Distributed Optimal Contention Window Control for Elastic Traffic in Wireless LANs

Transcription

1 Dstrbuted Optmal Contenton Wndow Control for Elastc Traffc n Wreless LANs Yalng Yang, Jun Wang and Robn Kravets Unversty of Illnos at Urbana-Champagn { yyang8, junwang3, rhk@cs.uuc.edu} Abstract Ths paper presents a theoretcal study on dstrbuted contenton wndow control algorthms for achevng arbtrary bandwdth allocaton polces and effcent channel utlzaton. By modelng dfferent bandwdth allocaton polces as an optmal contenton wndow assgnment problem, we desgn a general and fully dstrbuted contenton wndow control algorthm, called GCA (General Contenton wndow Adaptaton), and prove that t converges to the soluton of the contenton wndow assgnment problem. By examnng the stablty of GCA, we dentfy the optmal stable pont that maxmzes channel utlzaton and provde solutons to control the stable pont of GCA near the optmal pont. Due to the generalty of GCA, our work provdes a theoretcal foundaton to analyze exstng and desgn new contenton wndow control algorthms. I. INTRODUCTION Due to the shared nature of wreless channels and the ntrnsc scarcty of bandwdth n wreless LANs, nodes must contend for the channel and compete for bandwdth. Whle both contenton resoluton and bandwdth allocaton can be acheved through centralzed schedulng at a wreless LAN access pont, such centralzed control s not scalable to a large number of nodes, suggestng the use of dstrbuted algorthms for both contenton resoluton and bandwdth allocaton. Common dstrbuted contenton resoluton protocols, ncludng IEEE 82. [2], MACA [] and MACAW [3], use contenton wndows to control the channel access of nodes. Contenton wndows not only reduce network congeston, but also drectly affect the share of bandwdth that a node acheves durng competton for the channel. Therefore, t s natural to extend such algorthms to support bandwdth allocaton. Applcatons requrng bandwdth allocaton can ether be realtme traffc (e.g., vdeo/audo streamng) or elastc traffc [9] (e.g., fle transfer). Whle realtme traffc requres servce guarantees to ensure optmal bandwdth allocaton, elastc traffc always has backlogged packets and adjusts ts rate to fll the avalable bandwdth. Hence, competng flows wth elastc traffc are more concerned about farness and effcency of bandwdth allocaton. Whle effcency s defned by bandwdth utlzaton, farness must be defned by the goals of the partcular network, whch may mean unform bandwdth allocaton or weghted proportonal bandwdth allocaton or the hghest prorty node obtanng all bandwdth. The focus of ths paper s to use contenton wndow control to allocate bandwdth to elastc traffc so that both an arbtrary defnton of farness and effcent channel utlzaton are acheved. Usng contenton wndow control for servce guarantees for realtme traffc s beyond the scope of ths paper and can be found n our techncal report [22]. There have been extensve studes on contenton wndow control n wreless LANs. However, none of these approaches can support both an arbtrary defnton of farness and effcent use of bandwdth. The frst type of algorthm, ncludng IEEE 82.e [5] and [], assgns dfferent mnmum contenton wndow szes to dfferent types of nodes to acheve weghted farness. However, snce mnmum contenton wndow szes are pre-confgured and do not adapt to congeston, such approaches do not utlze the channel effcently. The second type of algorthm, ncludng AOB [5], MFS [3], [6] and [7], only focuses on effcent channel utlzaton n the context of unform bandwdth allocaton and the support for other defntons of farness s lmted. The thrd type of algorthm, ncludng PFCR [7], tres to provde a more general defnton of farness by modelng farness as an optmzaton problem of transmsson rate allocaton. However, the mappng between rate allocaton and contenton wndow adaptaton n PFCR s only approprate for a lmted set of farness defntons (See Secton VIII-A). The fnal type of algorthm, P-MAC [8], tres to acheve both proportonal farness and effcent utlzaton by estmatng the contenton wndows used by the competng nodes. Such estmaton requres that every node, wth or wthout packets for transmsson, must start P-MAC smultaneously and calculate the contenton wndow szes for all other nodes synchronously. Nodes wth outdated contenton wndow szes of the other nodes due to asynchronous startng tme or temporary falure may cause the algorthm to fal. Due to the lmtatons of the exstng approaches, we propose our dstrbuted contenton wndow control algorthm, called GCA (General Contenton wndow Adaptaton), whch can be used to acheve optmal bandwdth allocaton for competng wreless nodes n terms of effcent channel utlzaton and varous defntons of

2 2 farness. The goal of GCA s to provde a general soluton for desgn and analyss of dynamc contenton wndow control algorthms n wreless LANs. There are four major contrbutons of ths paper. Frst, we dentfy and model, for the frst tme, an arbtrary farness defnton as an optmzaton problem for contenton wndow assgnment. Second, even though a node s bandwdth share depends on the contenton wndow szes of all competng nodes, GCA does not requre any global knowledge. In GCA, a node only adjusts ts own contenton wndow sze based on locally avalable nformaton and the system automatcally converges to any gven farness defnton. Thrd, by studyng the propertes of the stable pont of GCA, we show that effcent channel utlzaton can also be acheved by controllng the stable pont. Fnally, we demonstrate that GCA provdes a systematc scheme to generalze and evaluate related approaches. Ths paper s organzed as follows. Secton II revews the IEEE 82. contenton resoluton and the relatonshp between bandwdth allocaton and contenton wndow sze. Secton III relates an arbtrary farness defnton to an optmal contenton wndow assgnment problem. Secton IV ntroduces our contenton wndow control algorthm, GCA. Secton V shows that GCA converges to the soluton of the optmal contenton wndow assgnment problem and Secton VI shows how to control the stable pont of GCA to acheve hgh channel utlzaton. Secton VII dscusses gudelnes for mplementng GCA. In Secton VIII, we use GCA to analyze several exstng approaches. Secton IX presents the evaluaton of GCA usng smulatons. Fnally, Secton X concludes and dscusses future research. II. BANDWIDTH ALLOCATION AND CONTENTION WINDOW SIZE To realze far bandwdth allocaton by adaptng contenton wndow szes, t s essental to understand the relatonshp between a node s bandwdth allocaton and ts contenton wndow sze. In ths secton, we frst brefly revew the contenton resoluton algorthm n IEEE 82. and then ntroduce the relatonshp between the contenton wndow sze and bandwdth allocaton. A. IEEE 82. DCF In IEEE 82. DCF [2], before a transmsson, a node must determne whether the medum s busy or dle. If the medum remans dle for DIFS tme unts, the node can transmt. If the medum was ntally busy or changed from dle to busy durng the DIFS, the node must defer ts transmsson. The frst part of the deferment perod s determned by the success of the last transmsson. If the last frame was successful, the node wats DIFS tme unts. If the last frame was not successful, the node wats EIFS tme unts. The second part of the deferment perod s determned as Backoff Tme = Random() aslottme, where Random() s a pseudo-random nteger unformly dstrbuted over [,W]. The contenton wndow, W, s an nteger n the range [mnmum contenton wndow (W mn ), maxmum contenton wndow (W max )]. For every dle aslott me, the backoff tmer s decremented by aslott me. The tmer s stopped when the medum s busy and restarted after the medum s dle for a DIFS. When the tmer expres, the node can transmt. The transmsson ncludes ether a fourway RTS-CTS-DATA-ACK handshake or just a two-way DATA-ACK handshake. After a successful transmsson, W s set to W mn. After an unsuccessful transmsson, W s doubled, up to W max. B. Bandwdth Allocaton vs. Contenton Wndow Sze To understand the relatonshp between bandwdth allocaton and contenton wndow sze, we use the followng result presented n [4]: L W mn j exponental ncrease of W s, = L jw mn after a collson, s j L W j L jw, W does not change after a collson, () where s s the absolute bandwdth allocated to Node n bts per second. L s the channel transmsson rate, S, multpled by the duraton of a successful transmsson at Node, ncludng the DIFS and RTS/CTS/DATA/ACK handshake. Snce all nodes carry elastc traffc and always have backlogged packets, the combned transmssons of all nodes consume the entre network capacty, C. Hence, s = C, (2) N where N s the set of all transmttng nodes. Combnng Equatons () and (2), the fracton of channel bandwdth allocated to Node, x, s x = s L /W mn exponental ncrease C = k N L, k/wk mn of W after a collson, L /W k N L, k/w k otherwse (3) Equaton (3) shows that the relatonshp between x and W s approxmately the same as the relatonshp between x and W mn. Therefore, n the rest of ths paper, we desgn GCA assumng no exponental ncrease of W after a collson. Our algorthm can also be used to dynamcally adjust W mn when exponental ncrease s used, whch we valdate through smulaton (see Secton IX). Notaton for the entre paper can be found n Appendx D. III. FAIRNESS FORMULATION In ths secton, to desgn a contenton wndow control algorthm that supports varous farness defntons, we frst

3 3 formulate the general farness requrement as an optmal bandwdth allocaton problem, and then translate t to an optmal contenton wndow assgnment problem. A. Optmal bandwdth allocaton (OP T BW) Followng [], [6], far bandwdth allocaton n wreless LAN can be modeled as an optmzaton problem. In ths formulaton, each Node s assumed to have a utlty of U (s ) when ts bandwdth allocaton s s. For elastc traffc, U (s ) s an ncreasng, strctly concave and contnuously dfferentable functon of s over the range s [9]. Assumng that the channel capacty s C, the optmal bandwdth allocaton can be formulated as follows: OP T BW (U, C) : max N U (s ) subject to N s C and s for N. Accordng to the Karush-Kuhn-Tucker optmalty condton [2], the unque soluton to OP T BW s gven by []: U (s ) = µ, for N µ(c N s ) =, (4) µ, where µ s the Lagrange multpler. Dfferent defntons of utlty functons result n dfferent solutons to OP T BW (U, C) and so acheve dfferent defntons of farness [6] (See examples n Secton VII-B). In wred networks, OP T BW (U, C) s solved usng a dstrbuted rate adaptaton algorthm []. Because wred networks are assumed to be pont-to-pont and/or have hgh lnk bandwdth, the sendng rate of a node s essentally ts own TCP congeston wndow sze over ts own round trp tme. Therefore, the rate adaptaton algorthm s easy to mplement n wred networks through TCP congeston wndow control. However, n wreless networks, the sendng rate of a node depends on the contenton wndow szes of all competng nodes, and so no node has drect control over ts sendng rate. Therefore, the same rate control algorthm can not be drectly appled to contenton wndow control (see an example n Secton VIII-A). To solve OP T BW (U, C), we must translate OP T BW (U, C) to a problem of contenton wndow assgnment, called OP T W IN. B. Optmal contenton wndow assgnment (OP T W IN) To translate the OP T BW (U, C) problem to the OP T W IN problem, we frst map U (s ) n OP T BW (U, C) to a functon of x by substtutng U (s ) wth Ũ(x ), where U (s ) = U (x C) = Ũ(x ). Smlar to U (s ), Ũ (x ) s an ncreasng, strctly concave and contnuously dfferentable functon of x over the range x. Then, based on Equaton (3) and usng the fact that N x, we replace x wth W n the OP T BW (U, C) and fnally get the OP T W IN problem as follows: OP T W IN(Ũ, C) : over max N Ũ( L W k N W > for N. L k ) W k By replacng s n Equaton (4) wth W based on Equaton (3) and usng the fact that ( N soluton to OP T W IN(Ũ, C) s gven by: L W k N ( ) Ũ L /W k N L = µ, k/w k for N µ. L k ), the W k It s very mportant to note that, although the soluton to OP T W IN(Ũ, C) s unque n terms of x L = /W k N L k/w k, t s not unque n terms of W. Consder contenton wndow szes W = {W : N } that solve OP T W IN(Ũ, C). When W s multpled by a constant factor a, the resultng contenton wndow assgnment aw = {aw : N } s also a soluton to OP T W IN(Ũ, C). Among the possble solutons to OP T W IN(Ũ, C) that satsfy the farness requrement, channel utlzaton can be qute dfferent. Therefore, the dentfcaton of the soluton of OP T W IN(Ũ, C) that maxmzes channel utlzaton s mportant (see Secton VI-A). IV. GENERAL CONTENTION WINDOW ADAPTATION ALGORITHM (GCA) In ths secton, we present GCA, our dstrbuted contenton wndow control algorthm that acheves both far bandwdth allocaton for arbtrary defntons of farness and hgh bandwdth utlzaton. Snce GCA s fully dstrbuted, each node needs only collect local nformaton and adjust ts contenton wndow sze accordngly. In addton, unlke prevous work on dynamc contenton wndow control [5], [6], [7], [7], [8], GCA can be used n a network where nodes have dfferent frame szes. In GCA, a Node adapts ts W followng dfferental equaton: (5) accordng to the Ẇ (t) = αw (t)[ũ (x ) f(ω)], (6) where α s a postve constant factor, Ẇ (t) s the tme dervatve of W, f( ) s a functon of a locally observable channel state Ω and Ω can be known by any node through montorng the channel locally. Although there are varous choces of Ω and f( ), GCA does make three assumptons about them. Frst, Ω must be observable by all nodes sharng the channel and all nodes must be pre-confgured wth the same f( ). Snce n the networks targeted by GCA, every node can hear each other

4 4 Channel State Backoff tmer of Node aslottme Busy DIFS aslottme aslottme aslottme Vrtual tme slots Fg.. Busy DIFS Vrtual slots aslottme aslottme Busy Node transmts Real tme Real tme and hence see the same channel state, the frst assumpton s not very restrctve. Second, to guarantee that the system stablzes at a unque pont (see detals n Secton V), the value of Ω must depend on the wndow szes and packet lengths of all nodes. In other words, denotng W = {W : N } and L = {L : N }, GCA requres Ω = Ω(W, L). Such Ω s not hard to fnd snce many channel states depend on W and L (e.g., packet transmsson delay, average length of an dle perod or collson probablty). Thrd, f(ω) must be strctly ncreasng wth respect to N W L nsde a certan set of system states. Gven the relatonshp between Ω and (W, L), by choosng the rght form of f( ), f(ω) can easly meet the thrd assumpton. As long as the three assumptons are satsfed, GCA s not lmted to any specfc Ω or f( ). We demonstrate n Secton VIII that these assumptons are easy to meet and GCA can be used to model dfferent dynamc contenton wndow control algorthms. To mplement GCA n a real system, the update algorthm n Equaton (6) must be translated to ts dscrete counterpart. To fnd the approprate tme nterval between updates, note that the state transton of a wreless LAN s a dscrete-tme Markov process as shown n the Banch model [4] of IEEE 82.. The tme unt for ths dscretetme process, called a vrtual slot, s the tme perod for a backoff node to decrement ts backoff tmer by one. At most one packet can be transmtted n a vrtual slot and a collson happens when there are multple transmsson attempts n the same vrtual slot. Fgure llustrates an example of ths model. The example shows that a vrtual slot can ether be exactly an aslott me perod durng the dle tme of the channel (e.g., the frst vrtual slot) or nclude a busy perod, a DIFS and an aslott me f the channel s busy (e.g., the second vrtual slot). Snce the network state only changes n the steps of vrtual slots, the effects of any contenton wndow update can not be seen n any smaller tme unt than a vrtual slot. Therefore, the update nterval of GCA should not be smaller than a vrtual slot. If a node updates ts contenton wndow sze at the end of every backoff slot, essentally at every vrtual slot, the dscrete verson of GCA becomes: W k+ = W k αw k [Ũ (x k ) f(ω k )]. (7) If a node only performs the wndow sze update for each packet transmsson, whch means that the average number of vrtual slots between each update s W 2, the dscrete verson of GCA becomes: W k+ = W k.5α(w k ) 2 [Ũ (x k ) f(ω k )]. (8) The GCA algorthm tself s smple and only requres local nformaton about the state of the network. Despte ths smplcty, GCA converges to the soluton of OP T W IN (see Secton V) and can also acheve effcent channel utlzaton (see Secton VI). V. CONVERGENCE AND FAIRNESS OF GCA In ths secton, we prove that GCA, as expressed n Equaton (6), asymptotcally converges to a unque pont that s a soluton to OP T W IN gven the three assumptons about f(ω). Our proof ncludes two theorems. Theorem states that under the frst assumpton of f(ω), GCA converges to an nvarant set [2] where each element of the set s a soluton to OP T W IN. Based on the second and thrd assumptons for f(ω), the second theorem shows that GCA converges to a unque pont that solves OP T W IN. Theorem : Startng from any ntal state of W, GCA Ẇ converges to an nvarant set Γ = {W : W = Ẇj W j, for, j N } and every element n Γ s a soluton to OP T W IN. In addton, nsde Γ, the Ũ (x ) n Equaton (6) remans a constant and Ũ ( L /W k N L ) = k/w Ũ k j ( L j/w j k N L ), k/w k for, j N. Proof: The proof conssts of three steps. At step one, for notaton smplcty, we translate GCA n Equaton (6) to an equvalent algorthm of Z = W, called GCA-Z. At step two, we prove that GCA-Z converges to an nvarant set R. At step three, usng the equvalence between GCA and GCA-Z, we fnd the nvarant set Γ of GCA and show that every pont n Γ s a soluton to OP T W IN. Step : From Z = /W, we have Ż = (W ) 2 Ẇ = Z 2 Ẇ. By replacng x n Equaton (6) wth W based on Equaton (3) and then replacng W and Ẇ wth Z and Ż, GCA s translated to GCA-Z as follows: ] Ż = αz [Ũ ( Z L k N Z ) f(ω). (9) Step 2: Defne a scalar functon V (Z) as: V (Z) = Z Ũ ( L N k N Z ), () where Z = {Z : N }. Accordng to Lemma n Appendx A, V s a Lyapunov functon for GCA-Z wth V and the zero values of V are obtaned n the set: { } Ż R = Z : = Żj,, j N. () Z Note that for a maxmzaton problem, the convergence condton s V [], [2].

5 5 Accordng to Lemma 2 n Appendx B, R s an nvarant set for GCA-Z and every element nsde R satsfes Ũ ( Z L k N Z ) = kl Ũ L j j( k k N Z ) = constant,, j N. (2) Therefore, whenever the system state evolves nto R, t remans n R. Usng the La Salle Invarant Set Prncple [2], we conclude that GCA-Z converges to R. Step 3: Due to the equvalence between GCA and GCA- Z, we conclude that GCA converges to an nvarant set Γ. Replacng W and Ẇ wth Z and{ Ż n Equatons () and } Ẇ (2), Γ can be expressed as Γ = W : W = Ẇj W j,, j N and every element nsde Γ satsfes Ũ ( L /W k N L ) = k/w k Ũ j ( L j/w j k N L ) = k/w k constant,, j N. Clearly, Γ matches the optmalty condton for OP T W IN n Equaton (5) and so any element n Γ solves OP T W IN. Theorem 2: If Ω = Ω(W, L) and f(ω) s strctly ncreasng wth respect t N W L nsde Γ, startng from any ntal state of W, GCA converges to a unque pont Ŵ Γ that solves OP T W IN. Proof: Snce Z = W, the assumptons n ths theorem are equvalent to that Ω = Ω(Z, L) and f(ω) s strctly ncreasng wth respect to N Z L. Accordng to Lemma 3 n Appendx C, wth these assumptons, GCA-Z has a unque equlbrum pont Ẑ n R and startng from any pont n R, GCA-Z converges to Ẑ. Based on the equvalence between GCA and GCA-Z, we conclude that startng from any pont n Γ, the algorthm of Equaton (6) converges to a unque equlbrum pont Ŵ n Γ. Note that Theorem already shows that startng from any ntal state of W, the algorthm of Equaton (6) converges to Γ and every element of Γ s a soluton to OP T W IN. Therefore, we conclude that n the context of the assumptons, startng from any ntal state of W, GCA converges to a unque pont Ŵ Γ that solves OP T W IN. Theorems and 2 demonstrate that the system s stable under the control of GCA and that GCA converges to a unque pont that solves OP T W IN. Therefore, GCA acheves arbtrary farness defntons. Next, we present how GCA can be used to acheve hgh channel utlzaton. VI. CHANNEL UTILIZATION OF GCA Theorem shows that the choce of utlty functons defnes the ratos of W s at the stable pont of GCA, and, therefore, the farness between nodes. However, multple assgnments of W may satsfy the same rato condton and ther channel utlzaton may be qute dfferent. Theorem 2 shows that the choce of f(ω) ensures that the system only has one stable pont and hence determnes channel utlzaton. If W at the stable pont s too large, channel bandwdth s not fully utlzed snce dle perods are too long. If W at the stable pont s too small, collsons ncrease, whch also results n neffcent use of bandwdth. Therefore, the problem of maxmzng channel utlzaton s essentally the problem of choosng the rght f( ) and Ω. Together wth the choce of utlty functons, ths should enable the system to stablze at a pont that acheves both the farness defnton and hgh channel utlzaton. A. Optmal Stable Pont To choose f(ω) to stablze the system at a pont that maxmzes channel utlzaton, we need to dentfy the optmal stable pont. In ths secton, we analyze the property of the optmal stable pont of GCA and show that at the optmal stable pont, the sum of the recprocals of all W s, denoted ω, s quas-constant regardless of the number of competng nodes and therefore can be pre-calculated. Usng ths property, we can desgn f(ω) to ensure that GCA converges around the optmal stable pont. To dentfy the property of ω at the optmal stable pont, assume there are n competng nodes belongng to m classes c, c 2,... c m and nodes n the same class have the same utlty functon. The fracton of nodes n each class s β, β 2,... β m, where m = β =. To capture the fact that the ratos between contenton wndow szes are determned by the choce of utlty functons and the fact that the nodes n the same class share the same utlty functon, we defne a new varable ϕ as follows: ϕ = /W n j= /W j = /W, n, (3) ω ϕ = ϕ j = ϕ ck,, j c k, (4) where n = ϕ = m k= nβ kϕ ck =. To maxmze channel utlzaton, the average tme between each successful transmsson, denoted as F, must be mnmzed. F ncludes two types of vrtual slots, dle slots and slots wth collsons. Gven the probablty that a slot s an dle slot, P I, and the probablty that a slot ncludes a collson, P c, the average number of vrtual slots n F s P P c P I. In average, a I P I +P c fracton of these slots s dle and a fracton ncludes collsons. Therefore P c P I +P c F = P c P I (aslott me P I + T cp c), (5) where aslott me s the duraton of an dle vrtual slot and T c s the duraton of a vrtual slot wth a collson. Based on the Banch model [4], the probablty that Node transmts n a vrtual slot s /(W /2 + ) and P I equals the probablty that no node transmts n a slot. Therefore, usng Equatons (3) and (4), P I = n ( W /2 + ) = m k= ( + 2ωϕc k) nβ. (6) k =

6 6 ω opt Confguraton Confguraton 2 Confguraton n Fg. 2. ω opt n three network confguratons. Confguraton has class, Confguraton 2 has 2 classes wth ϕ c : ϕ c2 = : 5 and β : β 2 =.5 :.5 and Confguraton 3 has 4 classes wth ϕ c : ϕ c2 : ϕ c3 : ϕ c4 = : 5 : : 2 and β : β 2 : β 3 : β 4 =.5 :.3 :.5 :.5 Snce a collson happens n a slot when more than one node transmts n that slot, P c = P I n n = W /2+ j=,j ( W ) j/2+ = ( + 2ω)P I. (7) Usng Equatons (6) and (7), Equaton (5) becomes [ F (ω) = m ] T c ( + 2ωϕ ck ) nβ k T c( + 2ω) + aslott me. 2ω k= Snce the ω that mnmzes F, denoted ω opt, satsfes F (ω opt ) =, by settng F (ω) =, we get: [ ] m ( 2ω optnβ k ϕ ck m ) (+2ω opt ϕ ck ) nβ k aslott me =. + 2ω opt ϕ ck T c k= k= (8) Snce m k= nβ kϕ ck =, when n, Equaton (8) becomes: ( 2ω opt )e 2ωopt = aslott me T c. (9) Solvng ths equaton gves the lower bound of ω opt. Fgure 2 depcts how ω opt changes as n ncreases. As we can see, for a large n, ω opt s a quas-constant and the dfferences between dfferent confguratons of classes are hard to dstngush. Therefore, we can pre-calculate ths quas-constant and pre-confgure GCA to converge around ths value by a proper desgn of f(ω). B. Choce of f(ω) Snce ω opt s a quas-constant, by controllng the system to stablze near ω opt, the channel utlzaton s close to the maxmum value. To acheve ths, f(ω) should be a large negatve value when the ω of the system s much larger than ω opt. Ths large negatve value of f(ω) forces the system to ncrease ts W (see Equaton (6)), drvng ts ω back to ω opt. Smlarly, when the ω of the system s much smaller than ω opt, f(ω) should be a large postve value to drag the system back to ω opt. Examples of f(ω) are presented n Secton VIII. Our smulaton results n Secton IX verfy the effectveness of ths approach. VII. IMPLEMENTATION CONSIDERATIONS In the prevous secton, we ntroduced GCA and analyzed ts farness and effcency. In ths secton, we address two mplementaton ssues of GCA: estmaton of x n Equaton (6) and choce of utlty functons. A. Estmaton of x Gven the network capacty C, a node can smply observe ts own sendng rate, s, to obtan x, snce x = s /C. However, the capacty of a wreless channel may vary due to outsde nterference, such as a mcrowave. Therefore, ths method s not practcal for use n real networks. However, a node can drectly estmate ts x by observng two states of the channel, the average number of dle vrtual slots between two busy vrtual slots, I, and the average length of a busy vrtual slot, T b. Snce n IEEE 82. networks a node montors the channel contnuously, I and T b can be obtaned at the MAC layer easly. In the rest of ths secton, IEEE 82. DCF RTS/CTS mode s used as an example to show how ths can be done. Let P b be the probablty that a vrtual slot s a busy slot. Snce I s the average number of dle slots between two consecutve busy slots, P b = P I = I +. (2) Snce a busy vrtual slot s caused ether by a successful transmsson or by a collson, the average length of a busy slot, T b, can be expressed as: N T b = T P P c s + T c, (2) P b P b where P s the probablty that Node successfully transmts n a vrtual slot, T s s the average length of a vrtual slot wth a successful transmsson and T c s the duraton of a vrtual slot wth a collson. T s can be expressed as T s = RT S + CT S + 3 SIF S + DAT A + ACK + DIF S + aslott me. Snce RTS/CTS exchange s used, collsons usually happen between RTS packets. Hence, T c = RT S + EIF S + aslott me. Based on IEEE 82. confguratons, T c s much smaller than T s. Therefore, as long as P c s not much larger than N P, usng Equaton (2), Equaton (2) can be smplfed to: N T b T P s = T s(i + ) P. (22) P b N Note that T s also satsfes T s = L P N S. Therefore, j N Pj from Equaton (22), we can get N P L = T b S/(I + ). Snce N P L s the average network throughput per vrtual slot and P L s Node s average throughput per vrtual slot, P x = L j N P P L (I + ). (23) jl j T b S

7 7 To calculate P from I, note that Node transmts n a slot successfully f and only f t s the only node that transmts n that slot. Therefore, P = W /2+ j N,j ( W ). Combnng the fact that P j/2+ b = P I and usng Equatons (6) and (2), P becomes P = W 2 ( I+ ). Integratng ths wth Equaton (23), we fnally obtan the estmaton of x based on I and T b : x B. Choce of Utlty Functons 2L I W T b S. (24) Dependng on the system goal, GCA supports a large range of utlty functons that defne a varety of farness. These utlty functons can be ether pre-confgured n nodes or selected by nodes at run tme accordng to applcaton requrements. In ths secton, we brefly revew several common utlty functons and ther correspondng farness defntons. How to enforce a node to use a certan utlty functon s beyond the scope of the paper. ) Strct Prorty: For a system that needs to acheve strct prorty (.e., the hghest-prorty nodes get all the bandwdth), we can use a weghted lnear utlty functon Ũ (x) = ρ x, where ρ s the prorty-based weght. The correspondng update algorthm s Ẇ = αw [ρ f(ω)]. Note that ths utlty functon does not satsfy the stablty condtons snce Ũ( ) s not strctly concave. Therefore, our update algorthm wll never converge to a certan W. However, snce the nodes wth hghest weght essentally drve f(ω) to be equal to max{ρ, N }, the other competng nodes nfntely ncrease ther W s. Therefore, the nodes wth the hghest weght quckly obtan all the bandwdth of the channel and our update algorthm acheves ths strct prorty between nodes. 2) Weghted Proportonal Farness: Some systems am to acheve weghted proportonal farness [] (.e., bandwdth allocatons satsfy x ρ = ρ x j,, j N, where ρ s the weght of Node ). The utlty functon for such a system s a weghted log functon U (x ) = ρ log x. Our update algorthm for ths system s: Ẇ = αw [ ρ x f(ω)]. 3) Mnmum Potental Delay: If the polcy of the system s to mnmze the total delay of fle transfers, the utlty functon can be expressed as U (x) = ρ x, where ρ s the sze of the fle that Node s transmttng. Our update algorthm for ths system s Ẇ = αw [ ρ x 2 f(ω)]. 4) Mxed Utlty: It s also possble that n a system, dfferent nodes have dfferent goals and hence dfferent utltes. In such stuatons, each node smply updates ts contenton wndow accordng to ts own utlty functon. The system automatcally converges to a stable pont where the aggregated utlty of all competng nodes s maxmzed. In general, the varety of choces of the utlty functons gve GCA the flexblty to be used n systems that have dfferent farness polces. VIII. CASE STUDY In the prevous sectons, we have analyzed the optmalty, stablty and optmal stable pont of GCA. Snce GCA s a general algorthm for contenton wndow control, these analyses can be used as a powerful tool to examne exstng approaches and desgn new algorthms. Due to space lmtatons, we can only present a bref analyss of three examples. (Addtonal examples can be found n our techncal report [23].) The frst example shows how to use GCA to check the farness of an exstng algorthm. The second case shows how to use GCA to analyze the stablty and effcency of an exstng algorthm. The fnal case shows how to use GCA to desgn a new contenton wndow control algorthm. A. Case : Farness Analyss In [7], t s proposed to drectly translate the rate adaptaton algorthm ṡ = α β Pc to a contenton Ũ wndow control algorthm, (s) Ż = α β Pc Ũ (Z ), (25) to solve OP T BW (U, C) (α and β are postve constants). A specal case of the algorthm wth a weghted log utlty functon s named PFCR. Assumng unform packet sze, t can be shown that ths algorthm can not acheves an arbtrary farness defnton. At the equlbrum pont of the algorthm, Ż =, whch results n: Ũ (Z ) = Ũ j () = βpc α,, j N. By replacng Z wth W, we get: Ũ ( ) = W Ũ j( ) = βpc,, j N, (26) W j α whch does not satsfy the optmalty condton for OP T W IN n Equaton (5), and hence can not acheve an arbtrary farness. Although, for log utlty functons (e.g. PFCR), when Equaton (26) s satsfed, the farness condton n Equaton (5) s also satsfed. However, such a property does not hold for many utlty functons (e.g Ũ (x ) = ρ x + logx ). B. Case 2: Stablty and Effcency Analyss In [5], an algorthm named AOB (Asymptotcally Optmal Backoff Algorthm) s proposed to dynamcally adjust contenton wndow to acheve maxmum bandwdth utlzaton. In ths secton, we examne AOB s stablty and effcency n a network wth a unform prorty and packet sze. It s also easy to show that AOB only acheves a specfc farness defnton n a mult-prorty network. Due to space lmtaton, ths analyss s not presented n ths paper and can be found n our techncal report [23].

8 8 In AOB, at every packet transmsson, a node sets ts contenton wndow sze to: [ ] Z k+ =.5 mn(, (I k ) m k, (27) + )2ω opt where ω opt s pre-computed and m k s the number of transmsson attempts for the current packet. The followng analyss shows that AOB s a specal form of GCA. Usng utlty functon Ũ(x) = x.5x2, whch s a strctly ncreasng concave functon n the range of [, ], Equaton (27) becomes: Z k+ Z k = 2(I+) [Ũ (2Z k (I k + )) mn(, (I k +) m k 2ω m k I)]. opt (28) By approxmatng k N Z k, the dscrete form of 2(I+) GCA-Z n Equaton (9) becomes: Z k+ Z k =.5α[Ũ (2Z k (I + )) f(ω)], (29) where each teratve step s a packet transmsson. Comparng Equatons (28) and (29) shows that the AOB algorthm s a specal case of GCA wth: ( ) f(ω) = mn, (I + ) m 2ωopt m I. (3) For f(ω) to satsfy the convergence condton of GCA, f(ω) must be strctly ncreasng wth respect to θ = N W L n the nvarant set Γ. Combnng Equatons (2) and (6), the relatonshp between I and W s: n I + = ( W j /2 ( )). (3) W j= j /2 + In addton, Theorem shows that nsde Γ, Ũ ( /W k N /W ) k remans a constant. Denotng the constant as ˆγ, we get W = /[Ũ (ˆγ)θ]. Combnng ths wth Equatons (3) and (3), f(ω) can be transformed to a functon of θ, whose dervatve can be shown to be larger than. Therefore, f(ω) n AOB satsfes GCA s convergence condton and hence AOB s a stable algorthm that converges to a unque pont. To understand the channel utlzaton of AOB, note that due to the unform prorty and packet sze, each node should have the same contenton wndow sze at AOB s stable pont, ndcatng Z = ω n. Snce at the stable pont, Z k+ Z k =, based on Equatons (27) and (3), 2ω n = ( I+ ) m, where n 2ω opt I + = + 2ω. (32) = n By settng n = and n, we obtan the bounds of ω as [ω, ω 2 ], where: ω 2ω = ( ) m, (+2ω )ω opt ( e 2ω 2 2ω opt ) m =. (33) Essentally, AOB bounds the ω of the system nsde a range that ncludes ω opt, whch explans why AOB can almost acheve maxmum channel utlzaton. f( W, L)=/(I I mn ) +/(I I max ) I I mn I max Fg. 3. f(ω) for Case 3 C. Case 3: New Algorthm Desgn In ths secton, we present an example of the process of desgnng a specal case of GCA. In ths example, we assume any utlty functon that s strctly ncreasng and concave and that the observed channel state Ω s I. Accordng to Equaton (3), for a large n, I Therefore, f(ω) can be defned as: f(ω) = λ/(i I mn ) + λ/(i I max),. e 2ω where I mn < e 2ω opt < Imax and the range of [I mn, I max] s small. Fgure 3 shows the shape of f(ω) wth I mn = 2 and I max = 6. Accordng to Secton VI-B, ths functon f(ω) ensures that at the converged pont, ω s near ω opt, so that the system utlzaton s close to the maxmum. The performance of ths algorthm s evaluated n Secton IX. IX. EVALUATION In ths secton, we evaluate the performance of two varants of GCA usng smulatons n ns2 [8]. GCA-EXP adjusts W mn, where W s exponentally ncreased after a collson. GCA-DIRECT drectly adjusts W, wthout exponental ncrease of W after a collson. The evaluaton focuses on three aspects: () support for dfferent defntons of farness, (2) mantanng farness and (3) mantanng effcency. Although GCA supports varous farness defnton, we only present the performance of GCA for strct prorty and proportonal farness n ths paper. These two types of farness represent opposte extremes, where strct prorty requres that all bandwdth s allocated to the node wth the hghest prorty whle proportonal farness requres that every node gets a fracton of bandwdth proportonal to ts prorty. To demonstrate the correctness of GCA, we use the newly desgned specal case of GCA dscussed n Secton VIII-C. The performance of AOB, whch s an exstng varant of GCA, can be found n [5]. Fnally, channel bandwdth s Mbps. A. System Evoluton In ths secton, we llustrate how GCA adapts contenton wndow sze to support far and effcent bandwdth allocaton, usng unform 52B packet sze.

9 9 To examne the behavor of GCA for proportonal farness, n a 7-second smulaton, fve competng nodes start transmttng at tme 5s, 5s, 25s, 35s and 45s respectvely and use weghted log utlty functons wth weghts to 5. Fgure 4(a) shows that as the number of competng nodes ncreases, GCA quckly ncreases the contenton wndow szes of all competng nodes to prevent congeston. Therefore, GCA keeps the system operatng near ts optmal pont and mantans a steady throughput regardless of the number of competng nodes (see Fgure 4(c)). At the same tme, GCA mantans the rato between contenton wndow szes to provde each node ts weghted far share of bandwdth (see Fgure 4(b)). To examne the behavor of GCA for strct prorty, n a s smulaton, fve competng nodes start to transmt bulk data at 5s and each s equpped wth a lnear utlty functons wth weghts rangng from to 5. Fgure 5(a) shows that at the begnnng of the smulaton, the node wth weght 5 has a very small contenton wndow sze whle the other nodes wth lower weghts keep on ncreasng ther contenton wndow szes. Therefore, the node wth weght 5 soon obtans all channel bandwdth (see Fgure 5(b)). After the node wth weght 5 fnshes ts transmsson, the contenton wndow sze of the node wth weght 4 drops down and grabs the channel. Then after the node wth weght 4 fnshes, the node wth weght 3 gets the channel. The process goes on untl only the node wth the lowest prorty s left, demonstratng GCA s ablty n achevng strct prorty between competng nodes usng weghted lnear utlty functons. B. Farness Next, we evaluate GCA s accuracy n achevng farness by Jan s farness ndex [9], a common measure of farness for bandwdth allocaton. Gven n competng nodes, Jan s farness ndex s Ψ = ( /[ n = s /r ) 2 n n = (s /r ) 2], where r s the share of bandwdth proportonal to Node s weght and s s Node s acheved bandwdth. The farness ndex s a real value between and wth values closer to ndcatng better proportonal farness. When perfect proportonal farness s acheved, the farness ndex equals. If, on the other hand, only one node out of n s allocated bandwdth, the farness ndex s /n. Snce bandwdth allocaton based on strct prorty ams to gve all the bandwdth to the node wth the hghest prorty, the farness ndex should be /n for a perfect strct prorty based bandwdth allocaton. ) Weghted proportonal farness: To examne GCA s ablty for achevng weghted proportonal farness, n ths smulaton, 5 to 5 competng nodes start n the frst s and use weghted log utlty functons wth weghts from to 5. GCA s performance under both heterogeneous packet szes rangng from 4B to B and homogeneous packet szes of 52B are examned. Fgure 6(a) shows that when packet szes of nodes are dfferent, both GCA-EXP and GCA-DIRECT acheve a much larger farness ndex than IEEE 82.e and the farness ndex s very close to regardless of the number of competng nodes. Snce n IEEE 82.e, the contenton wndow sze s ndependent of the packet sze, nodes that send larger packets obtan more bandwdth than ther far share, resultng n IEEE 82.e s severe unfarness. Wth unformed packet sze, the farness for IEEE 82.e s greatly mproved (see Fgure 6(b)), although ts performance s stll worse than GCA. The farness ndex of GCA- EXP s also slghtly smaller than GCA-DIRECT because the exponental ncrement of the contenton wndow after a collson changes the rato between contenton wndow szes and hence degrades the farness of bandwdth allocaton. However, snce GCA-EXP s able to adjust the mnmum contenton wndow to avod excessve collsons, t essentally lmts the effects of collsons on farness, resultng n better farness performance than IEEE 82.e. 2) Strct prorty: To demonstrate the ablty of GCA to acheve strct prorty, we use smlar set up as n Secton IX-B. except that the utlty functon s a weghted lnear functon. Fgure 6(c) shows that regardless of unform or dfferent packet szes, both GCA-EXP and GCA-DIRECT acheve a farness ndex that s very close to the deal allocaton of strct prorty polcy. C. Channel utlzaton Fnally, we evaluate GCA s ablty to acheve hgh channel utlzaton by comparng t wth IEEE 82., IEEE 82.e and the theoretcal network capacty. ) Weghted proportonal farness: In ths set of smulatons, the channel utlzaton of GCA s compared wth IEEE 82.e and the theoretcal maxmum network capacty. For GCA, the weghts of the log utlty functons range from to 5. For IEEE 82.e, there are fve classes of traffc, wth the mnmum contenton wndow szes of the classes beng 3, 37, 5, 75 and 5, respectvely. These contenton wndow szes are selected to ensure smlar weghted bandwdth allocaton to GCA. Each smulaton runs for s wth 5 to 5 competng nodes, each startng at s and sendng 52B packets. Fgure 7(a) shows that the throughput of GCA, normalzed to the theoretcal maxmum capacty of an IEEE 82. network, s very close to the theoretcal lmt, ndcatng effcent channel usage. Snce IEEE 82.e can not dynamcally adjust ts mnmum contenton wndow sze accordng to the congeston level, ts channel utlzaton degrades as the number of competng nodes ncreases.

10 Contenton Wndow Sze Node wth weght= Node wth weght=2 Node wth weght=3 Node wth weght=4 Node wth weght= Tme (second) Throughput (packets per second) Node wth weght= Node wth weght=2 Node wth weght=3 Node wth weght=4 Node wth weght= Tme (second) Throughput (packets per second) Total Channel Throughput Tme (second) (a) Contenton wndow (b) Per node throughput (c) Total network throughput Fg. 4. Evoluton of GCA: proportonal farness. Contenton Wndow Sze Node wth weght= Node wth weght=2 Node wth weght=3 Node wth weght=4 Node wth weght=5 Throughput (packets per second) Node wth weght= Node wth weght=2 Node wth weght=3 Node wth weght=4 Node wth weght= Tme (second) Tme (second) (a) Contenton wndow (b) Per node throughput Fg. 5. Evoluton of GCA: strct prorty Farness ndex Farness ndex Farness ndex Theortcal Lmt GCA-DIRECT (dff. pkt. sze) GCA-EXP (dff. pkt. sze) GCA-DIRECT (same pkt. sze) GCA-EXP (same pkt. sze).84 GCA-DIRECT.82 GCA-EXP IEEE 82.e Number of competng nodes.96 GCA-DIRECT GCA-EXP IEEE 82.e Number of competng nodes Number of competng nodes (a) Proportonal farness (dff. pkt. sze) (b) Proportonal farness (same pkt. sze) (c) Strct prorty Fg. 6. Farness ndex Normalzed total throughput of network Theoretcal max. capacty.6 IEEE 82.e GCA-DIRECT GCA-EXP Number of competng nodes Normalzed total throughput of network Theoretcal max. capacty.5 GCA-DIRECT (dff. pkt. sze) GCA-EXP(dff. pkt. sze).4 IEEE 82. (dff. pkt. sze) GCA-DIRECT (same pkt. sze).3 GCA-EXP(same pkt. sze) IEEE 82. (same pkt. sze) Number of competng nodes (a) Proportonal farness (b) Strct prorty Fg. 7. Channel Utlzaton 2) Strct prorty: Snce strct prorty requres that only the node wth the hghest prorty wns the bandwdth, we compare the channel utlzaton of GCA to a snglenode IEEE 82. network and the theoretcal maxmum network capacty, under both fxed and heterogeneous packet szes. Each smulaton runs for s. All nodes start n the frst s. Fgure 7(b) shows that the throughput of GCA (wth multple competng nodes) normalzed to the maxmum network capacty s very close to the theoretcal lmt of the network, whle the channel utlzaton of the

11 sngle-node IEEE 82. network s much lower than GCA due to ts nablty to adapt to the lght load n the network. X. CONCLUSION AND FUTURE WORK In response to the lmtatons of current algorthms, n ths paper, we provde a systematc method for desgnng stable contenton wndow control algorthms that can be used to acheve far and effcent bandwdth allocaton. We decompose the requrement for both farness and effcency to the problem of choosng proper utlty functons and functons of observable channel states. Due to the ncluson of a wde dversty of both of these types of functons, we essentally broaden the scope of desgnng dynamc contenton wndow control algorthms. The general dynamc contenton wndow control algorthm (GCA) proposed by us can be used to acheve both arbtrary farness and effcent channel utlzaton. For future work, we plan on comparng the performances of dfferent choces of f(ω) and extendng GCA nto multhop wreless networks to support far bandwdth allocaton along wth effcent channel utlzaton. 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[22] Yalng Yang and Robn Kravets. Dstrbuted QoS Guarantees for Realtme Traffc n Ad Hoc Networks. Techncal Report UIUCDCS-R , June 24. [23] Yalng Yang, Jun Wang, and Robn Kravets. Dstrbuted Optmal Contenton Wndow Control for Elastc Traffc n Wreless LANs. Techncal Report UIUCDCS-R , Aprl 24. APPENDIX A. Proof that V s a Lyapunov functon Lemma : Scalar functon V (Z) n Equaton () s a Lyapunov functon for GCA-Z n Equaton (9) wth V. The zero values of V Ż are obtaned n the set R = {Z : Z = Ż j,, j N }. Proof: Note that the dervatve of V (Z) to Z s: V Z = Ũ ( Z L k N k N Z ) L Z Z L 2 ( k N Z ) 2 j N,j Ũ j ( L j Z k N Z ) jl jl ( k N Z ) 2 L = ( k N Z ) 2 j N,j [Ũ ( Z L k N Z ) Ũ j ( L j k N Z )]Z j L j (From Equaton (9)) L = ( k N Z ) 2 j N,j {[ Ż αz + f(ω)] Ż j [ α + f(ω)]} L j L = α( k N Z ) 2 j N,j [ Z Ż (Denote η = Żj ] L j α( k N Z, η > ) ) 2 = η[( j N,j L j ) L Z Ż ( j N ŻjL j )L ]. Hence, the tme dervatve of V (Z) s: V = N Z V Ż = η,j N,j [ ZjLjL Z Ż 2 + ZLLj Żj 2 2ŻŻjL L j ] = η,j N,j [( ZjL jl ZL Z Ż L j Ż j ) 2 +2 Ż Ż j L L j 2ŻŻjL L j ] η,j N,j ( ZjL jl ZL Z Ż L j Ż j ) 2. The equalty holds f and only f Ż Z = Żj, for, j N.

12 2 B. Proof that R s an nvarant set { Ż Lemma 2: For GCA-Z, R = Z : Z = Żj, for, j } N s an nvarant set and nsde ths nvarant set, Ũ ( Z L k N Z ) = kl Ũ k j ( L j k N Z ) = constant,, j N Proof: Combnng Equaton (9) wth the defnton of R, we get that nsde R, Ũ ( Z L k N Z ) = kl Ũ k j ( L j k N Z ) holds. To prove that R s an nvarant set, note that: d dt Ũ ( Z L k N Z ) = [ j N Ũ ( Z L )] Żj k N Z = Ũ ( Z L L k N Z ) ( k N Z [ ) 2 j N,j ŻL j ] j N,j Z ŻjL j =.(Insde R, Ż = ŻjZ.) Therefore, f at any tme GCA-Z gets nsde R, Ũ ( Z L k N Z ) for N remans the same constant for all future tme. Combned wth Equaton (9), Ż j, for, j holds for all future tme. Hence, R s an nvarant set for GCA-Z. C. Proof for convergence nsde the nvarant set Ż Z = Lemma 3: If Ω = Ω(Z, L) and f(ω) s strctly ncreasng wth respect to N Z L nsde R, then startng from any pont n R, GCA-Z converges to a unque equlbrum pont Ẑ R. Proof: For ease of notaton, we defne θ = N Z L and use f(z, L) to represent f(ω(z, L)). The rest of the proof conssts of three steps. Frst, we translate f(z, L) to a functon of θ. Then, usng the property that f(z, L) s strctly ncreasng wth respect to θ, we dentfy the unque equlbrum pont Ẑ. Fnally, we prove that Ẑ s the unque stable pont and so the update algorthm n Equaton (9) converges to Ẑ. Step : Lemma 2 shows that when GCA-Z converges to R, Ũ ( Z L k N Z ) for N remans a constant. Defnng ths constant as ˆγ, Ũ ( Z L k N Z ) = kl Ũ ( Z L ) = ˆγ, for N. (34) k θ Next, by defnng g( ) = {Ũ ( ) : N } and L = { L : N }, Z can be expressed as ({ } means set): { } Ũ (ˆγ)θ Z = {Z } = = θ[g(ˆγ)] T L. (35) L Therefore, f(z, L) = f(θ[g(ˆγ)] T L, L). Step 2: From Equatons (9) and (34), when Ż =, the value of θ, denoted ˆθ, satsfes: f(z, L) = f(ˆθ[g(ˆγ)] T L, L) = ˆγ. (36) Snce f(z, L) s strctly ncreasng wth respect to θ, ˆθ s unque. Therefore, the unque equlbrum pont n R s Ẑ = ˆθ[g(ˆγ)] T L = {Ẑ : N } and: ˆθ = k N Ẑ k L k. (37) Step 3: To show that Ẑ s the unque stable pont of the system, we defne a scalar functon V 2 (Z) as follows: V 2 (Z) = Z N Ẑ σ (Ẑ σ)l dσ. The followng proof shows that V 2 (Z) s a Lyapunov functon, and therefore GCA converges to Ẑ. V 2 = N( V 2 )Ż = (Ẑ Z )L Ż. (38) Z Z N Note that nsde R, accordng to Equatons (9) and (34), Ż = αz (Ũ ( Z L ) f(z, L)) = αz (ˆγ f(z, L)). (39) θ Combnng Equatons (38) and (39), V 2 = N α(ẑ Z )L (ˆγ f(z, L)) = α( N ẐL N Z L )(ˆγ f(z, L)) (Usng Equatons [ (36) and (37) ) ] = α(ˆθ θ) f(ˆθ[g(ˆγ)] T L, L) f(θ[g(ˆγ)] T L, L). Snce f(θ[g(ˆγ)] T L, L) s strctly ncreasng wth respect to θ n R, V2. Ths equalty holds f and only f θ = ˆθ. Therefore, we have shown that the algorthm n Equaton (9) converges to a unque pont Ẑ R. D. Notaton N: the set of transmttng nodes, 2 n C: the network capacty P : the probablty that Node successfully transmts n a vrtual slot s : the bandwdth allocated to Node n bts per second S: the channel sendng rate of IEEE 82. L : S the average duraton of a successful transmsson at Node, ncludng the DIFS and the RTS/CTS/DATA/ACK handshake x : the fracton of channel bandwdth of Node W : the contenton wndow sze of Node : the mnmum contenton wndow sze of Node W mn W: {W : N } L: {L : N } P: {P : N } Ω: locally observable channel state Z : W θ: L N = W k N Z Γ: the nvarant set of GCA R: the nvarant set of GCA-Z F: the average tme between successful packet transmssons I: the average number of dle vrtual slots between two busy vrtual slots ω: N W ϕ : /W ω T b : the average length of busy vrtual slot T c: the average duraton of a vrtual slot ncludng a collson P I : the probablty that a vrtual slot s an dle slot P c: the probablty that a collson happens n a vrtual slot