Fixed Point Analysis of Single Cell IEEE e WLANs: Uniqueness, Multistability and Throughput Differentiation

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1 Fxed Pont Analyss of Sngle Cell IEEE 82.e WLANs: Unqueness, Multstablty and Throughput Dfferentaton Venkatesh Ramayan ECE Department Indan Insttute of Scence Bangalore, Inda Anurag Kumar ECE Department Indan Insttute of Scence Bangalore, Inda Etan Altman INRIA Sopha-Antpols France ABSTRACT We consder the vector fxed pont equatons arsng out of the analyss of the saturaton throughput of a sngle cell IEEE 82.e wreless local area network wth nodes that have dfferent back-off parameters, ncludng dfferent Arbtraton InterFrame Space (AIFS) values. We consder balanced and unbalanced solutons of the fxed pont equatons arsng n homogeneous and nonhomogeneous networks. We are concerned, n partcular, wth () whether the fxed pont s balanced wthn a class, and () whether the fxed pont s unque. Our smulatons show that when multple unbalanced fxed ponts exst n a homogeneous system then the tme behavour of the system demonstrates severe short term unfarness (or multstablty). Implcatons for the use of the fxed pont formulaton for performance analyss are also dscussed. We provde a condton for the fxed pont soluton to be balanced wthn a class, and also a condton for unqueness. We then provde an extenson of our general fxed pont analyss to capture AIFS based dfferentaton; agan a condton for unqueness s establshed. An asymptotc analyss of the fxed pont s provded for the case n whch packets are never abandoned, and the number of nodes goes to. Fnally the fxed pont equatons are used to obtan nsghts nto the throughput dfferentaton provded by dfferent ntal back-offs, persstence factors, and AIFS, for fnte number of nodes, and for dfferentaton parameter values smlar to those n the standard. Categores and Subect Descrptors C.2.5 [Computer-Communcaton Networks]: Local and Wde-Area Networks Access schemes; I.6.4 [Smulaton and Modelng]: Model Valdaton and Analyss General Terms Performance Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. SIGMETRICS 5, June 6, 25, Banff, Alberta, Canada. Copyrght 25 ACM /5/6...$5.. Keywords Performance of Wreless LANs, Short term Unfarness, QoS n Wreless LANs, EDCF Analyss. INTRODUCTION A new component of the IEEE 82.e medum access control (MAC) s an enhanced DCF (EDCF), whch provdes dfferentated channel access to packets by allowng dfferent backoff parameters (see [2]). Several traffc classes are supported, the classes beng dstngushed by dfferent back-off parameters. Thus, whereas n the legacy DCF all nodes have a sngle back-off state machne all wth the same back-off parameters (we say that the nodes are homogeneous), n EDCF the nodes can have multple back-off state machnes wth dfferent parameters, and hence are permtted to be nonhomogeneous. Ths paper s concerned wth the saturaton throughput analyss of sngle cell networks wth nonhomogeneous nodes. We lmt our study to the case n whch each node has one EDCF queue; the further generalsaton to multple traffc classes per node can also be done usng the same technques but s not reported here for lack of space. Thus n the nonhomogeneous case our analyss s applcable to a sngle cell ad hoc network of IEEE 82.e nodes (sngle cell meanng that all nodes are wthn control channel range of each other), wth each node offerng traffc of a sngle class. We consder an deal channel (wthout capture, fadng or frame error) and assume that packets are lost only due to collson of smultaneous transmssons. Much work has been reported on the performance evaluaton of EDCF to support dfferentated servce. Most of the analytcal work reported has been based on a decouplng approxmaton proposed ntally by Banch ([3]). Whle keepng the basc decouplng approxmaton, n [] Kumar et al. presented a sgnfcant smplfcaton and generalsaton of the analyss of the IEEE 82. back-off mechansm. Ths analyss led to a certan one dmensonal fxed pont equaton for the collson probablty experenced by the nodes n a homogeneous system (.e., one n whch all the nodes have the same back-off parameters). In ths paper we consder multdmensonal fxed pont equatons for a homogeneous system of nodes, and also for a nonhomogeneous system of nodes. The nonhomogenety could arse due to dfferent ntal back-offs, or dfferent back-off multplers, or dfferent amounts of tme that nodes wat after a transmsson before restartng ther back-off counters (.e., the AIFS (Arbtra-

2 ton InterFrame Space) mechansm of IEEE 82.e). We consder balanced solutons of the resultng multdmensonal fxed pont equatons (.e., solutons n whch all the coordnates are equal), and also unbalanced fxed ponts. The man contrbutons of ths paper are the followng:. We provde examples of homogeneous systems n whch, even though a unque balanced fxed pont exsts, there can be multple unbalanced fxed ponts, thus suggestng multstablty. We demonstrate by smulaton that, n such cases, sgnfcant short term unfarness can be observed and the unque balanced fxed pont fals to capture the system performance. 2. Next, n the case where the backoff ncreases multplcatvely (as n IEEE 82.), we establsh a smple suffcent condton for the unqueness of the soluton of the multdmensonal fxed pont equaton n the homogeneous and the nonhomogeneous cases. 3. We perform an analytcal study of the dfferentaton provded by each of the three mechansms that we model. We then provde an asymptotc analyss of the servce dfferentaton (as the number of nodes become large), and also some approxmate results for a fnte number of nodes. A survey of the lterature: There has been much research actvty on modelng the performance of IEEE 82. and IEEE 82.e medum access standards. The general approach has been to extend the decouplng approxmaton ntroduced by Banch ([3]). Wthout modelng the AIFS mechansm, the extenson s straghtforward. Only the ntal back-off, and the back-off multpler (persstence factor) are modeled. In [4], [5] and [6], such a scheme s studed by extendng Banch s Markov model per traffc category. In ths paper, n Secton 3, we wll provde a generalsaton and smplfcaton of ths approach. We wll provde examples where nonunque fxed ponts can exst, the consequences of such nonunqueness, and also condtons that guarantee unqueness. AIFS technque s a further enhancement n IEEE 82.e that provdes a sort of prorty to nodes that have smaller values of AIFS. After any successful transmsson, whereas hgh prorty nodes (wth AIFS = DIFS) wat only for DIFS (DCF Interframe Space) to resume countng down ther backoff counters, low prorty nodes (wth AIFS > DIFS) defer the ntaton of countdown for an addtonal AIFS DIFS slots. Thus a hgh prorty node decrements ts back-off counter faster than a low prorty node and also has fewer collsons. Among the approaches that have been proposed for modelng the AIFS mechansm (for example, [7], [8], [9], [], [] and [2]) the ones n [] and [2] come much closer to capturng the servce dfferentaton provded by the AIFS feature. In [] the authors propose a Markov model to capture both the dfferent back-off wndow expanson approach and AIFS. AIFS s modelled by expandng the state-space of the Markov chan to nclude the number of slots elapsed snce the prevous transmsson attempt on the channel. In [2] the authors observe that the system exsts n states n whch only nodes of certan traffc classes can attempt. The approach s to model the evoluton of these states as a Markov chan. The transton probabltes of ths Markov chan are obtaned from the assumed, decoupled attempt probabltes. Ths approach yelds a fxed pont formulaton. Ths s the approach we wll dscuss n Secton 6. Relaton of the exstng lterature to our work: We note that the analyses n [] and [2] are based on Banch s approach to modelng the resdual back-off by a Markov chan. In ths paper, we have extended the smplfcaton reported n [] (whch was for a homogeneous system of nodes) to nonhomogeneous nodes wth dfferent backoff parameters and also AIFS based prorty schemes. Thus, n our work, we have provded a smplfed and ntegrated model to capture all the essental servce dfferentaton mechansms of IEEE 82.e. In the prevous lterature, t s assumed that the collson rate experenced by a node of each traffc category s constant over tme. There appears to have been no attempt to study the phenomenon of short term unfarness n the fxed pont framework. Also, all the exstng work assumes that the collson probabltes of all the nodes of a gven traffc category are the same. Thus there appears to have been no earler work on studyng the possblty of unbalanced solutons of the fxed pont equatons. In addton, the possblty of nonunqueness of the soluton of the fxed pont equatons arsng n the analyses seems to have been mssed n the earler lterature. In our work, we study the fxed pont equatons for IEEE 82.e networks and take nto account all these possbltes. Outlne of the paper: In Secton 2 we revew the generalsed back-off model that was frst presented n []. In Secton 3 we develop the multdmensonal fxed pont equaton for the homogeneous and nonhomogeneous cases, and obtan the necessary and suffcent condtons satsfed by the solutons to the fxed pont equatons. We provde examples n Secton 4 to show that even n the homogeneous case there can exst multple unbalanced fxed ponts and show the consequence of ths. In Secton 5., we analyse the fxed pont equatons for a homogeneous system of nodes and obtan a condton for the exstence of only one fxed pont. In Sectons 5.2 and 6, we extend the analyss to nonhomogeneous system of nodes, wth dfferent back-off parameters. In Secton 7 we provde an analytcal study of the servce dfferentaton provded by the varous mechansms. Secton 8 concludes the paper and dscusses future work. The proofs of all lemmas and theorems, f not n the paper, are provded n [8]. 2. THE GENERALISED BACK-OFF MODEL There are n nodes, ndexed by, n, each wth one EDCF queue. We adopt the notaton n [] whose authors consder a generalsaton of the back-off behavour of the nodes, and defne the followng back-off parameters (for node ) K := At the (K + )th attempt ether the packet beng attempted by node succeeds or s dscarded b,k := The mean back-off (n slots) at the kth attempt for a packet beng attempted by node, k K Defnton 2.. A system of n nodes s sad to be homogeneous, f the back-off parameters of the nodes K, b,k, k K are the same for all, n. A system of nodes s called nonhomogeneous f the back-off parameters of the nodes are not dentcal.

3 Remark: IEEE 82.e permts dfferent backoff parameters to dfferentate channel access obtaned by the nodes n an attempt to provde QoS. The above defntons capture the possblty of havng dfferent CW mn and CW max values, dfferent exponental back-off multpler values and even dfferent number of permtted attempts. For ease of dscusson and understandng, we wll postpone the topc of AIFS untl Secton 6. Hence n the dscussons up to Secton 5.2, all the nodes wat only for a DIFS after a busy channel. It has been shown n [] that under the decouplng assumpton, ntroduced by Banch n [3], the attempt probablty of node (condtoned on beng n back-off) for gven collson probablty γ s gven by, G (γ ) := + γ + + γ K b, + γ b, + + γ K b,k () Remark: When the system s homogeneous then we wll drop the subscrpt from G ( ), and wrte the functon smply as G( ). 3. THE FIXED POINT EQUATION It s mportant to note that n the present dscusson all rates are condtoned on beng n the back-off perods;.e., we have elmnated all duratons other than those n whch nodes are countng down ther back-off counters (see []). Ths suffces to obtan the collson probabltes. Now consder a nonhomogeneous system of n nodes. Let γ be the vector of collson probabltes of each node. Wth the slotted model for the back-off process and the decouplng assumpton, the natural mappng of the attempt probabltes of other nodes to the collson probablty of a node s gven by γ = Γ (β, β 2,..., β n) = n =, ( β ) where β = G (γ ). We can now expect that the equlbrum behavour of the system wll be charactersed by the solutons of the followng system of equatons. For n, γ = Γ (G (γ ),, G n(γ n)) We wrte these n equatons compactly n the form of the followng multdmensonal fxed pont equaton. γ = Γ(G(γ)) (2) Snce Γ(G(γ)) s a composton of contnuous functons t s contnuous. We thus have a contnuous mappng from [, ] n to [, ] n. Hence by Brouwer s fxed pont theorem there exsts a fxed pont n [, ] n for the equaton γ = Γ(G(γ)). Consder the th component of the fxed pont equaton,.e., γ = ( G (γ )) or equvalently, ( γ ) = n, n, ( G (γ )) Multplyng both sdes by ( G (γ )), we get, ( γ )( G (γ )) = ( G (γ )) n Thus a necessary and suffcent condton for a vector of collson probabltes γ = (γ,, γ n) to be a fxed pont soluton s that, for all n, n ( γ )( G (γ )) = ( G (γ )) (3) = where the rght-hand sde s seen to be ndependent of. Defne F (γ) := ( γ)( G (γ)). From Equaton 3 we see that f γ s a soluton of Equaton 2, then for all,,, n, F (γ ) = F (γ ) (4) Notce that ths s only a necessary condton. For example, n a homogeneous system of nodes, the vector γ such that γ = γ for all n, satsfes Equaton 4 for any γ, but not all such ponts are solutons of the fxed pont Equaton 2. Defnton 3.. We say that a fxed pont γ (.e., a soluton of γ = Γ(G(γ))) s a balanced fxed pont f γ = γ for all, n; otherwse, γ s sad to be an unbalanced fxed pont. Remarks 3... It s clear that f there exsts an unbalanced fxed pont for a homogeneous system, then every permutaton s also a fxed pont and hence, n such cases, we do not have a unque fxed pont. 2. In the homogeneous case, by symmetry, the average collson probablty at each node must be the same for every node. If the collson probabltes correspond to a fxed pont (see 3, next), then ths fxed pont wll be of the form (γ, γ,, γ) where γ solves γ = Γ(G(γ)) (snce Γ ( ) = Γ( ) and G ( ) = G( ) for all n). Such a fxed pont of γ = Γ(G(γ)) s guaranteed by Brouwer s Fxed Pont. The unqueness of such a balanced fxed pont was studed n []. We reproduce ths result n Theorem There s, however, the possblty that even n the homogeneous case, there s an unbalanced soluton of γ = Γ(G(γ)). By smulaton examples we observe n Secton 4 that when there exst unbalanced fxed ponts, the balanced fxed pont of the system does not characterse the average performance, even f there exsts only one balanced fxed pont. In Secton 5., we provde a condton for IEEE 82. type nodes wth geometrc backoff under whch there s a unque balanced fxed pont and no unbalanced fxed pont for a homogeneous system of nodes. In such cases, t s now well establshed, that the unque balanced fxed pont accurately predcts the saturaton throughput of the system. 4. For the homogeneous case the back-off process can be exactly modeled by a postve recurrent Markov chan (see []). Hence the attempt and collson processes wll be ergodc and, by symmetry, the nodes wll have equal attempt and collson probabltes. In such a stuaton the exstence of multple unbalanced fxed ponts wll suggest short term unfarness or short term multstablty. We wll observe ths phenomenon n Secton 4.

4 5. Consder a system of homogeneous nodes havng unbalanced solutons for the fxed pont equaton γ = Γ(G(γ)) (.e., there exsts, such that γ γ ), then from Equaton 4, we see that F (γ ) = F (γ ), or the functon F s many-to-one. Hence for a homogeneous system of nodes, f the functon F s one-to-one then there cannot exst unbalanced fxed ponts. In Secton 5.2 we use ths observaton to obtan a suffcent condton for the unqueness of the fxed pont n the nonhomogeneous case. 4. NONUNIQUE FIXED POINTS AND MUL- TISTABILIT: SIMULATION EXAMPLES 4. Example Consder a homogeneous system (let us call t System-I) wth n = nodes. The functon G( ) of the nodes s gven by, G(γ) = + γ + γ 2 + γ γ + γ 2 + γ (γ 4 + γ ) The system corresponds to the case where K =, b = b = b 2 = b 3 = and b 4 = b 5 = b 6 = = 64. From the form of functon G, we can see that a node whch s currently at backoff stage s more lkely to reman at that stage as t takes 4 successve collsons to make the attempt rate of the node <. Lkewse, a node that s n the larger back-off stages b 4 = b 5 = = 64, wll retry contnuously wth mean nter-attempt slots of 64 untl t succeeds. Observe that only one node can be at backoff stage at any tme. Ths leads to the apparent multstablty of the system. Fgure plots G(γ), the correspondng F (γ) = ( γ)( G(γ)) and shows the balanced fxed pont of the system for n = nodes. The balanced fxed pont of the system shown n the fgure s obtaned usng the fxed pont equaton γ = ( G(γ)) 9. Observe that the functon F ( ) s not one-to-one (the functon F ( ) not beng one-to-one does not necessarly mply that there exst multple fxed pont solutons; see Remarks 3., 5). Fgure 2 shows the exstence of unbalanced fxed ponts for System-I. These fxed ponts are obtaned as follows. Assume that we are nterested n fxed ponts such that γ γ 2 = = γ n. Gven γ 2 = = γ n, the attempt probablty of the nodes s gven by G(γ 2). Hence, the collson probablty of node s gven by γ = ( G(γ 2)) n. The attempt probablty of node would then be G(γ ). Usng the decouplng assumpton, the collson probablty of any of the n nodes would then be, ( G(γ 2)) n 2 ( G(γ )) = γ 2. Thus we obtan a fxed pont equaton for γ 2 (and hence for all the other γ, 3 n). In Fgure 2 we plot ( G(γ)) 8 ( G( ( G(γ)) 9 )) (plotted as lne marked wth dots), the ntersecton of whch wth the y=x lne shows the solutons for γ 2 = = γ n. In the same way, by elmnatng γ 2 from the multdmensonal system of equatons, we can obtan a fxed pont equaton for γ. Ths functon s also plotted n Fgure 2 (usng pluses and lnes) and the nterseton of ths curve wth the y=x lne shows the solutons for γ. We see that there are three solutons n each case. The smallest values of γ (approx..4) pars up wth the largest value of γ 2 = = γ n (approx..97). Notce that the balanced fxed pont of the system s also a fxed pont n the plot (compare wth Fgure ). Then there G(γ) ( γ)( G(γ)) ( G(γ)) 9 Balanced Fxed Pont collson probablty (γ) Fgure : Example System-I: The balanced fxed pont. Plots of G(γ), F (γ) = ( γ)( G(γ)) and ( G(γ)) 9 vs. the collson probablty γ; we also show the y=x lne. s one remanng unbalanced fxed pont whose values can be read off the plot. We note that there could exst many other unbalanced fxed ponts for ths system of equatons, as we have consdered only a partcular varety of fxed ponts that have the property that γ γ 2 = = γ n. In order to examne the consequences of multple unbalanced fxed ponts we smulated the back-off process wth the back-off parameters of System-I. The followng remarks summarse our smulaton approach n ths paper. Remarks 4. (On the Smulaton Approach used).. All the smulaton results reported n ths paper are based on smulatons of the coupled multdmensonal back-off processes of the varous nodes. We are not smulatng the actual Wreless LAN system (as s done n an ns2 smulaton). The man am of the smulatons s to understand the backoff behavour of the nodes wth respect to the dfferent backoff parameters. From the pont of vew of performance analyss, t may also be noted that once the back-off behavour s correctly modelled the channel actvty can easly be added analytcally, and thus throughput results can be obtaned (see [3] and []). Note that a good match between analyss that uses a decoupled Markov model of the back-off process and ns2 smulatons has already been reported n earler work (see the lterature survey n Secton ). 2. Thus our smulaton s programmed as follows. The system evolves over back-off slots. All the nodes are assumed to be n perfect slot synchronsaton. The actual coupled evoluton of the backoff process s modeled. The backoff dstrbuton s unform and the resdual backoff tme s the state for each node. At every slot, dependng on the state of the back-off process, there are three possbltes: the slot s dle, there s a successful transmsson, or there s a collson. Ths causes further evoluton of the back-off process.

5 collson probablty (γ ) γ 2... γ γ Balanced Fxed Pont short term average collson probablty Node Node 2 Avg collson prob collson probablty (γ ) Fgure 2: Example System-I: Demonstraton of unbalanced fxed ponts. Plots of ( G(γ)) 8 ( G( ( G(γ)) 9 )) (the curve drawn wth dots and lnes) and the functon for the unbalanced fxed pont equaton for γ (see text). P k= C () 3. In Fgures 3 and 5, for the purpose of reportng the short term unfarness results, the entre duraton of smulaton s dvded nto k frames, where the sze of each frame s, slots. The short-term average of the collson probablty of each node, n, s calculated as C () A where C () () and A () correspond to the number of collsons and attempts n frame, k, for node. The long-term aver- P age s smlarly calculated as n P n = where k= A () n s the number of nodes. Notce that the long-term average collson rate s a batch based average of the short-term collson rates. Hence, when lookng at the graphs, t wll be ncorrect to vsually average the short-term collson rate plots n an attempt to obtan the long-term average collson rate. Ths s because when a node s shown to have a low collson probablty, t s the one that s attemptng every slot (whle the other nodes attempt wth a mean gap of 64 slots), and hence t sees a low probablty of collson. In ths case A ( ) s large and C ( ) A ( ). On the other hand, when a node s shown to have a hgh collson probablty t s attemptng at an average rate of 64 and almost all ts attempts collde wth the node that s then attemptng n every slot. In ths case A ( ) s small and C ( ) A ( ). Thus, n obtanng the lnear average, t s essental to account for the large varaton n A ( ) between the two cases. In Fgure 3 we plot a (smulaton) snap shot of the short term average collson probablty of 2 of the nodes of System-I and the average collson probablty of the nodes (The average s calculated over all frames and all nodes. Snce the nodes are dentcal, the average collson probablty s the same for all the nodes). Observe that the short term average has a huge varance around the long term average. It s evdent that over s of slots one node or the other monopolses the channel (and the remanng nodes see a collson probablty of durng those slots). Ths could be tme n slots Fgure 3: Example System-I: Snap-shot of short term average collson probablty of 2 of the nodes. Also plotted s the average collson probablty of the nodes (averaged over all slots and nodes). The 95% confdence nterval for the average collson probablty les wthn.7% of the mean value. descrbed as short term multstablty. A look nto the farness ndex (see Fgure 6) plotted as a functon of the frame sze used to calculate throughput suggests that System-I exhbts sgnfcant unfarness n servce even over reasonably large tme ntervals. Implcaton for the use of the balanced fxed pont: Notce also that the average collson rate shown n Fgure 3 s about.25, whereas the balanced fxed pont shown n Fgure shows a collson probablty of about.62. Hence we see that n ths case, where there are multple fxed ponts, the balanced fxed pont does not capture the actual system performance. 4.2 Example 2 Let us now consder yet another homogeneous example (let us call t System-II) wth n = 2 nodes. The functon G( ) of the nodes s gven by, G(γ) = + γ + γ γ 7 + 3γ + 9γ γ γ 7 The system corresponds to the case where K = 7, b =, p = 3 and b k = p k b for all k K. We notce that n ths example the way the back-off expands s smlar to the way t expands n the IEEE 82. standard, except that the ntal back-off s very small ( slot), and the multpler s 3, rather than 2. We observe that, smlar to Example System-I, ths system also has multple (unbalanced) fxed ponts and exhbts short-term unfarness n servce (A detaled comment on System-II s provded n [8]). Dscusson of Examples and 2: From the smulaton examples, we can make the followng nferences. When there are multple unbalanced fxed ponts n a homogeneous system then the system can dsplay short term multstablty, whch manfests tself as sgnfcant short term unfarness n channel access. 2. When there are multple unbalanced fxed ponts n a homogeneous system then the collson probablty

6 Balanced Fxed Pont G(γ) ( γ)( G(γ)) ( G(γ)) 9 short term average collson probablty Node Node 2 Avg collson prob collson probablty (γ) Fgure 4: Example System-III: Plots of G(γ), F (γ) = ( γ)( G(γ)) and ( G(γ)) 9 vs. the collson probablty γ; the lne y=x s also shown. obtaned from the balanced fxed pont may be a poor approxmaton to the long term average collson probablty. 3. Smlar conclusons can be drawn for nonhomogeneous systems when the system of fxed pont equatons have multple solutons. It appears that the exstence of multple-fxed ponts s a consequence of the form of the G( ) functon n the above examples, where G( ) s smlar to a swtchng curve; see, for example, Fgure where there s a very hgh attempt probablty at low collson probabltes and a very low attempt probablty at hgh collson probabltes. 4.3 Example 3 Consder a homogeneous system n whch backoff ncreases multplcatvely as n IEEE 82. DCF (let us call t System- III), wth n = nodes. The functon G( ) s gven by, G(γ) = + γ + γ γ γ + 64γ γ 7 The system corresponds to the case where K = 7, p = 2 and b = 6 and b k = p k b for all k K. Fgure 4 plots G( ), the correspondng F (γ) = ( γ)( G(γ)) and the unque balanced fxed pont of the system (Notce that F s one-to-one and unqueness of the fxed pont wll be proved n Secton 5.) The balanced fxed pont of the system s obtaned usng the fxed pont equaton γ = ( G(γ)) 9. The balanced fxed pont yelds a collson probablty of approxmately.29. Fgure 5 plots a snap shot of the short term average collson probablty (from smulaton) of 2 of the nodes and the average collson probablty of the nodes of the Example System-III. Notce that the short term average collson rate s close to the average collson rate (the vertcal scale n ths fgure s much fner than n the correspondng fgures for System-I and System-II, see [8]). Also, the average collson rate matches well wth the balanced fxed pont soluton obtaned n Fgure 4. Remark: Thus we see that n a stuaton n whch there s a unque fxed pont not only s there lack of short term tme n slots Fgure 5: Example System-III: Snap-shot of short term average collson probablty of 2 of the nodes. Also plotted s the average collson probablty obtaned by the nodes. The 95% confdence nterval of the average collson rate les wthn.2% of the mean value. multstablty, but also the fxed pont soluton yelds a good approxmaton to the long run average behavour. 4.4 Short Term Farness n Examples, 2, 3 ( P n = τ ) 2 P n= τ 2 Fgure 6 plots the throughput farness ndex n (where τ s the average throughput of node over the measurement frame, see [5]) aganst the frame sze used to measure throughput. The farness ndex s obtaned for each frame and s averaged over the duraton of the smulaton. Also plotted n the fgure s the 95% confdence nterval. We note that values of ths ndex wll le n the nterval [, ], and smaller values of the ndex correspond to greater unfarness between the nodes. The performance of all the three example systems are compared. Notce that Example System-III (smlar to IEEE 82. DCF) has the best farness propertes. The system acheves farness of.9 over s of slots (or packets). However, for Example System-I and II, smlar performance s acheved only over, and,, slots (or packets). The unfarness of Example Systems-I and II can be attrbuted to ther apparent mult-stablty. In the subsequent sectons we establsh condtons for the unqueness of the solutons to the multdmensonal fxed pont equaton. 5. ANALSIS OF THE FIXED POINT 5. The Homogeneous Case The followng two results are adopted from []. Lemma 5.. G(γ) s nonncreasng n γ f b k, k, s a nondecreasng sequence. In that case, unless b k = b for all k, G(γ) s strctly decreasng n γ. Theorem 5.. For a homogeneous system of nodes, Γ(G(γ)) : [, ] [, ], has a unque fxed pont f b k, k, s a nondecreasng sequence. Remark: The fxed pont (γ, γ,, γ) s the unque balanced fxed pont for γ = Γ(G(γ)). From Equaton 4, we

7 Jans farness measure Example System I Example System II Example System III averagng nterval (n slots) Fgure 6: Jan s throughput farness ndex s plotted aganst the number of slots used to measure throughput. The dotted lnes mark the 95% confdence nterval. see that a necessary condton for the exstence of unbalanced fxed ponts n a homogeneous system of nodes s that the functon F (γ) = ( γ)( G(γ)) needs to be manyto-one. In other words, f the functon ( γ)( G(γ)) s one-to-one and f γ = (γ, γ 2,..., γ n) s a soluton of the system γ = Γ(G(γ)), then γ = γ for all,. Consder the multplcatvely ncreasng back-off model for whch G( ) s gven by, G(γ) = + γ + γ γ K b ( + pγ + p 2 γ p K γ K ) Clearly, G(γ) s a contnuously dfferentable functon and so s F (γ) = ( γ)( G(γ)). The followng smple lemma s a consequence of the mean value theorem. Lemma 5.2. F (γ) s one-to-one f F (γ) for all γ. Remarks 5.. When F ( ) s one-to-one, the followng hold () F (γ) = ff γ =, () F () >, snce G(γ) for all γ, and () F (γ) s a decreasng functon of γ. Now the dervatve of F s F (γ) = + G(γ) G (γ)( γ) Lemma 5.3. If K, p 2 and G( ) s as n Equaton 5, then G (γ) 2p b for all γ. Clearly, G(γ) b and G (γ) and ( γ) for all γ. Substtutng nto the expresson for F (γ), we get, F (γ) + + 2p b The followng result s then mmedate. Theorem 5.2. For a functon G( ) defned as n Equaton 5 f K, p 2 and b > 2p +, then the system γ = Γ(G(γ)) has a unque fxed pont whch s balanced. (5) Remark: It can be shown that f Lemma 5.3 holds for G( ) as n Equaton 5 t also holds for any case n whch b k = p k b for k m K and b k = p m b for m < k K. The latter s the stuaton n the IEEE 82. standard (wth b = 6, p = 2, K = 7, m = 5). Hence a homogeneous IEEE 82. WLAN has a unque fxed pont whch s also balanced. In general, f the functon G( ) s arbtrary (as n Equaton ) but monotone decreasng, there exsts a unque balanced fxed pont for the system as long as the functon ( γ)( G(γ)) s one-to-one. 5.2 The Nonhomogeneous Case In ths secton, we wll extend our results to systems wth nonhomogeneous nodes. AIFS wll be ntroduced n Secton 6. Nonhomogenety s ntroduced by varyng b, p and K of the nodes. Consder a nonhomogeneous system of n nodes, wth G ( ) a monotoncally decreasng functon and the functon ( γ)( G (γ)) beng one-to-one for all. Let there be two fxed pont solutons γ = (γ,, γ,2,..., γ,n) and γ 2 = (γ 2,, γ 2,2,..., γ 2,n) for the above system. From the necessary condton (Equaton 4) we requre that, for all, and for some J > and J 2 >, ( γ,)( G (γ,)) = J ( γ 2,)( G (γ 2,)) = J 2 Snce ( γ)( G (γ)) s one-to-one, we requre J J 2. Wthout loss of generalty, assume J < J 2. Hence, γ, > γ 2, for all. Usng Equaton 3 we have, γ 2, = ( G (γ 2,)) ( G (γ,)) = γ, a contradcton. Hence, we requre J = J 2 or there exsts a unque fxed pont. Notce that the arguments above mmedately mply the followng result. Theorem 5.3. If G (γ) s a decreasng functon of γ for all and ( γ)( G (γ)) s a monotone functon n [, ], then the system of equatons β = G (γ ) and γ = Γ (β,..., β,..., β n) has a unque fxed pont. Where nodes use exponentally ncreasng back-off, the next result then follows. Theorem 5.4. For a system of nodes wth G ( ) as n Equaton 5 that satsfy K, p 2 and b > 2p +, there a exsts a unque fxed pont for the system of equatons γ = Q ( G(γ)) for n. Remark: The above result has relevance n the context of the IEEE 82.e standard where the proposal s to use dfferences n back-off parameters to dfferentate the throughputs obtaned by the varous nodes. Whle Theorem 5.4 only states a suffcent condton, t does pont to a cauton n choosng the back-off parameters of the nodes. 6. ANALSIS OF THE AIFS MECHANISM Our approach for obtanng the fxed pont equatons s the same as the one developed n [2]. However, we develop

8 the analyss n the more general framework ntroduced n []. We show that under the condton that F ( ) s one-toone there exsts a unque fxed pont for ths problem as well. The analyss s presented here for the two prorty class case, but can be extended to any number of classes. Let us begn by recallng the basc dea of AIFS based servce dfferentaton (see [3]). In legacy DCF, a node decrements ts back-off counter and attempts to transmt only after t senses an dle medum for more than a DCF nterframe space (DIFS). However, n EDCF (Enhanced Dstrbuted Coordnaton Functon) based on the access category of a node (and ts AIFS value), a node attempts to transmt only after t senses the medum dle for more than ts AIFS. Hgher prorty nodes have smaller values of AIFS (though not less than DIFS), and hence obtan a lower average collson probablty, snce these nodes can decrement ther back-off counters, and even transmt, n slots n whch lower prorty nodes (watng to complete ther AIFSs) cannot. Thus, nodes of hgher prorty (lower AIFS) not only tend to transmt more often but also have fewer collsons compared to nodes of lower prorty (larger AIFS). 6. The Fxed Pont Equatons Let us consder two classes of nodes of two dfferent prortes. The prorty for a class s supported by usng AIFS as well as b, p and K. All the nodes of a partcular prorty have the same values for all these parameters. There are n () nodes of Class and n () nodes of Class. Class corresponds to a hgher prorty of servce. The AIFS for Class s DIFS, and for Class the AIFS s DIFS+l slots. Thus, after every transmsson actvty n the channel, only Class nodes attempt to transmt n the frst l slots followng an dle DIFS, whle Class nodes wat to complete ther AIFS. Also, f there s any transmsson actvty (by Class nodes) durng those l slots, then agan the Class nodes wat for another l slots followng an dle DIFS, and so on. As n [3] and [], we need to model only the evoluton of the back-off process of a node (.e., the back-off slots after removng any channel actvty such as transmssons or collsons) to obtan the collson probabltes. For convenence, let us call the slots n whch only Class nodes can attempt as Excess AIFS slots, whch wll correspond to the subscrpt EA n the notaton. In the remanng slots (correspondng to the subscrpt R n the notaton) nodes of ether class can attempt. Let us vew such groups of slots, where dfferent sets of nodes contend for the channel, as dfferent contenton perods. Let us defne β () := the attempt probablty of a Class node for all, n (), n the slots n whch a Class node can attempt (.e., all the slots) β () := the attempt probablty of a Class node for all, n (), n the contenton perods durng whch Class nodes can attempt (.e., slots that are not Excess AIFS slots) Note that n makng these defntons we are modelng the attempt probabltes for Class as beng constant over all slots,.e., the Excess AIFS slots and the remanng slots. Ths smplfcaton s ust an extenson of the basc decouplng approxmaton, and has been shown to yeld results that match well wth smulatons (see [2]). q EA q EA q EA q EA l q q EA EA q R Fgure 7: AIFS dfferentaton mechansm: Markov model for remanng number of AIFS slots. Now the collson probabltes experenced by nodes wll depend on the contenton perod that the system s n. The approach s to model the evoluton over contenton perods as a Markov Chan over the states (,, 2,, l), where the state s, s (l ), denotes that an amount of tme equal to DIFS +s slots has elapsed snce the end of the prevous channel actvty. These states correspond to the Excess AIFS perod n whch only Class nodes can attempt. In the remanng slots, where the state s s = l, all nodes can attempt. In order to obtan the transton probabltes for ths Markov chan we need the probablty that a slot s dle. Usng the decouplng assumpton, the dle probablty n any slot durng the Excess AIFS perod s obtaned as, q EA = n () = ( β () ) (6) Smlarly, the dle probablty n any of the remanng slots s obtaned as, q R = n () = n () ( β () ) ( β () ) (7) = The transton structure of the Markov chan s shown n Fgure 7. As compared to [2], we have used a smplfcaton that the maxmum contenton wndow s much larger than l. If ths were not the case then some nodes would certanly attempt before reachng l. In practce, l s small (e.g., slot or 5 slots; see [2]) compared to the maxmum contenton wndow. Let π(ea) be the statonary probablty of the system beng n the Excess AIFS perod;.e., ths s the probablty that the above Markov chan s n states, or, or, or (l ). In addton, let π(r) be the steady state probablty of the system beng n the any of the remanng slots,.e., state l of the Markov chan. Solvng the balance equatons for the steady state probabltes, we obtan, π(ea) = π(r) = + q EA + q 2 EA + + q l EA + q EA + q 2 EA + + ql EA + ql EA q R q l EA q R (8) + q EA + qea ql EA + ql EA q R Average collson probablty of a node s then obtaned by averagng the collson probablty experenced by a node over the dfferent contenton perods. Average collson prob- q R

9 ablty for Class nodes s gven by, for all, n (), γ () = π(ea)( + π(r)( ( n () =, n () =, ( β () )) ( β () n () ) = ( β () ))) (9) Smlarly, the average collson probablty of a Class node s gven by, for all, n (), n () = ( ( β () ) γ () = n () =, ( β () )) () Our analyss n the remanng secton now generalses the analyss of [2] and also establshes unqueness of the fxed pont and the property that the fxed pont s balanced over nodes n the same class. Defne G () ( ) and G () ( ) as n Equaton (except that the superscrpts here denote the class dependent back-off parameters, wth nodes wthn a class havng the same parameters). Then the average collson probablty obtaned from the prevous equatons can be used to obtan the attempt rates by usng the relatons β () = G () (γ () ), and β () = G () (γ () ) () for all n (), n (). We obtan fxed pont equatons for the collson probabltes by substtutng the attempt probabltes from Equaton nto Equatons 9 and (and also nto Equatons 6 and 7). We have a contnuous mappng from [, ] n() +n () to [, ] n() +n (). It follows from Brouwer s fxed pont theorem that there exsts a fxed pont. 6.2 Unqueness of the Fxed Pont Lemma 6.. If F ( ) s one-to-one, then collson probabltes of all the nodes of the same class are dentcal;.e., the fxed ponts are balanced wthn each class. Theorem 6.. The set of Equatons 9, and (together wth 8, 6 and 7), representng the fxed pont for the AIFS model, has a unque soluton f the correspondng functons F () ( ) and F () ( ) are one-to-one. Remark: It follows from the earler results n ths paper (see, for example, Theorem 5.2) that f G () ( ) and G () ( ) are of the form n Equaton 5, and f K (), p () 2, and b () 2p () +, for =,, then the fxed pont wll be unque. 7. THROUGHPUT DIFFERENTIATION: AN ANALTICAL STUD It should be noted that all the results n ths secton are for the fxed pont soluton. Hence, when we use the term collson probablty and attempt rate t s only n so far as a good match between the fxed pont analyss and smulaton has already been reported n earler lterature (see Secton ). We wll consder two alternatves for K, the maxmum retransmsson attempts allowed for a packet, namely K = and K fnte. In ths secton, for the fnte K case, the form of the functon G(γ), for all γ, γ s, G(γ) = + γ + γ γ K b ( + pγ + p 2 γ p K γ K ) (2) It s clear that for fnte K the attempt rate of a node s lower bounded, and hence as the number of nodes ncreases to nfnty the collson probablty of any node goes to. Hence, for ths case, we wll obtan nsghts regardng performance dfferentaton only for a fntely large number of nodes. For the nfnte K case, however, we wll study (as n []) the asymptotcs of performance dfferentaton as the number of nodes tends to. In the K = case, the functon G(γ) smplfes to, G (γ) = ( ( γp) b ( γ) γ < p γ p (3) In the nonhomogeneous case we wll wrte G () (γ) and G () (γ). For the homogeneous case wth K =, the (balanced fxed pont) asymptotc analyss as n was performed n []. Consder a set of nodes, dvded nto two classes, Class and Class, wth Class correspondng to a hgher prorty of servce. For smplcty, we assume that n () and n (), the number of nodes of Class and Class respectvely, are related as, n () = αn, n () = ( α)n for some n and α, < α <. Let γ () (K, n) and β () (K, n) be the fxed pont solutons for the collson probablty and attempt rate of a Class node for a gven K and total number of nodes n. Smlarly, let γ () (K, n) and β () (K, n) be the correspondng values for a Class node. We wll study three cases: Case : b () < b (), p() = p () = p, AIF S () = AIF S () = DIF S Case 2: b () = b () = b, p () < p (), AIF S () = AIF S () = DIF S Case 3: b () = b () = b, p () = p () = p, AIF S () < AIF S () Note that n the analyss n earler sectons, we used the Bnomal model for the number of attempts n a slot. Wth n, n ths secton, we wll use the Posson batch model for the number of attempts n a slot (as n []). 7. Case : Dfferentaton by b 7.. K =, Asymptotc Analyss as n Wth the random number of attempts of each class n a back-off slot beng modeled as Posson dstrbuted, the collson probabltes γ ( ) (, n) and the attempt rates β ( ) (, n) are related by γ () (, n) = e ((n() )β () (,n)+n () β () (,n)) γ () (, n) = e (n() β () (,n)+(n () )β () (,n)) (4) Substtutng β ( ) (, n) = G ( ) (γ ( ) (, n)) n the above equatons gves the desred fxed pont equatons governng the system. Trvally, we see that, ( γ () (, n))e β() (,n) = ( γ () (, n))e β() (,n) (5)

10 Lemma 7.. For {, }, F () (γ) := ( γ)e G() (γ) s one-to-one for all γ, γ f b 2p +. Theorem 7.. In Case, wth K =, when F () one-to-one for {, },. γ () (, n) < γ () (, n) for all n 2. lm n γ () (, n) p, lmn γ() (, n) p 3. lm n (n () β () (, n)+n () β () (, n)) ln( p p ) Theorem 7.2. In Case, wth K =, the rato of the throughputs of Class and Class 2 converges to b() n. s p as b () p Remark: Thus, for example, f b () = 6, b () = 32, and p = 2 then the rato of the Class to Class node throughput wll be approxmately 3/4 for large n Fnte K, Approxmate Analyss for Large n Wth fnte K, as the number of nodes ncreases, the collson probablty of ether class ncreases to (snce the attempt rate s lower bounded) and G ( ) s small (snce t decreases lke, see Equaton 2). Then the dfference b p K+ between the collson probabltes (we drop the arguments K and n n the followng) γ () γ () = (G () (γ () ) G () (γ () )) ( G () (γ () )) (n() ) ( G () (γ () )) (n() ) also becomes nsgnfcant. Hence, we can assume that γ () γ (). For equal packet length transmsson, the rato of the throughputs of a Class node to a Class node corresponds to the rato of ther success probabltes, hence the throughput rato s gven by, G () (γ () )( G () (γ () )) n() ( G () (γ () )) n() G () (γ () )( G () (γ () )) n() ( G () (γ () )) n() = G () (γ () ) ( G () (γ () )) G () (γ () ) ( G () (γ () )) (6) Usng γ () γ (), wrtng ths as γ, and usng the fact that G ( ) (γ) for large n, we have, G () (γ) ( G () (γ)) G () (γ) ( G () (γ)) G() (γ) G () (γ) = b() b () It follows that when servce dfferentaton s provded by the back-off wndow, for a large number of nodes, the throughput rato roughly corresponds to b() b (), whch, for large values of b () and b () s almost that same as that obtaned for the asymptotc analyss wth K = n Theorem 7.2 Remark: For fnte K case, ths observaton (throughput rato s approxmately equal to b() b () ) s well known. Ths result has been shown analytcally (usng smlar approxmatons) and also has been observed n smulatons (see [6], [] and [4]). It has been observed n [] that for a gven number of nodes, n, there wll exst a K(n) such that the system performance wll not vary much for all K > K(n). Hence, an asymptotc analyss would suffce for such cases. Moreover, we have obtaned ths result n a much more general settng, usng the functon G( ). 7.2 Case 2: Dfferentaton by p It may be noted that n the current verson of IEEE 82.e standard ths mechansm no longer exsts [2] K =, Asymptotc Analyss as n The fxed pont equaton governng the collson probablty and the attempt rate s the same as Equaton 4. The followng theorem summarzes the man results for Case 2. Theorem 7.3. In Case 2, wth K =, when F () one-to-one for {, }, the followng hold:. γ () (, n) < γ () (, n) for all n 2. lm n γ () (, n) p (), lm n γ () (, n) p () 3. lm n n () β () (, n) ln( p() p () ) 4. lm n n () β () (, n) = Remark: Thus we see that, wth K = and a large number of nodes, unlke ntal back-off based dfferentaton, the persstence factor based dfferentaton completely suppresses the class wth the larger value of p Fnte K, Approxmate Analyss for Large n For fnte K, wth the approxmaton γ () γ () and the fact that G ( ) (γ ( ) ), the throughput rato approxmates to (+p() γ+p () 2 γ 2 + +p ()K γ K ) (see Equaton 6). Hence, as (+p () γ+p () 2 γ 2 + +p ()K γ K ) the collson probablty of the system ncreases wth load, the rato of the throughputs of Class to Class also ncreases (dependng on p (), p () and the value of K). We note that as n, the throughput rato for the fnte K case s fnte, unlke the asymptotc case (K = ). However, the rato tends to nfnty when we consder K. 7.3 Case 3: Dfferentaton by AIFS 7.3. K =, Asymptotc Analyss for n In ths case servce dfferentaton s provded only by AIFS and we let G () = G () = G (.e., the back-off parameters b and p are the same). Wth the assumpton that the number of attempts n each slot s Posson dstrbuted, the fxed pont equatons for the AIFS model are (see Equatons 9 and ) γ () (, n) = π(ea)( e (n() )β () (,n) ) + π(r)( e (n() )β () (,n) n () β () (,n) ) γ () (, n) = ( e n() β () (,n) (n () )β () (,n) ) Theorem 7.4. In Case 3, wth K =, when F () one-to-one for {, },. γ () (, n) < γ () (, n) for all n 2. lm n γ () (, n) p, lmn γ() (, n) p s s

11 3. lm n n () β () (, n) ln( p p ) 4. lm n n () β () (, n) = Remark: Agan we see that usng AIFS for dfferentaton, when K = and large n, completely suppresses the class wth the larger value of AIFS. Observe that Parts 3 and 4 of Theorem 7.4 mply that the ndvdual node attempt rato β () (,n) goes to as n. Some nsght nto ths result β () (,n) wll be obtaned from the analyss n the followng secton Fnte K, Approxmate Analyss rato of throughput AIFS = /, b = 6/6, n () = n () AIFS = /, b = 6/32, n () = n () AIFS = /, b =6/6, n () = 5 AIFS = /, b = 6/32, n () = n () Lemma 7.2. In Case 3 for fnte K, wth l =, f the fxed pont collson probabltes are γ () and γ (), then the rato of the throughputs of Class to Class s gven by G () (γ () ) ( G () (γ () )) G () (γ () ) ( G () (γ () )) Usng ths result and approxmatng ( G () (γ () )) as before, the rato of throughput equals G () (γ () ) ( G () (γ () )) G () (γ () ) ( G () (γ () )) q R G() (γ () ) (7) q R G () (γ () ) q R For general l, we can expect a factor lke n the prevous q R l expresson. For low loads, when q R s not close to, the domnatng term n the prevous expresson s G() (γ () ). At G () (γ () ) hgh loads, both the terms contrbute to throughput dfferentaton dependng on the values of n () and n (). 7.4 Numercal Study and Dscusson In Fgure 8 we plot throughput ratos obtaned from a smulaton of the coupled back-off processes of two classes of nodes (the smulaton approach s explaned n Remarks 4.). We note that ths s the throughput rato f the packet szes of the two classes are equal. If the packet szes are unequal then we only need to multply the throughput rato plotted here by the rato of the packet lengths of the two classes. The followng remarks help n nterpretng the results n Fgure 8. Remarks 7... For fnte K the attempt rates are bounded below, and the term G() (γ () ) s bounded, but as G () (γ () ) (n() + n () ) the dle probablty q R ensurng (see Equaton 7) that the ndvdual node throughput rato goes to for fnte K as well (smlar to the asymptotc results n Theorem 7.4). In addton, when n () ncreases, π(ea) ncreases to. Hence, the lower prorty nodes (wth larger AIFS) rarely get a chance to attempt and the throughput rato goes to nfnty; ths s demonstrated by the smulaton results n Fgure 8, plots wth + and. When n () s kept constant and n () s ncreased (whch s more typcal), the collson probablty of Class nodes ncreases to and ther success probablty tends to. However, the collson probablty of Class nodes remans much less than n () + n () Fgure 8: Rato of the throughput of a Class (hgher prorty) node to the throughput of a Class node (lower prorty). Analyss results (sold lnes) and smulaton results (symbols). Four cases are consdered: +: dfferentaton only by AIFS wth equal number of nodes, n () = n () ; : dfferentaton by AIFS and by b wth equal number of nodes, n () = n () ; : dfferentaton only by b wth equal number of nodes, n () = n () ; : dfferentaton only by AIFS wth, 5 = n () n (). In all cases p = 2 and K = 7 for ether class. For the smulaton results, the 95% confdence nterval les wthn % of the average value. dependng on the value of n () and hence agan the throughput rato tends to (see Fgure 8, plots wth ). Fgure 8 also shows the throughput rato when only b s used for dfferentaton (plots wth ); notce that, as shown earler, the throughput rato s ust the recprocal of the ratos of the ntal back-off duratons, and does not change wth n. 2. For Case 3, n general, γ () and γ () are dfferent, unlke n Cases and 2. Ths s captured by the frst term n the expresson G() (γ () ) G () (γ () ) q R. 3. Notce that the above results for AIFS hold even when the functons G () and G () are not dentcal (see Fgure 8, plot wth ). A comparson between the plots wth + and n Fgure 8 shows the effect of usng both b and AIFS for throughput dfferentaton. The b based dfferentaton causes the entre curve to shft up (n favour of the hgher prorty class), and AIFS stll causes the rato to ncrease wth ncreasng n. 8. SUMMAR In ths paper we have studed a multdmensonal fxed pont equaton arsng from a model of the back-off process of the EDCF access mechansm n IEEE 82. and e Wreless LANs. Our frst concern was the consequences of the nonunqueness of the fxed pont soluton and condtons for unqueness. We demonstrated va examples of homogeneous systems that even when the balanced fxed pont s unque, the exstence of unbalanced fxed ponts coexsts wth the observaton of severe short term unfarness n smulatons.