and a the nodes woud eventuay be n the back-off stage M the network. We aow for the p n wth probabty 1. The above choce of p n precudes ths possbty. Note that the way transmsson probabtes are chosen n the IEEE DCF, Assumpton 1 woud mpy that c = 2c 1 = = 2 M c M. We need some preparaton before we can state our man resut. Henceforth, we use x to denote the L 2 norm of x. For x n E n, et Ix n M 1 p n xn. = Strcty speakng, the functon I s not reay the same for dfferent n; for the sake of brevty, we w contnue to foow the above notaton. We start wth the foowng smpe resut: Lemma 1: Consder x n E n and x m E m. Suppose Assumpton 1 hods. Then, Inx n Imx m 2c M x n x m, whenever n and m are arge enough. Proof: Wthout oss of generaty, suppose m n. Usng the defnton of Ix n, and observng that for M > 1 e 2Mc M1 1 p n n 1 p m m, we have for n arge enough: Inx n M Imx m = = 1 p n nxn 1 p m mxm M 1 p n = M 1 p n = nxn nx m n xn x m 1 p n Mn xn x m e 2cM xn x m. Now we have Inx n Imx m = Inx n 1 Imxm Inx n 1 Imxm Inx n 1 e 2cM xn x m 2c M x n x m, provng the cam. The foowng coroary s an easy consequence of the proof of Lemma 1. Coroary 1: Consder x n E n and x m E m. Suppose Assumpton 1 hods. Then, Inx n 1 p n Imxm 1 p m 2c M x n x m, {, 1,..., M}, whenever n and m are arge enough. For each n, defne a functon F n x n on E n by settng F n x n = f n nx n. We have the foowng resut: Lemma 2: Suppose Assumpton 1 hods. Then the sequence {F n } s unformy bounded,.e., there exsts a constant C < such that F n x n C for a x n E n and for a n. Moreover, for x m E m, x n E n, and m, n arge enough, we have F n x n F m x m ηc, M x n x m, whenever x n x m < 1/c M, where ηc, M s a constant that depends ony on c and M. Proof: To prove that {F n } s unformy bounded, observe that n vew of Assumpton 1, we have F n x n c for F n x n M = {1,..., M 1} and for a n. Thus, we have x n c M 1 for a F n

9 9 x n E n and for a n. Now observe that nx n p n mx m p m c x n x m. Usng Eqs. 1, 11, and 12, aong wth Coroary 1, for 1 M 1 we have F n nx n x n F m x m nx n p n mx m p m nx n 1 pn 1 In x n 1 p n 1 1 pn 1 mxm 1 pm 1 mxm 1 pm 1 Im x m 1 p m 1 2c x n x m In x n nx n 1 p n 1 pn 1 mxm 1 pm 1 1 I n x n 1 pm 1 1 p n 1 = 3c 2c 2 M x n x m mx m Im x m 1 p m 1 Smary, t can be shown that F n x n F m x m 2c c M2c 2 M2c2 M2 x n x m and F n F m M M xn xm 3c 4c 2 M xn x m. Now snce F n x n F m x m M = F n x n F m x m, the resut foows by takng ηc, M = 2c 4c M4c 2 M 4c 2 M2. Remark 2: The above emma mpes that for a Cauchy sequence {x n } n E, the sequence {F n x n } s Cauchy n R M1. Defne a functon Fx on E as foows: Fx = m n F n x n, 19 where {x n } s any sequence n E satsfyng x n E n and x n x. The exstence of the mt n Eq. 19 foows from Remark 2. To prove the unqueness, et {x n } and {y n } be two sequences n E, satsfyng: x n, y n E n, and m n xn = m n yn = x. Then for n arge enough, we woud have y n x n ǫ < 1/c M, whch, n vew of Lemma 2, mpes that F n x n F n y n ηc, Mǫ, showng that m n F n x n = m n F n y n. Remark 3: The defnton of Fx and Lemma 2 mpy that Fx C for a x E. Remark 4: An aternatve, but equvaent, way of defnng the functon F coud be to frst defne for a n 1 a functon ˆF n on E by settng ˆF n x = f n nx, for x E, and then take F as the pontwse mt of the sequence of functons ˆF n x. The foowng resut s a drect consequence of Lemma 2, the defnton of Fx, and the boundedness of Fx see Remark 3. Lemma 3: Suppose Assumpton 1 hods. Then the fucnton Fx s Lpschtz contnuous,.e., there exsts a constant K < such that for a x, y E, we have Fx Fy K x y. Next, we obtan a cosed form expresson for the functon Fx: Lemma 4: Suppose Assumpton 1 hods. Then the functon Fx = F x,..., F M x, defned by Eq.19, satsfes: F x = M c x Lx x c, 2 = F x = x 1 c 1 1 Lx x c, for = 1,..., M 1, 21 F M x = x M 1 c M 1 1 Lx x M c M Lx, 22 where c = m n np n for {,..., M}, and Lx = M = e cx. Proof: Consder x E, wth ratona co-ordnates,.e., x = p /q for {, 1,..., M}, where p, q are nonnegatve ntegers. Let q = LCMq,..., q M, where LCM denotes the east commom mutpe. Observe that x E n for n = qk, where k 1 s an nteger. In vew of the defnton of Fx, we have that Fx = m k F qk x = m k fqk qkx, whch, n vew of Eqs.1-12 satsfes Eqs For an rratona x E, the resut now foows by appeang to the Lpschtz contnuty of Fx see Lemma 3. The foowng resut whch s smar to the noton of unform convergence s now an easy consequence of the defnton of F n, Eqs.1-12, and Lemma 4. Lemma 5: Suppose Assumpton 1 hods. Then there exsts a sequence {δ n } of numbers satsfyng: sup x n E n F n x n Fx n δ n and For Z M1, et β n foowng resut: m δ n =. n sup x Sn P n x. We have the Lemma 6: For n 1, et g n = βn and h n = 2 β n. Then the sequences {g n } and {h n } are unformy bounded,.e., there exsts a constant G, H < such that g n G and h n H. Proof: Consder the set of states S = { : M}. Observe that for a S, we have M for {, 1,..., M}. Thus, can take at most 2M 1 vaues, and therefore the tota number of states n S s no more than 2M 1 M1. Hence, we have β n 2M 1 M1. S Now consder the set of states S k = { : km k 1M} for k 1. A smar argument as above shows that the number of states n S k can be no more than 2kM 1 M1. Aso, note that for a jump of magntude km to occur, more than km/2 nodes must transmt durng the current

10 1 sot; the probabty of whch s smaer than n c km/2 2c km/2, 23 km/2 n km/2! for km/2 n. Thus, we have Let and β n S k 2kM 1 M1 2c km/2. km/2! D 2M 1 M1, D k 2kM 1 M1 2c km/2 km/2! for k 1. Now observng that β n = for a such that > 2n, we obtan g n = D k G <, : 2n β n k=1 provng the cam regardng {g n }. The cam regardng {h n } can be proved n a smar fashon. We are now ready to prove the amost sure convergence of the sequence {Z n t} to a determnstc process. Theorem 1: Suppose Assumpton 1 hods, m n Z n = z, and Zt satsfes: Zt = z Then for every t, we have m sup n s t FZsds for t. Z n s Zs = a.s. Proof: From Theorem 4.1 n [27, Chapter 6, pp. 327], we have that the jump Markov process Z n t wth ntenstes q n n k,k/n = np nk satsfes for t ess than the frst nfnty of jumps: Z n t = Z n n 1 Y n P n nz n sds, 24 where Y u are ndependent standard Posson processes. Now for each E n, et Ŷ u Y u u, then Ŷu s a Posson process centered at ts mean. It s we known that Ŷ u satsfes: m sup n u v n 1 Ŷ nu = a.s., for a v. 25 Now observe that for x n E n, we have F n x n = f n nx n = and therefore, Z n t = Z n n 1 Ŷ n :nx n S n P n nx n, P n nz n sds F n Z n sds. 26 Let ǫ n t sup s t Z ns Z n then usng Lemma 6, we obtan ǫ n t = sup n 1 Ŷ n s t n 1 sup Ŷ s t n 1 Y nβ n s s F n Z n udu, P n nz n udu nβ n s 27 t nβ n t. The strong aw of arge numbers apped to the ndependent ncrement process Y., the unform boundedness of the sequence {g n } = { βn } see Lemma 6, and the domnated convergence theorem, together mpy that m ǫnt X Z t www m n n n 1 sup w n β n s t wŷ ds«w = a.s. Usng the Lpschtz contnuty of F and Lemma 5, we have for t that Z nt Zt Z n z ǫ nt Z t F n Z ns FZs ds Z n z ǫ nt Z t F n Z ns FZ ns FZ ns FZs ds Z n z ǫ nt δ nt Z t K Z ns Zs ds. Appeang to Gronwa s Inequaty see, for exampe, [27, Appendx 5, pp. 498], t foows that Z n t Zt Z n z ǫ n t δ n te Kt. The resut now foows by notng that m Z n z ǫ n t δ n t =. n Our goa s to prove a resut smar to Theorem 1 for the sequence of stochastc processes {Y n t}. We w do ths by comparng {Y n t} wth {Z n t} as foows: Theorem 2: Suppose Assumpton 1 hods. Then the sequences {Y n t} and {Z n t}, defned by Eqs.17 and 18, respectvey, satsfy: m sup n s t Y n s Z n s = a.s., for a t. Proof: Let γ n t sup s t Y n s Z n s. Note that γ n t = sup X n Nns X n [ns]. s t We need to prove that m n γ n t = a.s. for a t.

11 11 Let β 1/16 and α 7/8. We have P γ n t > n β P P γ n t > n β sup sup Nns [ns] > n α s t Nns [ns] n α s t P sup Nns ns > n α 1 s t P γ n t > n β sup Nns [ns] n α s t 28 Now observng that Nns ns for s s a martnage, t foows that Nns ns for s s a submartngae. Usng the L p maxmum nequaty for p = 4, we obtan E{ sup Nns ns s t 4 } E{Nnt nt 4 }. Usng the Markov Inequaty, and observng that E{Nnt nt 4 } = nt 2n 2 t 2, we obtan pr sup Nns ns > n α 1 E{Nnt nt4 } s t n α nt 2n 2 t 2 3 n α n 3/2 t 2, 29 for arge enough n. Now we cam that for a x S n and a p, the Markov chan X n t satstes: «P sup X nq x > n 1 β X np = x 2n α e n1/8 /16 p q pn α To prove the above cam observe that for the event { } sup X n q x > n 1 β p q pn α 3 to occur, there must be at east one tme sot, out of the n α tme sots foowng the p th tme sot, n whch n 1 α β /2 = n 1/16 /2 or more nodes transmt. Snce the probabty of a node transmttng s no bgger than p n = c /n, we have that the random varabe N = Bernoun, c /n stochastcay domnates the random varabe correspondng to the number of nodes that transmt durng a tme sot. A standard appcaton of Chernoff Bound shows that: PN > n 1/16 /2 2e n 1/8 /16, and Eq.3 foows by usng the unon bound. Now observe that f { } sup Nns [ns] n α s t occurs, then the tota number of jumps upto tme t of the processes Z n t and Y n t combned, s no bgger than 2nt n α. Appeang to the unon bound once agan, we have that the second term n Eq.28 s no bgger than 2n α 2nt n α e n1/8 /16 t/n 3/2, 31 for arge enough n. Combnng Eqs.31 and 29, we obtan P γ n t > n β 12t2 t n 3/2, for arge enough n, whch mpes that P γ n t > n β <, n=1 and m n γ n t = a.s. now foows from the frst Bore- Cante Lemma. Remark 5: Usng the L p maxmum nequaty for p = 4γ, γ > 1, and makng approprate changes to the proof of Theorem V, one can show that P γ n t > n β.o. = for a β < 1/2. Thus for any β < 1/2, there exsts a correspondng nteger N β < such that γ n t < n β for n N β. Combnng the resuts n Theorems 1 and V, gves the desred resut: Theorem 3: Suppose Assumpton 1 hods, m n Y n = y, and Y t satsfes: Y t = y Then for every t, we have m sup n s t FY sds for t. 32 Y n s Y s = a.s. Remark 6: The Lpschtz contnuty of F guarantees that for a y E, there exsts a unque souton to the nta vaue probem IVP correspondng to Eq.32. Theorem 3 shows the convergence of {Y n t} to Y t, over bounded ntervas of tme. For fnte, but arge n, Remark 5 shows that the dfference between Y n t and Z n t s On β for a β < 1/2. Next, we w characterze the error nvoved n approxmatng Z n t wth Y t, foowng the approach gven n [28]. Set W n u = n 1/2 Ŷ nu and et V n t = nz n t Xt. Then, Eq.26 can be rewrtten as V n t = W n P n nz n sds nf n Z n s F n Xsds, whch suggests the foowng mtng equaton: V t W β Xsds FXsV sds, 33 where β x = m n P n nx n for x E, and {x n } satsfes: x n E n for n = 1, 2,..., and m n xn x. The exstence and unqueness of the above mt can easy be shown. Let Φ be the souton of the matrx equaton and et Φt, s = FXtΦt, s, Φs, s = I, 34 t Ut W β Xsds.

12 12 Then, we have V t = Φt, sdus. Observe that snce Ut s Gaussan wth zero mean, V t s Gaussan wth zero mean and covarance matrx where CovV t, V r = r Cx = Φt, scxsφr, s T ds, T β x. From Coroary 6, we have that sup x E Cx H <. Thus V t s we defned. Let D R M1[, = {x : [, R M1 for a t m s t xs = xt and m s t xs exsts},.e., the space of rght contnuous functons havng eft mts. Henceforth, we w use the symbo to denote the convergence n dstrbuton n D R M1[,, or equvaenty, weak convergence n P D R M1[, - the set of Bore probabty measures on D R M1[,. For the sake of defnteness, the metrc used on P D R M1[, can be assumed to be the Prohorov metrc see, for exampe, [27, Chapter 3]; and the metrc used on D R M1[, coud be the one specfed n [27, Chapter 3] that nduces the Skorohod topoogy on D R M1[,. For a detaed dscusson of these metrcs and reated concepts, we refer the reader to [29]. The foowng theorem characterzes the error nvoved n approxmatng Z n t wth Xt: Theorem 4: Let V n t and V t be as above, then V n t V 1n t V 2n t, where V 1n V and V 2n t = Ot/ n. Remark 7: A consequence of the above resut s that for arge n, X n t can be we approxmated by ny t/n n 1/2 V t/n. In vew of Remark 5, the error n such an approxmaton s amost surey bounded by On β for any β > 1/2. Aso, snce V t has a fnte varance for a t, the error n approxmatng X n t wth ny t/n s aso amost surey bounded by On β for any β > 1/2. where Proof: We have V n t = V 1n t n 1/2 Ŷ n n P n nz n sds nf n Z n s F n Xsds = V 1n t V 2n t, n 1/2 Ŷ n n FXsV n sds P n nz n sds and V 2n t [ nfzn s FXs FXsV n s ] where Fx s gven by Eqs We woud now ke ds, to further nvestgate the behavor of Y t for arge t. In n FXs F n Xs ds n F n Z n s FZ n s ds. Usng Eqs.1-12, Eqs.2-22, and notng np n c K see Remark 1, t foows that there exsts a constant K 1 < such that for a n arge enough, we have sup x E n F n x Fx K 1 /n. Thus V 2n t 2K 1 t/ n = Ot/ n. Now turnng to V 1n t, et U n t n 1/2 Ŷ n n P n nz n sds, and e n t V 2n t [ n FZn s FXs FXsV n s ] ds Usng the resuts n [27, Chapter 4], t can be shown that U n U, wth U as above. Usng the Lpschtz contnuty of F, we have V n t U n t and hence usng Gronwa s nequaty sup s t K V n s ds, V n s sup U n s e Kt. s t Snce U n U and U s contnuous, t foows that sup s t U n s sup s t Us, and hence the V n are stochastcay bounded on bounded ntervas. Furthermore, t s easy to see from Eqs.2-22 that F s contnuous and bounded, whch together wth the fact that V n are stochastcay bounded on bounded ntervas mpes that e n. Wth Φ as above, we have V nt = U nt e nt Z t Φt, s FXsU ns e nsds. Fnay, notng that the mappng J : D R M1[, D R M1[, gven by Jθt = θt Φt, s FXsθsds s contnuous, the resut foows from the contnuous mappng theorem see, for exampe, [27, Chapter 3, pg. 13]. APPENDIX B In the prevous secton, we proved the convergence of the sequence of stochastc processes {Y n t} to the determnstc process Y t satsfyng: Y t = Y FY sds for t,

13 13 partcuar, we woud ke to determne whether the vector dfferenta equaton d Y t = FY t, 35 dt has an equbrum pont. Supposng t does, we woud ke to fnd out whether that equbrum pont s unque. If the equbrum pont does exsts and s unque, we woud ke to determne f the process Y t started from an arbtrary nta state woud converge to the equbrum pont. Exstence of Equbrum Ponts In ths secton, we w prove that the dfferenta equaton specfed by 35 has at east one equbrum pont. The ssue of the unqueness w be deat wth n the next secton. Defne a functon fx on E as foows: fx = x Fx, for x E. From Eqs.2-22, t s easy seen that the functon fx maps E nto tsef. Snce E s a compact subset of R M1, Brouwer s fxed pont theorem guarantees the exstence of at east one fxed pont of f: Proposton 1: The fucton f has at east one fxed pont n E. Remark 8: Note that any fxed pont of f s an equbrum pont of the vector dfferenta equaton specfed by 35. To see ths, suppose x E s a fxed pont of f. Then fx = x, mpyng that Fx = ; thus showng that x s ndeed an equbrum pont of the vector dfferenta equaton specfed by 35. Smary, we have that any equbrum pont of the vector dfferenta equaton specfed by 35 s a fxed pont of f. Unqueness of Equbrum Pont We w now estabsh the unqueness of the equbrum pont: Proposton 2: The vector dfferenta equaton specfed by 35 has a unque equbrum pont. Proof: Let us suppose that the vector dfferenta equaton specfed by 35, has more than one equbrum ponts. Then the functon f must have more than one fxed ponts. Let x and y be two dfferent fxed ponts of f. Then, n vew of Eqs.2-22, we have that x must satsfy: x c = M c x Lx, 36 = x c = x 1 c 1 1 Lx, for = 1,..., M 1, 37 x M c M Lx = x M 1 c M 1 1 Lx, 38 and y must satsfy a smar set of equatons. Now the foowng possbtes can arse: x 1 Lx = Ly. In ths case, we have x 1 = y y 1 for a {1, 2,..., M}. Snce M = x = M = y = n, we have x = y, whch contradcts our nta assumpton that x y. x x 1 < 2 Lx > Ly. In ths case, we have a {1, 2,..., M}. If x y, then x < y, for a {1, 2,..., M}. However, ths s not possbe snce M = x = M = y = n. Hence, we must have x > y. Now et k mn { : x < y } {,1,...,M} = y y 1 for Aso, et a x y. From the defnton of k, t foows that a for {, 1,..., k 1}, and a < for {k, k 1,..., M}. Snce M = x = M = y = n, we have M = a =. In partcuar, we have k 1 M a = =k Usng the defnton of Lx and Ly, we have that M M M Lx = e cx = e cya = Ly = k 1 = Ly = < Lye c k 1 = M e ca e ca a =k P k 1 = a e c P m k =k a = Lye c k 1 c k P k 1 = a < Ly, = e ca whch contradcts our nta assumpton that Lx > Ly. 3 Lx < Ly. In ths case aso, one arrves at a contradcton, ke n the prevous case. Snce one of the above cases must occur, we have proved that f can have at most one fxed pont, and, n vew of Proposton 2, the resut foows. Convergence to Equbrum Pont In ths secton, we w nvestgate whether the process Y t, started from any arbtrary nta state n E, converges to the unque equbrum pont. We have the foowng resut for M = 1: Proposton 3: Suppose M = 1. Then the process Y t started from any arbtrary nta state n E, converges to the unque equbrum pont ŷ satsfyng Fŷ =. Proof: For M = 1, the set of equatons gven by 35 smpfy to dy t = c 1 y 1 tly y tc 1 Ly, 39 dt dy 1 t = y tc 1 Ly y 1 tc 1 Ly. 4 dt Now ŷ satsfes ŷ c 1 Lŷ = c 1 y 1 Lŷ. Observe that for a yt wth y t > ŷ, we have dyt dt = dy1t dt <. Now consder the Lyapunov functon λyt = yt ŷ T yt ŷ. It s straghtforward to show that d dt λyt = yt ŷt d dt yt < for yt ŷ, whch mpes that m t yt = ŷ.