Far and Effcent User-Network Assocaton Algorthm for Mult-Technology Wreless Networks Perre Coucheney, Cornne Touat and Bruno Gaujal INRIA Rhône-Alpes and LIG, MESCAL project, Grenoble France, {perre.coucheney, cornne.touat, bruno.gaujal}@mag.fr Abstract Recent moble equpment (as well as the norm IEEE 802.2) offers the possblty for users to swtch from one technology to another (vertcal handover). Ths allows flexblty n resource assgnments and, consequently, ncreases the potental throughput allocated to each user. In ths paper, we desgn a fully dstrbuted algorthm based on tral and error mechansms that explots the benefts of vertcal handover by fndng far and effcent assgnment schemes. On the one hand, mobles gradually update the fracton of data packets they send to each network based on the rewards they receve from the statons. On the other hand, network statons send rewards to each moble that represent the mpact each moble has on the cell throughput. Ths reward functon s closely related to the concept of margnal cost n the prcng lterature. Both the staton and the moble algorthms are smple enough to be mplemented n current standard equpment. Based on tools from evolutonary games, potental games and replcator dynamcs, we analytcally show the convergence of the algorthm to far and effcent solutons. Moreover, we show that after convergence, each user s connected to a sngle network cell whch avods costly repeated vertcal handovers. To acheve fast convergence, several smple heurstcs based on ths algorthm are proposed and tested. Indeed, for mplementaton purposes, the number of teratons should reman n the order of a few tens. I. INTRODUCTION The overall wreless market s expected to be served by sx or more major technologes (GSM, UMTS, HSDPA, WF, WMAX, LTE). Each technology has ts own advantages and drawbacks and none s expected to elmnate the rest. Moreover, rado access equpment s becomng more and more mult-standard, offerng the possblty of connectng through two or more technologes concurrently, usng norm IEEE 802.2. Swtchng between networks usng dfferent technology s referred to as vertcal handover. Ths s currently done n UMA, for nstance, whch gves an absolute prorty to WF over UMTS whenever a WF connecton s avalable. In ths paper, n contrast, we address the problem of computng an optmal and far assocaton through a dstrbuted algorthm. The contrbutons of the paper are: - We propose an teratve dstrbuted algorthm wth guaranteed convergence to a Nash equlbrum, based on real-tme measurements (as opposed to off-lne data). - Based on tools from potental games, we show that, by approprately settng up the reward measure, the resultng equlbra can be made Pareto effcent (optmal for the prce of anarchy [] and the SDF (Selfsh Degradaton Factor) [2]), and can correspond to any α-far pont (defned n cooperatve game theory [3]), for arbtrary chosen value of the parameter α. Ths wde famly of farness crtera ncludes Ths work was performed at the INRIA -ALU Bell Labs. jont laboratory. n partcular max-mn farness and proportonal farness and can be generalzed to cover the Nash Barganng Soluton [4]. - We show that the obtaned equlbrum s always pure: after convergence each user s assocated to a sngle technology. - We valdate our results through extensve smulatons of several mplementatons of the algorthm n the practcal settng of a geographcal area covered by a global WMAX network overlappng wth several local WF cells. Evolutonary games [5], [6], or the closely-related populaton games, are based on Darwnan-lke dynamcs. The evolutonary game lterature ncludes several so-called populaton dynamcs, whch model the evoluton of each populaton as tme goes by. In our context, a populaton could be a set of ndvduals adoptng the same strategy (.e. connected to the same network cell and adoptng dentcal network parameters). Recent work [7] have shown that, consderng the so-called replcator dynamcs, an approprate choce of the ftness functon (that determnes how well a populaton s adapted to ts envronment) leads to effcent equlbra. However they do not provde wth algorthms that follow the replcator dynamcs (and hence converge to the equlbra). Addtonally they do not justfy the use of evolutonary games. Indeed, such games assume a large number of ndvduals, each of them havng a neglgble mpact on the envronment and the ftness of others. Ths assumpton s not satsfed here, where the number of actve users n a gven cell s on the order of a few tens. The arrval or departure of a sngle one of them hence sgnfcantly mpacts the throughput allocated to others. As the number of players s lmted, we are hence dealng wth another knd of equlbra, namely the Nash Equlbra. In the context of load balancng, a few algorthms (see, for nstance [8], [9]) have been shown to converge to Nash Equlbra. It has been ponted out that ths class of algorthms has smlar behavor and convergence propertes as replcator dynamcs n evolutonary game theory. Yet, such algorthms may converge to mxed strategy Nash equlbra, where each user randomly pcks up a decson at each tme epoch. Such equlbra are unfortunately not nterestng n our case, as they amount to perpetual handover between networks. In the present paper, we revst prevous works n evolutonary games, wth addtonal farness consderatons, whle proposng an Nash learnng algorthm that can be mplemented on future moble equpments. In addton, our work present a novel result whch s that our algorthm converges to pure (as opposed to mxed) equlbra, preventng undesred repeated handovers between statons. Due to space lmtaton, the proofs of ths work have been omtted. The nterested reader should refer to [0] for the full verson.
II. FRAMEWORK AND MODEL We present below the model and notatons of the paper. A. Interconnecton of Heterogeneous wreless networks We consder a set C of network cells, that can be of varous technologes, and a fxed set I of actve users. Any of the I users can connect to a specfc subset of these cells and network technologes, dependng on her geographcal locaton, wreless equpment and operator subscrpton. B. User throughput and cell load By throughput, we refer to the rate of useful nformaton avalable for a user, n a gven network, sometmes also called goodput n the lterature. It depends on both the user s own parameters and the ones of others. These parameters nclude geographcal poston (nterference and attenuaton level) as well as wreless card settngs (codng schemes, TCP verson,...). As done n prevous papers [], we dscretze the cells of networks nto zones of dentcal throughput. Ths means that users n the same zone wll receve the same throughput. We denote by Z c the set of zones n cell c. The dstrbuton of users and ther number n the zones of a network s called load of the network (see Fg ). Zone separator B Zone 2 Zone WF cell WMAX cell Fg. : Heterogeneous system made of a wde WMAX cell and several WF hot-spots (n grey). As user B (n zone ) s closer to the WMAX antenna, she uses a more effcent codng scheme than A (n zone 2) (eg QAM nstead of QPSK). More formally, we suppose that each user has a set of network cells she can connect to, and a specfc zone assocated to each of them. An admssble choce a for user s a par a = (a c, az ) A. The set of all possble choces s A def = I A. The user s decson s denoted def by A = (A c, Az ). Then, we denote A the vector of users decsons A def = (A ) I, and call t an admssble assocaton. Hence, an assocaton s a functon from the set of users to the set of possble choces. For each assocaton A, the load on zone z of network c s denoted by l (c,z) (A), and s the number of users n ths zone usng cell c: c C, z Z c, l c,z (A) def = δ A,(c,z), wth δ the Kronecker delta I (δ a,b = f a = b and 0 otherwse). Hence, the load l c on cell c s a vector of sze Z c whose components correspond to the load on a partcular zone: l c (A) def = (l c,z (A)) z Zc. Assumpton : The throughput t c,z of cell c n zone z s a functon dependng only of the vector load l c (A) of cell c. Wth ths notatons, the throughput receved by user when she takes decson A = a s t a (l a c (A)). A III. NASH LEARNING ALGORITHM In ths secton, we buld an teratve algorthm wth guaranteed convergence to the Nash equlbrum of the system. A. Nash Equlbra (NE): Defntons Let x be a vector. We denote by x s the s th component of vector x, and by x s the set of the other components. It follows that up to some re-orderng, x = (x s, x s ). We also ntroduce e a a vector of sze Z a c defned by e a [s] def = f s = a z, and 0 otherwse. Defnton (Pure strategy NE): An assocaton scheme A = (A ) I s.t., A = a s a pure strategy NE for reward r f, I, a a, r a (l a c (A)) r a (l a c (A) + e a ). A mxed strategy for user s the choce of a vector probablty q over her possble choces. Each q,a s the probablty wth whch she takes acton a A. Equvalently, a mxed strategy of a user s the choce of the percentage of packets or sessons she sends to each network she has access to. Let S be the set of strateges for user : S = {q [0, ] A s.t. a A q,a = }, and Q S = I S the strategy matrx for all users, Q = (q ) I. When A = a, we denote by r a (l a c (A)) the reward receved by user (as for the throughput, t depends on the choces of the other mobles of the system, reflected n the assocaton vector A). For mxed strateges, R s the random varable correspondng to the reward of user (dependng on probabltes Q). Its mean s E Q [R ] def = E[R (A(Q)) Q = Q]. Defnton 2 (Mxed strateges NE): A set of I probablty vectors q of sze C s a NE f for reward R,, q q, E q,q [ R ] E q,q [ R ]. As the number of users and ther set of choces (networks) s fnte, ths s a fnte game and t admts (at least one) mxed strategy NE [2]. A and T are random varables correspondng respectvely to the decson and the throughput of user (dependng on probabltes Q). Then, the expected throughput for user s wrtten E Q [T ]. (Strctly speakng, T s a functon of the load, tself dependng on the strategy Q. For smplcty however, we omt the load L n the notatons.) B. Our Nash Learnng Algorthm A Nash learnng algorthm s an teratve algorthm on the strategy set that converges to a NE. Based on [8], we consder Algorthm, where b (σ) R s the step of user at tme σ. Snce I, q must reman n the strategy space S, the step sze s constraned by: m b (σ)r c,z (σ), where m = max a ( max( q,a, q ),a ) 0. q,a q,a Algorthm Learnng Algorthm Intalze arbtrarly vectors q (0) for all users At each tme epoch σ, forall user do Take decson â def = A (σ) wth probablty q (σ) Receve reward râ (lâ (A(σ))) Update strategy vector: a A, q,a (σ + ) q,a (σ) + b (σ)r a (l a (A(σ)))(δâc,a c q,a (σ)) () In the followng we use the term users and mobles nterchangeably.
Eq. determnes the update mechansm, whch we call system dynamcs. Reward functon: Selfsh behavor may lead to neffcent use. To crcumvent ths, we ntroduce some rewards that are notfed to users. Thus, nstead of competng for throughput, we consder an algorthm reflectng a non-cooperatve game between users that compete for maxmzng ther rewards. We consder the margnal cost prcng [3], whch asserts that each user on a network should pay a tax balancng the loss of throughput caused by her presence. Accordng to the random decson A, we defne the reward functon for user as: r A (l A c (A)) = G ( t a (l A c (A)) ) ( ) G(u,j (A)) G(v j (A)), wth (2) j δ A c j,a c u,j (A) = t Aj (l A c j (A) e A ), and v j (A) = t Aj (l A c j (A)). IV. ALGORITHM PROPERTIES In ths secton, we study the convergence propertes of our Nash learnng algorthm. A. Convergence Consder b def = sup σ max I b (σ), and Q b (θ) def = Q(σ), θ [σb, (σ+)b) the pecewse-constant nterpolaton matrx of Q(σ). Recall that a sequence of random varables (A t ) t R weakly converges to a random varable A f for any contnuous and bounded functon f: E[f(A t )] E[f(A)]. Then: Theorem : When b 0, the sequence {q,a b (.)} weakly converges to the soluton of a dfferental equaton, belongng to the replcator dynamcs famly: dq,a = q,a [f,a (Q) f (Q)], (3) dθ wth f,a (Q)=E Q [R A =a ] and f (Q)= q,a f,a (Q). a B. Effcency and Farness A As we consder elastc or data traffc, the Qualty-of-Servce of each user s her experenced throughput. We hence seek at Pareto optmal NE,.e. matrces Q such that Q Q, I s. t. E Q [T ] > E Q [T ], j I, E Q [T j ] < E Q [T j ]. Further, our NE should not only be Pareto optmal but also α-far [3],.e. satsfes max Q I E Q [G(T )] wth G(x) def = x α α. (4) Note that n pure strateges, for each moble such that A = a, then E Q [T ] = t a (l a c (A)). So, we am at fndng a set of decsons A = (A ) I that reaches max a A I G(t a (l a c (A))). When α=0, the correspondng soluton s a socal optmum. The lmt α s the proportonal far pont (or Nash Barganng Soluton) and α, the max-mn far pont. Parameter α hence gves flexblty n the choce of Pareto optmal ponts: from fully effcent to perfectly far allocatons. Consder a dynamcs V (Q) = dq. We say that Q s a dθ statonary pont for V f V (Q) = 0. Also, Q s asymptotcally stable f there exsts a neghborhood U of Q such that: Q(θ) U Q(θ) Q. Then, one can show that the dynamcs θ (3), (and hence the algorthm when b 0) can only converge to a NE of the system. Followng an dea of [4], we consder the summaton of all users expected α-far throughput: F (Q) = q,a E Q [G(T ) A = a ]. (5) I a A Then, we have the followng proposton: Proposton : F s a potental functon for the dynamcs (3),.e. : I, a A, f,a (Q) = F q,a (Q). For replcator dynamcs (Eq. 3), the potental functon F s a Lyapunov functon (t ncreases along the trajectores) [5], [6]. Hence, the global α-far throughput converges to an optmal pont when the dfferental equaton trajectory approaches a stable state. Ths s summarzed n the followng theorem. Theorem 2: Usng reward functon (2), the dynamcs (3), and hence Algorthm converge to an asymptotcally stable state that maxmzes the total expected α-far throughput. C. Pure Strateges Unlke mult-homng between WF systems (see [7], [4]), mult-homng between dfferent technologes (e.g. WF and WMAX) nduces several complcatons: the dfferent technologes may have dfferent delays, packet szes or codng systems,... and re-constructng the messages sent by the mobles may be hazardous. Durng the convergence phase, each moble s usng mxed strateges. Yet, studyng the stablty of equlbrum ponts of Eq. 3, one can show that our algorthm converges -after a transtonal state- to NE n whch each user uses a sngle network,.e. pure strategy equlbra (full proof avalable n [0]). From Theorem 2 and the fact that the algorthm converges to a pure equlbrum, we can conclude: Theorem 3: Algorthm wth rewards 2 converges to a pure NE whch corresponds to the α-far pont of the system. Probablty q,a 0.8 0.6 0.4 0.2 0 0 500 000 500 2000 Tme epoch σ Fg. 2: Convergence of the probablty values for each of the 5 possble choces of one user. Consder a typcal run of algorthm over a system made of 0 users wth 5 choces over 0 networks (Fg. 2). As, A = 5, then, a A, q,a = 0.2 (ntal equprobablty of all choces). As σ grows, all probabltes tend to 0 except correspondng to the optmal acton at the pure NE. V. IMPLEMENTATION AND VALIDATION Frst, notce that each user only needs to know her own reward to update her step sze and strategy vector. Second,
each base staton only needs her own load to compute the rewards, hence allowng for a fully dstrbuted algorthm. In the prevous sectons, convergence of the algorthm has been shown when the step sze b tends to 0. Ths secton shows numercal tests we performed to study possble practcal heurstcs for the step sze computaton. Indeed, whle the step szes should be small enough to ensure convergence, larger values speed up convergence. A. The Dfferent Heurstcs for the Steps Each heurstc actually conssts of two parts: A roundng up test: As tme ncreases, the probabltes of choosng each acton tends ether to 0 or. To speed up convergence, we consder thresholds ɛ m and ɛ M such that: { q,a (σ + ) 0 f q I, a A,,a (σ) < ɛ m q,a (σ + ) f q,a (σ) > ɛ M. After roundng up a strategy, the vector s normalzed to reman n S. In the tests, ɛ m = 0.05 and ɛ M = 0.3. A step sze computaton: : dfferent schemes to compute b (σ) are consdered: ) Constant Step Sze (CSS): the steps are predefned: I, σ, b (σ) = b. Numercal values are b = 0.0 (CSS L, slow but optmal), b= (CSS H ) and the ntermedate b=0. (CSS M ). 2) Constant Update Sze (CUS): the steps are such that the changes of probabltes are bounded by a predefned value q (fxed to 0. n the experments): I, a A, abs (q,a (σ + ) q,a (σ)) q. 3) Decreasng Step Sze (DSS): potental pure NE are detected durng a few large steps teratons and then confrmed or nfered by smaller step teratons. We mplemented 2 varants: a) DSS-SA: Inspred from smulated annealng, we consder cyclc decreasng step sze: b = 3/(σ mod 0). b) DSS-CSS: A DSS phase to stablze most users followed by a constant large step sze to speed up convergence of the others: b = 4/σ f σ < 20 and b = 4 otherwse. B. System Scenaro We consder a smple scenaro of an operator provdng subscrbers wth a servce avalable ether through a large WMAX cell or a seres of WF access ponts. For each smulaton, a topology s chosen randomly, accordng to 3 parameters: number of users, of WF access ponts and of possble choces for each user A. More precsely, for each user: The frst choce s the WMAX cell and one of the 8 possble zones (cf Secton V-C), pcked at random (unformly). All other A choces are one of the C cells, pcked up accordng to a unform law. As explaned n Secton V-C, we do not consder zones n the WF cells. The strategy vector s ntalzed wth equal probabltes: I, a A, q,a (0) = / A. C. Throughput of TCP sessons n WLAN and WMAX To valdate our algorthm, we mplemented n our smulator the average mean throughputs of TCP connectons avalable n the ltterature obtaned through flud approxmatons. Equatons of throughput n WF cells: Based on [7], we consder that the throughput of connecton s L T CP t (l c ) = l c (T DAT A + T ACK + 2T T BO (l c ) + 2T W (l c )) where L T CP = 8000 bts, T ACK =.09 ms, T DAT A =.785 ms. Then, T W and T T BO are soluton of the fxed pont equaton gven n [7]. WMAX: We consder a far sharng of N bc carrers [8]: f p users are present n the cell, each of them wll receve NbC/p sub-carrers. Hence, the throughput of a user n zone z s roughly the fracton / z Zc l c,z of the throughput she would obtan f she were alone n the cell. For a sngle user, we follow expermental values obtaned n [9] for IEEE WMAX 802.6d for ts eght zones: Modulaton QAM64 3/4 QAM64 2/3 QAM6 3/4 QAM6 /2 TCP goodput 9.58 8.88 6.80 4.50 Modulaton QPSK 3/4 QPSK /2 BPSK 3/4 BPSK /2 TCP goodput 3.37 2.2.65.08 D. Comparsons between Heurstcs Fgure 3(a) dsplays the performance (n terms of global throughput) obtaned by the sx heurstcs as a functon of the total number of users I. For a gven load, all heurstcs have been tested on the same topology to allow a far comparson. The small constant step sze (CSS L wth b = 0.0), provdes the best performance. It s even tested optmal for the small values of I, up to 20. Most heurstcs stay wthn 0 % of the optmal (except for DSS-CSS whose performance can be poor). Also note that the total capacty of the system s less than 36 (0 2.6(WF)+9.58 (WMAX)) Mbt/s. Thus, the best heurstc s always wthn 5 % of the optmal. Fnally, t should be noted that the medum constant step sze (CSS M ) wth b = 0. s always very close to the best (CSS L ) and that the constant update sze (CU S) performs better and better when the number of users grows. As for the number of teratons, t vares wdely between the dfferent heurstcs, even on a logarthmc scale (Fgure 3(b)). The CU S heurstc s a clear wnner here (wth an average number of teratons never above 80). Meanwhle, CSS L does not always converge wthn the lmt of 20,000 teratons set n the program. Under hgh loads, CU S provdes the best compromse wth very fast convergence and reasonable performance. Under lght load, the constant step sze of medum sze (CSS M ) s also an nterestng choce, for ts performance s almost optmal and ts number of teratons remans below 00. E. Impact on Farness Consder the followng scenaro: a set of 20 users, each havng 3 avalable choces among 0 cells. The WMAX cell s numbered 0 and ts 8 zones are numbered from 0 to 7. The set of choces of the users are A = {{0, }, {8}, {}} {{0, 5}, {6}, {4}} {{0, }, {6}, {9}} {{0, 2}, {2}, {6}} {{0, 3}, {8}, {9}} {{0, 6}, {4}, {9}} {{0, 7}, {3}, {6}} {{0, 4}, {}, {2}} {{0, 6}, {6}, {9}} {{0, 5}, {3}, {4}} {{0, 6}, {3}, {}} {{0, 7}, {9}, {6}} {{0, 3}, {8}, {}} {{0, 6}, {4}, {7}} {{0, 6}, {9}, {5}} {{0, 0}, {6}, {5}} {{0, 5}, {4}, {}} {{0, 6}, {6}, {4}} {{0, 3}, {3}, {4}} {{0, 3}, {8}, {4}}. The optmal assocaton scheme, for α = 0 (effcent scheme) and α = 2 (far schemes) are respectvely: A eff = {2,, 2,,,,, 2, 2, 2,,, 2, 2, 2, 0, 2,,, }, A far = {0,, 0,, 0, 2,, 2,,, 2,,, 2, 2, 2,, 2, 0, }.
Throughput (Mb/s) 35 34 33 4 5 Number of teratons 00000 4 0000 32 3 30 29 28 27 26 2 3 6 5 20 25 30 35 40 45 50 Load 000 00 0 (a) Average performance (b) Average number of teratons Fg. 3: Comparson of the heurstcs (CUS, DSS SA, DSS CSS, CSS L, CSS M and CSS H resp.) under dfferent loads (wth 5% confdence ntervals). The nduced throughputs are: T eff = 0.824,.225, 0.824,.225,.225,.225, 0.824,.225, 0.824,.225, 0.824, 0.824, 0.824, 2.245, 2.246, 9.58, 0.824,.225, 0.824,.225. T far = 2.22,.225, 2.22,.225,.25,.225,.225,.225,.225,.225, 2.245,.225,.225, 2.246,.225,.225,.225,.225,.25,.225. The effcent scheme acheves a total throughput of 3.29 Mb/s. The far scheme suffers a degradaton of slghtly less than 0%, wth a total throughput of 28.34 Mb/s. Yet a closer look at the fgures ndcates that the effcent scheme leads to hgh dfferences between users (user only obtans a throughput of 0.8 Mb/s whle user 6 s granted 9.58 Mb/s). Meanwhle, n the far assocaton scheme, all users beneft from throughputs hgher. Mb/s. As n bandwdth allocaton mechansms n wred systems [4], the parameter α hence allows one to fnely tune the compromse between maxmum global throughput and farness between users. F. Seamless Adaptaton to User Arrvals and Departures The assocaton algorthm has to be run at every arrval or departure of a user n a cell. We now dscuss the mpact of such events. Frst, typcal tme scales compare ncely: whle arrvals or departures of users n WMAX or WF cells occur every mnute or so, the assocaton algorthm converges n less than a second n most cases, when the best heurstc (CUS) for the step szes s used. Second, the convergence of the algorthm s much faster when the ntal state s chosen close to the former optmal soluton for all old users and one change occurs (an arrval of a departure). Ths s llustrated n Fg. 4: n the frst phase, 30 users wth 3 choces over 0 cells run the algorthm usng CUS step updates. Startng wth unform probabltes (/3, /3, /3), convergence occurs after 60 teratons. Upon arrval of a new user, the algorthm s reset the startng ponts s (/3, /3, /3) for the newcomer and (/4, /4, /2) for all other users, wth probablty /2 gven to the prevously chosen cell. Then, convergence only requres 27 teratons. VI. CONCLUSION AND FUTURE WORKS In ths paper, we have desgned a dstrbuted algorthm that converges to an optmal (n terms of farness or effcency) network assocaton n heterogeneous wreless networks. Ths opens the way to several nterestng future works, such as the mplementaton of such methods n modern moble devces n collaboraton wth Alcatel-Lucent. 2 3 5 6 Throughput 5 20 25 30 35 40 45 50 34 32 30 28 26 24 22 20 Convergence Arrval New convergence 0 0 20 30 40 50 60 70 80 90 Load Iteratons Fg. 4: Convergence speed after arrval of a new user. REFERENCES [] E. Koutsoupas and C. Papadmtrou, Worst-case equlbra, n Proc. of STACS, 998. [2] A. Legrand and C. Touat, Non-cooperatve schedulng of multple bag-of-task appplcatons, n Proc. of INFOCOM, May 2007. [3] J. Mo and J. Walrand, Far end-to-end wndow-based congeston control, n Proc. of SPIE, I. Symp. on Voce, Vdeo & Data Comm., 998. [4] C. Touat, E. Altman, and J. Galter, Generalsed Nash barganng soluton for banwdth allocaton, Computer Networks, vol. 50, no. 7, pp. 3242 3263, Dec. 2006. [5] J. W. Webull, Evolutonary Game Theory. MIT Press, 995. [6] J. Hofbauer and K. Sgmund, Evolutonary game dynamcs, Bull. Amer. Math. Soc., vol. 40, pp. 479 59, 2003. [7] S. Shakkotta, E. Altman, and A. Kumar, Multhomng of users to access ponts n WLANs: A populaton game perspectve, IEEE J. on Sel. Areas of Comm., vol. 25, no. 6, pp. 207 25, 2007. [8] D. Barth, O. Bournez, O. Boussaton, and J. Cohen, Dstrbuted learnng of Wardrop equlbra, n Proc. of UC, ser. LNCS, vol. 5204, 2008. [9] P. Sastry, V. Phansalkar, and A. Thathachar, Decentralzed learnng of Nash equlbra n mult-person stochastc games wth ncomplete nformaton, IEEE Transactons on System, man, and cybernetcs, vol. 24, no. 5, pp. 769 777, 994. [0] P. Coucheney, C. Touat, and B. Gaujal, A dstrbuted algorthm for far and effcent user-network assocaton n mult-technology wreless networks, INRIA, Tech. Rep., 2008. [] M. Coupechoux, J.-M. Kelf, and P. Godlewsk, Network controlled jont rado resource management for heterogeneous networks, n Proc. of IEEE VTC Sprng, 2008. [2] J. F. Nash, Equlbrum ponts n n-person games, Proceedng of the Natonal Academy of Scences, vol. 36, pp. 48 49, 950. [3] R. Cole, Y. Dods, and T. Roughgarden, Prcng network edges for heterogeneous selfsh users, n Proc. of STOC, 2003. [4] S. Shakkotta, E. Altman, and A. Kumar, The case for non-cooperatve multhomng of users to access ponts n IEEE 802. WLANs, n Proc. of INFOCOM, 2005. [5] D. Monderer and L. S. Shapley, Potental games, Games and Economc Behavor, vol. 4, pp. 24 43, 996. [6] W. H. Sandholm, Potental games wth contnuous player sets, Journal of Economc Theory, vol. 24, pp. 8 08, 200. [7] D. Morand, A. A. Kheran, and E. Altman, A queueng model for HTTP traffc over IEEE 802. WLANs, IEEE Computer Networks, vol. 50, pp. 63 79, 2006. [8] C. Tarhn and T. Chahed, System capacty n OFDMA-based WMAX, n Proc. of the Int. Conf. on Syst. & Network Comm., 2006. [9] F. Yousaf, K. Danel, and C. Wetfeld, Performance evaluaton of IEEE 802.6 WMAX lnk wth respect to hgher layer protocols, n Proc. of the Int. Symp. on Wreless Communcaton Systems, 2007, pp. 80 84.