Computer Communications

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1 PREPRINT VERSION Contents lsts avalable at ScenceDrect Computer Communcatons journal homepage: A game theoretcal study of access pont assocaton n wreless mesh networks A. Argento, M. Cesana, N. Gatt, I. Malanchn Dpartmento d Elettronca e Informazone, Poltecnco d Mlano, Mlan, Italy artcle nfo abstract Artcle hstory: Receved 1 August 21 Receved n revsed form 14 December 21 Accepted 18 December 21 Avalable onlne 24 December 21 Keywords: Wreless mesh networks Assocaton Competton The process of assocaton to wreless access networks s often drven only by the qualty of the wreless lnk between the accessng moble staton and the wreless access pont; such qualty s manly assessed through the Receved Sgnal Strength Indcator (RSSI), whch captures the receved sgnal strength to/ from the wreless access pont. More refned measures have been proposed under dfferent wreless standards to capture traffc-orented parameters as the current load of the wreless access ponts. Stll the assocaton s mostly drven by local measures only whch represent the qualty of the wreless access pont only. In case connectvty s provded through a wreless mesh network (WMN), the qualty perceved by the user upon assocaton depends also on global network-wde parameters. These parameters nclude the length of the mult-hop wreless path to reach up to the Internet gateway, and the nterference level perceved by the traffc flow along such path. Ths paper addresses access pont assocaton n WMNs by resortng to game theoretc tools. The assocaton problem s formalzed as a non-cooperatve game where accessng statons selfshly play to mnmze ther own perceved assocaton cost (dually, maxmze the perceved assocaton qualty) whch shall account for the characterstcs of the entre path to reach the WMN gateway. The equlbra of the assocaton game are numercally derved and analyzed under two dfferent cost functons capturng n dfferent ways the assocaton nterference. Fnally, a leader follower game formulaton s gven for the case where wreless mesh network operators compete among themselves by properly settng the routng n ther managed wreless backbones. Ó 21 Elsever B.V. All rghts reserved. 1. Introducton The rapd dffuson of wreless technologes s nowadays promptng the end users wth multple and heterogeneous connectvty opportuntes. Wreless connectvty s largely avalable n urban and metropoltan areas whch actually feature prvate, publc, and commercal wreless pont of access. On the network s sde, the most common wreless access paradgm s stll the one based on hot-spots (e.g., WF-based) operatng as brdges between the wreless realm (connecton to the users) and the wred one (connecton to the Internet). However, such access paradgm may have both economcal and techncal shortcomngs under some network scenaros; as an example, the capllary coverage of large metropoltan areas may requre hgh captal nvestments for cablng up all the requred wreless access ponts, or cablng tself may smply be mpossble or externally constraned (e.g., hstorcal buldngs, etc.). To overcome the aforementoned lmtatons, the paradgm of wreless mesh networkng has been wdely recognzed as a cost effectve soluton for provdng wreless connectvty/access to Correspondng author. Tel.: E-mal addresses: mmargo@hotmal.t (A. Argento), cesana@elet.polm.t (M. Cesana), ngatt@elet.polm.t (N. Gatt), malanchn@elet.polm.t (I. Malanchn). moble users [1,2]. Such success s manly due to the hgh flexblty of the mesh networkng paradgm whch has many advantages n terms of self-confguraton capabltes and reduced nstallaton costs. Several WMNs deployments and ntatves have flourshed worldwde wth dfferent goals and applcaton targets [3,4]. Wreless mesh networks (WMNs) are composed of a mx of fxed and moble nodes nterconnected through mult-hop wreless lnks. Dfferent from flat Moble Ad hoc Networks (MANETs), the WMNs often feature a herarchcal archtecture. At the top of the herarchy, Mesh Gateways (MGs) are equpped wth wreless/wred connectvty cards and act as gateways toward the wred backbone. The Mesh Routers (MRs) have twofold functonaltes: they act as classcal Access Ponts (APs) towards the end users (access segment), and they have the capablty to set up a wreless backbone by connectng to other mesh routers through pont to pont wreless lnks to transport the access traffc to/from the MGs. The users, or user STAtons (STAs), can consequently get access to the network servces through mult-hop paths towards one or more MGs. Regardless the specfc wreless access network (cellular systems, WLANs, WMNs), the STA always has to make decsons on network assocaton. Namely, network assocaton refers to the dynamc and automatc choce of the best connectvty opportunty, that s, the cellular base staton n classcal cellular systems, the access /$ - see front matter Ó 21 Elsever B.V. All rghts reserved. do:1.116/j.comcom

2 542 A. Argento et al. / Computer Communcatons 35 (212) pont n WF WLANs, and the Mesh Router (MR) n WMNs. In classcal hot-spot-lke access networks (e.g., WLAN), the access decson s (almost) completely drven by the qualty of the last-mle lnk. Indeed, the STAs decde to assocate wth the WLAN hot-spot provdng the best wreless lnk accountng for the Sgnal to Interference and Nose Rato (SINR), the average load, the nomnal data rate, etc. In contrast, n case the access network s a WMN, the user-perceved qualty depends on the overall qualty of the entre path towards the specfc entry/ext MG (s); thus, the assocaton to the specfc MR n a WMNs should account for global qualty parameters, other than only local ones. The assocaton process s ntrnscally compettve n that each STA s wllng to act to maxmze her own perceved qualty. Moreover, the sngle assocaton choce taken by a STA mpacts the qualty of the other STAs. Roughly speakng, f a STA uses a wreless resource (access pont, wreless lnk, frequency band, etc.), t produces nterference to all the other STAs on the same resource. To ths extent, t becomes natural to study such compettve dynamcs resortng to non-cooperatve game theoretcal tools [5]. In ths work, we model the problem of assocaton n wreless mesh networks as a specal case of congeston game [6] where each STA acts selfshly to maxmze her perceved qualty of servce. To capture the end-to-end qualty perceved by each STA, we ntroduce two smple but consstent metrcs that capture the contenton perceved by the user s flow along a gven path of the WMN backbone (from the access MR to the MG). The frst metrc relates the perceved contenton to the number of wreless devces nterferng the gven path; the second metrc leverages the artme parameter ntroduced n the IEEE 82.11s standard. After formalzng the game, we show how to fnd ts Nash Equlbra, and we dscuss how the qualty of the equlbra can be characterzed n terms of Prce of Anarchy (PoA) and Prce of Stablty(PoS); we further expermentally study realstc sample network scenaros showng numercal results on the qualty of the equlbra. Fnally, we consder the case where also network operators compete among themselves to capture the largest number of accessng STAs by properly settng the mult-hop paths n ther managed WMN backbones. We then formalze ths scenaro as a multleader/mult-follower two-stage game where n the frst stage all the operators (.e., the leaders) play by choosng ther routng strateges, whle n the second stage all the STAs (.e., the followers) play the aforementoned assocaton game. The equlbra of ths game are fnally characterzed for sample network nstances. To summarze, the man contrbutons of the present work can be lsted as follows: () we formalze the problem of assocaton to wreless mesh networks by defnng a non-cooperatve game where the STAs selfshly select the least congested path; () we characterze the equlbra of the assocaton game and we obtan numercal results for sample network scenaros; () we formalze a mult-leader/mult-follower two-stage game model to capture also the competton among the network operators. The paper s organzed as follows: Secton 2 overvews the related work n the feld and comments further on the novel contrbutons of the present work. In Secton 3, we formalze the assocaton game for WMNs, whose propertes are hghlghted n Secton 4. A mathematcal programmng formulaton to fnd the Nash Equlbra of the assocaton game s proposed n Secton 5. In Secton 6, we dscuss numercal results on the qualty of the equlbra under realstc network scenaros. Secton 7 ntroduces network operators nto the game scenaro by employng a multleader/mult-follower game model. Secton 8 concludes the paper. 2. Related work Most of the currently deployed WLANs operate accordng to the IEEE standard [7], whch defnes access pont assocaton procedures based on the Receved Sgnal Strength Indcator (RSSI), only. Wthout gong nto the detals of the assocaton protocols, the STAs assocate to the access pont wth the hghest RSSI. It s commonly known that RSSI s not the proper metrc to be used n the assocaton phase [8]. Indeed, the relaton between RSSI and actual throughout perceved by the STA s not often easly predctable, and hgh RSSI values do not necessary mean hgh access throughput. The man shortcomng of RSSI-based assocaton comes from the fact that RSSI does not drectly account for the contenton/congeston at the specfc access pont, whch can often lead to poor resource utlzaton. To ths extent, much work has been carred out to mprove the metrcs used n the assocaton phase n WLANs [9] by leveragng measures/estmates on access nterference [1], on the actual access pont bandwdth [11,12], on user farness, and network-wde load balancng [13,14]. All the aforementoned references focus on the case of hotspot-lke access patterns, and consequently the metrcs for assocaton capture the local qualty of the wreless access segment. On the other hand, the ssue of desgnng effectve assocaton schemes n WMNs s fundamentally dfferent, snce the metrc drvng the assocaton must account also for global qualty parameters of the overall network. Wthn ths feld, the most common approach n desgnng assocaton algorthms/protocols leverages the network artme metrc to assess the qualty of mult-hop paths [15]. The artme of a wreless lnk n the wreless backbone measures the transfer tme of a test frame ncludng the tme overhead ntroduced by the access protocol and the lnk qualty n terms of frame error rate. Athanasou et al. propose n [16] and later n [8] an assocaton scheme whch couples the qualty and the current load of the access lnks wth a modfed verson of the artme metrc accountng for both the uplnk and downlnk segments. The authors further propose a practcal cross-layer protocol to mplement the assocaton procedures. Along the same lnes, Wang et al. propose n [17] a dynamc assocaton protocol whch couples the plan artme metrc for the backbone lnks, formalzed later n Eq. (2), wth an estmate of the access artme at the mesh access ponts. On the other hand, the work n [18] focuses on the defnton of metrcs to assess the qualty of the access segment. Namely, two metrcs are proposed accountng respectvely for the expected transmsson tme and the average load of the mesh access ponts. A modfed verson of the artme s consdered also n [19], whereas Ashraf et al. propose an assocaton metrc whch ncludes also the average load (queue length) of the gateways [2]. All the aforementoned approaches for WMNs share the common goal to desgn practcal assocaton protocols. In contrast, n ths work we are nterested n studyng the dynamcs and equlbra of the assocaton process n wreless mesh networks by leveragng tools and concepts of game theory. Game theory s wdely used to nvestgate the competton, cooperaton, and nteracton of multple agents. Related to the current work, non-cooperatve games have been prevously used to study network selecton dynamcs n [21 23]. In prevous work [24], we model network selecton through a non-cooperatve congeston game where the metrc drvng the access s the load of the WLAN access pont. The notable work n [25] also resorts to congeston games to analyze the assocaton process n WLAN, but the used metrc s the access artme, defned n Eq. (2). The man novel contrbuton of the current work s n the fact that we also account for network-wde metrcs n the game defnton. To our best knowledge, ths s one of the frst attempts to formalze the problem of assocaton n wreless mesh networks through a game.

3 A. Argento et al. / Computer Communcatons 35 (212) Reference scenaro and problem formulaton We consder a generc wreless mesh network composed of N network devces, ncludng mesh access ponts, mesh routers and gateways. Unless dfferently specfed, n the followng, we wll use access ponts, routers and gateways to refer to mesh access ponts, mesh routers and mesh gateways, respectvely. In partcular, A s the set of the access ponts and G s the set of gateways. The network provdes mult-hop access to a set of STAs U. All the traffc s assumed to flow to/from the gateway, whereas crosstraffc among STAs s neglected at ths stage. 1 Each access pont s connected through mult-hop paths to at least one gateway. The set P ncludes all the avalable routng paths n the network. The STAs can be potentally covered by multple access ponts, and can consequently decde to get connectvty to/from one specfc access pont, whch, n turn, wll reach the gateway through a specfc mult-hop path. Upon assocaton to a specfc access pont, the qualty perceved by the STA depends on the qualty of the mult-hop path to/from the gateway. The set P # P denotes all the routng path avalable for STA. The qualty of wreless mult-hop path s a long-debated ssue and depends on multple parameters affectng/nvolvng dfferent layers of the communcaton protocols: the wreless propagaton condtons, the specfc wreless technologes, the path-wse nterference of concurrent/contendng flows. We are not nterested n defnng here a complete and refned end-to-end metrc, but rather we are focusng on the study of the dynamcs of assocaton n WMNs. To ths extent, we leverage a smplfed but consstent cost functon c j for the STA over the routng path j, defned as: c j ¼ L j þ I j beng L j the path length (number of hops n the path), and I j the path contenton cost, that s, the contenton level experenced by the flow generated by STA along path j. In other words, the cost term I j of STA depends on the choces of the other STAs n the mesh network. Two dfferent approaches to model ths parameter are presented later on n ths secton. In the wreless mesh network assocaton problem the STAs need to make wse decsons on whch access pont to connect to (and consequently on whch path to use), by mnmzng the perceved overall assocaton cost. The decson process s ntrnscally dstrbuted (no coordnaton s allowed n the assocaton phase), and compettve, n that each STA selfshly mnmzes ts assocaton cost. Therefore, we model the wreless mesh assocaton problem as a non-cooperatve game n whch STAs are ratonal players amng at mnmzng ther costs, ther strategy space s composed by all the avalable routng paths 2 and ther payoffs are the assocaton cost whch account for the contenton along the mult-hop path to the gateway. The contenton-aware Wreless Mesh Network Assocaton Game (WMNAG) can be formally defned as: WMNAG ¼hU; fp g 2U ; fc j g 2U;j2P ð1þ A soluton for a game s a strategy profle r =(r 1,...,r ) wth 2 U specfyng one strategy r per player such that each player cannot reduce ts experenced cost by devatng unlaterally from r. We assume here that STAs playng the WMNAG have knowledge of the qualty of the entre path behnd an access pont; namely, the path length to reach the Internet gateway (L j ), and the average nterference perceved along that path ði j Þ s known by each STA. Such nformaton could be easly broadcasted by the very same access ponts through perodc beaconng. As an example, the beacons used n WF hot spots can be enhanced to nclude addtonal nformaton on the qualty of the entre path to the Internet gateway. In the followng sectons, we ntroduce two approaches to model the parameter I j whch yeld two dfferent cost functons n the WMNAG Cardnalty-based cost functon The contenton level experenced by a gven user flow along a wreless mult-hop path s deally proportonal to the number of wreless devces whch nterfere wth that path [26]. To ths extent, a frst, smple metrc to capture the path contenton cost can be based on the well-known concept of nterferng set. For every network devce n N, the nterferng set ncludes all the other devces whch nterfere wth the gven one and are currently used to delver STAs traffc. The path contenton cost I j can be calculated as the sum of the cardnaltes of all the nterference sets of all the nodes on path j. InFg. 1, dotted lnes represent nterference relatons, whereas bold lnes represent traffc flows; STA A s assocated to access pont 1 and reaches gateway 1 through the path ; STA B s assocated to access pont 2 and reaches gateway through path Focusng on STA B, the nterference set of devce 5 s composed of nodes 4 and 7, whereas the nterference set of node 8 s composed of node 7, only. Note that node 6 s not accounted n the nterference set of devce 5 snce t s not servcng any traffc. The overall nterference cost for path caused by traffc of STA A over path wll then be equal to 3. The same concept of nterference set can be appled also to those cases where paths are not dsjont by properly defnng avrtualnterference graph to be used for calculatng the nterference cost. Fg. 2 reports the case of two flows (to/from two STAs A and B) partally sharng the same path to the gateway, whereas Fg. 3 reports the vrtual graph used to calculate the nterference cost of the two flows Artme-based cost functon A lmtaton of the prevous metrc s n the fact that dfferent lnks (e.g., n terms of modulaton, dstance, transmtted power) Each STA maxmzes her own pay-off, regardless of the status of the other STAs,.e., each STA selects the routng path j 2 P whch mnmzes the experenced cost c j : j ¼ arg mn c j j2p 1 Nevertheless the proposed approach can be extended to the case of cross-traffc. 2 As mentoned before, each STA actually chooses one access pont and not a path. However, we assume a one-to-one correspondence between access ponts and paths and here we consder paths snce they are related to the cost functon. Fg. 1. Sample topology of a wreless mesh network.

4 544 A. Argento et al. / Computer Communcatons 35 (212) Fg. 4. Sample topology of a wreless mesh network. Fg. 2. Topology of a mesh network wth non-dsjont paths. cost perceved by STA along path j,.e., I j, can be calculated as the sum of the artme values of all the wreless lnks n the nterference sets of all the nodes on path j. As an example, Fg. 4 reports a network confguraton where two STAs select dfferent but nterferng paths to get connectvty to mesh gateways. 3 Namely, STA A uses path (path 1), whereas STA B goes through path (path 2). Let T j be the artme value for the wreless lnk between devce and devce j. The overall nterference contrbuton perceved by STA A s gven by the sum of the artme values of all the nodes n the nterference set of node 3,.e., node 4 and node 6. Then, t can be expressed by: I a 1 ¼ T 46 þ T 67 Smlarly, the nterference perceved by STA B s gven by the sum of the artme values for the nterference set of node 4 and node 6. Snce node 4 and node 6 have the same nterferng node,.e., node 3, the nterference cost for STA B s: I b 2 ¼ T 35 þ T Exstence of pure strateges assocaton equlbra Fg. 3. Vrtual graph to calculate the path contenton cost n the case of non-dsjont paths. nduce the same nterference on a gven lnk. To overcome ths smplfcaton, we extend here the cost metrc by leveragng the artme parameter proposed by the IEEE 82.11s standard as a consstent lnk qualty measure. The artme of the generc wreless lnk (,j) n the wreless backbone s defned as: T j ¼ O ca þ O p þ B t 1 ð2þ rð; jþ 1 e pt ð; jþ beng r(,j) the nomnal data rate of lnk (,j), e pt (,j) the frame error rate, B t the reference frame sze, O ca and O p the overhead related to the channel access and the protocol, respectvely. Intutvely, the artme captures the channel occupaton tme for a test transmsson over a wreless lnk. Ideally, wreless lnks wth longer artme values take longer tme to delver nformaton, and thus produce nterference onto the other lnks for longer tme. To ths extent, n the calculaton of the nterference contrbuton of the path contenton metrc t s reasonable to gve a hgher cost to lnks wth hgher artme value. Therefore, the path contenton Formally, the WMNAG ntroduced n the prevous secton s a mult-choce weghted congeston game wth player-specfc cost functons. In congeston games [6], the cost perceved by a player selectng a specfc resource strctly depends on the number of other players dong the same choce. In our case, the resources are the wreless lnks. Each STA, selectng a path, may nterfere multple wreless lnks (mult-choce), and the congeston of each STA to each lnk depends on the cardnalty of the nterference set (weghted). Furthermore, the cost functon s player-specfc snce the cost of a STA depends only on a subset of the nterfered lnks. In [27], the author proves that a weghted congeston game n whch players have equal payoff functons always admts at least one pure strategy equlbrum. 4 In [28], the authors prove that congeston games n whch each player s affected only by the congeston of a subset of resources always admt pure strategy 3 Interference relatons among wreless devces are ndcated wth dashed lnes. 4 The study of the exstence of pure strategy equlbra s very mportant. The basc assumpton behnd the employment of game tools for congeston problems, such as the one we are studyng, s that end users repeatedly play ther best response untl a stable state s not acheved. When the congeston games exhbt some propertes, the players are granted to converge to a pure strategy equlbrum. In these stuatons, the lterature lmts to study the stable states. When nstead no pure strategy exsts, the end users contnues to play ther best response wthout achevng any stable state

5 A. Argento et al. / Computer Communcatons 35 (212) equlbra. In ths case, the game s assumed to be non-weghted. In [29], the authors show that both weghted congeston games and games wth player-specfc delay functons possess pure Nash Equlbra n the case of sngle choce games,.e., when the players can choose only one resource. To the best of our knowledge, no result deals wth weghted games when utlty functons are player-specfc. The proposed WMNAG wth general nterference relatons among wreless devces may not admt any Nash Equlbrum n pure strateges. We prove t by provdng a smple counterexample. Let us consder the network topology reported n Fg. 5 composed of 3 access ponts, 6 mesh routers and 1 gateway. The network provdes connectvty to two STAs A and B whch can choose to assocate to each on of the 3 access ponts. The two STAs play usng the cardnalty-based cost functon. Let j be the path provded by AP j. The three paths n the network are nterference free except for the three mesh routers 7, 8 and 9, whch reach the gateway through drectonal antennas wth cone of nterference depcted n Fg. 5. The equvalent topology for the calculaton of the access cost s also reported n Fg. 5. Note that the drectonal nterference relatons are represented through arrows. If STA A assocates to AP 1 (.e., path 1) and STA B assocates to AP 2 (.e., path 2), the respectve assocaton costs are: c A1 ¼ L 1 þ I A 1 ¼ 3 þ 1 ¼ 4 c B2 ¼ L 2 þ I B 2 ¼ 3 þ 2 ¼ 5 STA B has ncentve to move to AP 3 gettng a lower assocaton cost: c B3 ¼ L 3 þ I B 3 ¼ 3 þ 1 ¼ 4 Assocatng to AP 3, STA B starts generatng nterference towards path 1 (from MR9 to MR7). Then A perceve a new assocaton cost: c A1 ¼ L 1 þ I A ¼ 3 þ 2 ¼ 5 Consequently, STA A has ncentve to move to AP 2 trggerng a new move of STA B. The two STAs keep changng ther played strategy never reachng a steady state regardless of the ntal strategy profle. It s easy to extend the very same example to the artme-based cost functon by properly settng the artme of the wreless lnks. The pecularty of the presented game scenaro s the asymmetry n the nterference relatons among wreless lnks. Even f NE may not exst n the WMNAG (as shown), t s worth to pont out here that n our expermental evaluatons we have found a NE for all the assocaton games wth symmetrc nterference patterns. 5. Fndng equlbra The WMNAG presented n the prevous secton may admt n general multple equlbra that dfferentate n terms of costs experenced by the STAs. Usually, the equlbra are characterzed n terms of ther socal cost, defned as the cumulatve costs of all the STAs. Obvously, no equlbrum has a socal cost smaller than the socal cost of the optmal assocaton found by usng a centralzed approach. One of the most common approach to analyze the dfference between equlbra and the optmal assocaton s based on two ndces, namely the Prce of Stablty [3] and the Prce of Anarchy [31]. They are, respectvely, the rato between the socal cost of the best/worst equlbrum and the socal cost of the optmal soluton. To fnd optmal equlbra (mnmzng/maxmzng the socal cost) and the optmal assocaton (mnmzng the socal cost wthout equlbrum constrants) we provde a set of mxed nteger lnear mathematcal programmng formulatons for the cardnalty-based and artme-based cost functons. We take a constructve approach and focus n ths secton on the smplfed case of a WMN featurng devces geared wth sngle rados and operatng on a sngle channel under the cardnalty-based cost functon. For such scenaro, we ntroduce the related mathematcal programmng formulaton to fnd the NE. A smlar formulaton have also been derved for the most general case of artme-based cost metrc and generc WMN topologes featurng multple rados and multple frequency channels. For the sake of brevty and readablty, we report the general formulaton n Appendx A. To represent the WMN topology we ntroduce the followng parameters of the model: 1 f node l belongs to path j s lj ¼ otherwse 1 f node l and node m nterfere a lm ¼ otherwse 1 f STA can choose path j d j ¼ otherwse We further ntroduce n our model the followng varables, n order to assocate each STA to a specfc access pont (and then to a specfc routng path): 1 f STA chooses path j y j ¼ otherwse The constrants of the model are the followng: y j ¼ U ð3þ j2p y j 6 d j 8 2 U; 8j 2 P ð4þ Constrants (3) ensure that each STA selects only one path (.e., access pont) whereas constrants (4) force each STA to select a path only among the set of the avalable ones. We defne the optmal soluton as the one that mnmzes the socal cost,.e., the sum of all the STAs costs c j. To ths extent, the followng objectve functon s used: mn c j y j ð5þ 2U j2p Fg. 5. Sample of network topology whch does not admt any pure strategy Nash Equlbrum (left) and correspondng nterference graph (rght). As defned before, the cost c j s composed by two terms: c j ¼ L j þ I j ð6þ

6 546 A. Argento et al. / Computer Communcatons 35 (212) where L j ¼ s lj 1 l2n I j ¼ r2u u2p r l2ðnnaþ m2ðnngþ s lj s mu a lm y ru Constrants (8) refer to the case of uplnk traffc, only. Indeed, t consders the set N n A (access ponts never suffer nterference because they never receve) and the set N n G (gateways never cause nterference to other nodes because they never transmt). A smlar constrant can be wrtten for the downlnk traffc, as well as for mxed downlnk/uplnk traffc patterns. The objectve functon n Eq. (5) s not lnear, however t can be easly lnearzed as follows: mn! s lj y j 1 þ I ð9þ 2U j2p l2n I P s lj s mu a lm y ru þ Mðy j 1Þ 8 2 U; 8j 2 P r2u u2p l2ðnnaþ m2ðnngþ r ð1þ where M s a bg number such that whenever y j =,l =. Note that I s the path contenton cost of STA for the path j she has actually chosen at the soluton,.e., when y j =1. The mxed nteger/lnear formulaton wth objectve functon (9) and constrants defned n (3), (4), and (1) can be used to characterzed the global optmal soluton. The very same formulaton can be also used to fnd the best Nash Equlbra of the game, by addng the followng constrants: Mðy j þ d h 2Þþ s lj þ s lj s mu a lm y ru l2n r2u u2p l2ðnnaþ m2ðnngþ 6 s lh þ l2n r2u u2p r 8 2 U; 8j; h 2 P : h j r l2ðnnaþ m2ðnngþ s lh s mu a lm y ru ð7þ ð8þ ð11þ that enforce the defnton of Nash Equlbrum. Indeed, constrants (11) requres each STA to select the path wth the mnmum assocaton cost, gven the other players strateges,.e., paths. Fnally, to fnd the worst Nash Equlbrum, we replace the objectve functon (9) and constrants (1) wth the followng maxmzaton of the socal cost: max! s lj y j 1 þ I 2U j2p l2n I 6 s lj s mu a lm y ru þ Mð1 y j Þ r2u u2p l2ðnnaþ m2ðnngþ r 6. Numercal results ð12þ 8 2 U; 8j 2 P ð13þ In ths secton, we provde numercal results to assess the qualty of the equlbra n realstc sample network topologes under the cardnalty-based and the artme-based cost functons. Two types of performance evaluaton are carred out: frst, we characterze the assocaton game by numercally computng the PoS and PoA under dfferent network topologes, then we compare the outcome of the assocaton game under dfferent assocaton polces. The man fndngs out of ths analyss for both cost functons can summarzed as follows: () the equlbra of the assocaton game are smlar n qualty to the optmal soluton,.e., PoA and PoS are pretty close to 1 n all the tested scenaros; () assocaton strateges accountng for end-to-end contenton n the WMN backbone lead to lower nterfered assocaton paths compared wth nterference-agnostc strateges Performance evaluaton settng To evaluate the outcome of the assocaton games, we have developed a network topology generator for WMNs whch deploys parametrc topologes wth dfferent number of STAs N, number of access ponts, number of routers and number of gateways. The software randomly deploys all the network devces and STAs n a square area of sde L and calculates the shortest paths between all the access ponts and the gateways usng the hop count as well as the artme metrc. Every devce has a crcular coverage/nterference regon wth radus r, and can feature up to 4 network nterface cards whch can be tuned to dfferent orthogonal channels. STAs are randomly scattered throughout the square network arena wth the constrant to be covered by at least two access ponts (and paths), and the end-to-end connectvty from each access pont to at least one gateway s enforced by the tool. When consderng the artme-based cost functon, upon the deployment of the WMN devces, the tool properly assgns artme values to all the lnks n the wreless topology. The artme calculaton utlty works under the followng assumptons: wreless devces are equpped wth omndrectonal antennas; transmssons are performed at the maxmum allowed power n each reference band (EIRP = 1 mw); transmssons are affected by addtve whte Gaussan nose (AWGN). Moreover, the followng SNR/dstance relatons have been used at 2.4 and 5 GHz, respectvely. SNR 2:4 GHz ¼ logðdþ SNR 5 GHz ¼ logðdþ ð14þ ð15þ From the two equatons above, the bt error rate has been obtaned accordng to the IEEE standards. Namely, the IEEE 82.11a at 54 Mbps (OFDM wth 64-QAM) has been adopted for the backbone transmsson and the IEEE 82.11g at 12 Mbps (OFDM wth QPSK) for the access network. The tables reportng the relatons between SNR, BER, dstance and artme values are reported n Appendx B. 5 The game nstances generated by the topology tool are fed nto the optmzaton models presented n Secton 5. In partcular, the models have been mplemented n AMPL [32] and solved wth CPLE [33]. Hereafter, all the reported results have been obtaned averagng over 2 nstances of the same type of the game (.e., users and network devces postons), unless dfferently specfed Equlbra under cardnalty-based cost functon Table 1 refers to a small-sze WMN deployment featurng a varable number of access ponts (6, 7 and 8), 1 mesh routers, 3 gateways, L = 1 m, r = 2 m and a varable number of STAs (N). The man result comng from the table s that the equlbra of the assocaton game are almost of the same qualty and very much close to the optmal assocaton,.e., PoS and PoA are smlar and very close to 1. Sad n other words, f the STAs play the assocaton game selfshly, accordng to the defned cost functon, the dstrbuted equlbrum s smlar n qualty to the centralzed optmal soluton. We note here that the network deployments of Table 1 may feature a low number of strateges for the end STAs, that s, the 5 Note that the proposed soluton technque s ndependent on the specfc rule/ procedure to assgn artme values to wreless lnks.

7 A. Argento et al. / Computer Communcatons 35 (212) Table 1 PoS and PoA for network topologes wth 1 mesh routers and 3 gateways under dfferent number of APs and STAs. N =15 N =25 N =35 N =45 N =55 N =65 6 APs PoS PoA APs PoS PoA APs PoS PoA average number of access ponts coverng each STA s around 2.5 for most of the cases. To ths extent, t s worth studyng whether smlar results on the qualty of Nash Equlbra hold true for larger strategy spaces. Table 2 reports the PoS and PoA for network topology constraned to have at least 2 and 3 assocaton alternatves per STA. The qualty of the equlbra s stll close to the optmum, even f a slght ncrease n the PoS and PoA can be apprecated as the number of STAs and the number of strateges per STAs ncrease. Fgs. 6 8 compare the qualty of the contenton-aware assocaton game aganst two other assocaton polces, under the very same settng of Table 1. Namely, ShortestPath refers to the case where the STA chooses the access pont closest to the gateway, whereas the assocaton polcy favors the access pont closest to the STA. The fgures report the average nterference perceved by the STAs. As expected, the equlbra of the contentonaware assocaton game provde an assocaton cost close to the optmum, whereas nterference-agnostc assocaton strateges always lead to hgher assocaton costs. Table 2 PoS and PoA for network topologes wth 4 access ponts, 9 mesh routers and 4 gateways under dfferent number of assocaton strateges per STA. 2 Strateges 3 Strateges N PoS PoA PoS PoA Equlbra under artme-based cost functon Smlar results have been obtaned for games featurng the artme-based cost functon. Table 3 reports the PoA and PoS for WMNs of dfferent szes. As n the case of cardnalty-based cost functon, the NE of the WMNAG are pretty close to the optmum, that s, also consderng the more general cost functon based on the artme, the compettve dynamc among the STAs leads to solutons whch are very close to the socal optmum. Fgs compare the average assocaton cost (average artme) for dfferent assocaton strateges. As clear from the fgures, the NE of the WMNAG are characterzed by assocaton costs whch are extremely close to the socal optmum. On the other hand, the other assocaton polces perform poorly leadng to an ncrease n the assocaton cost (and consequently n the perceved artme) whch can go up to 4%. Notably, the assocaton cost ncreases wth the number of STAs n the network. Fnally, t s worth analyzng the case where multple orthogonal channels can be used n the wreless backbone. A behavor smlar to the one wth a sngle channel has been observed n the multchannel case wth respect to the PoS and PoA. On the other hand, the avalablty of multple channels allows one to reduce the nterference among mult-hop paths n the wreless backbone. Fgs report the average assocaton cost (artme) perceved by the end STAs under dfferent assocaton polces. As expected, the average perceved cumulatve artme n the case of multple channels (four channels n the wreless backbone) s lower than n the sngle channel case. Moreover, nterference-agnostc 2 2 Average Cardnalty-based Interference Average Cardnalty-based Interference N=15 N=25 N=35 N=45 N=55 N=65 N=15 N=25 N=35 N=45 N=55 N=65 Fg. 6. Average perceved nterference under dfferent assocaton polces for WMNs wth 6 access ponts, 1 mesh routers and 3 gateways. Fg. 7. Average perceved nterference under dfferent assocaton polces for WMNs wth 7 access ponts, 1 mesh routers and 3 gateways.

8 548 A. Argento et al. / Computer Communcatons 35 (212) Table 3 PoS and PoA for network topologes wth varable number of access ponts, 1 mesh routers and 3 gateways under dfferent number of STAs. N =15 N =25 N =35 N =45 N =55 N =65 6 APs PoS PoA APs PoS PoA APs PoS PoA Average Artme [ms] Average Cardnalty-based Interference N=15 N=25 N=35 N=45 N=55 N=65 Fg. 1. Average perceved artme under dfferent assocaton polces for WMNs wth 7 access ponts, 1 mesh routers and 3 gateways N=15 N=25 N=35 N=45 N=55 N=65 Fg. 8. Average perceved nterference under dfferent assocaton polces for WMNs wth 8 access ponts, 1 mesh routers and 3 gateways. Average Artme [ms] N=15 N=25 N=35 N=45 N=55 N=65 Average Artme [ms] Fg. 11. Average perceved artme under dfferent assocaton polces for WMNs wth 8 access ponts, 1 mesh routers and 3 gateways. assocaton strateges lead to hgher assocaton costs wth respect to artme-based assocaton metrcs The technology for all test case 2 1 N=15 N=25 N=35 N=45 N=55 N=65 Fg. 9. Average perceved artme under dfferent assocaton polces for WMNs wth 6 access ponts, 1 mesh routers and 3 gateways. In order to analyze a realstc scenaro, we consder the network topology of the Technology For All (TFA) ntatve [3]. The network topology s reported n Fg. 15. Such topology features 17 access ponts and mesh routers, and 4 gateways. 6 5 and 75 STAs have been drawn randomly n the net- 6 The real TFA topology features 1 gateway drectly connected through dedcated pont-to-pont lnks to three more remote gateways. For the sake of the assocaton dynamcs, we have consdered the addtonal remote gateways as gateways.

9 A. Argento et al. / Computer Communcatons 35 (212) Multple Channels Sngle Channel 5 Multple Channels Sngle Channel 4 4 Average Artme [ms] 3 2 Average Artme [ms] Fg. 12. Average perceved artme under dfferent assocaton polces for WMNs wth 6 access ponts, 35 STAs, 3 gateways and 1 mesh routers. Fg. 14. Average perceved artme under dfferent assocaton polces for WMNs wth 8 access ponts, 35 STAs, 3 gateways and 1 mesh routers. 5 Multple Channels Sngle Channel 4 Average Artme [ms] 3 2 Fg. 15. Reference TFA network scenaro [3]. 1 Fg. 13. Average perceved artme under dfferent assocaton polces for WMNs wth 7 access ponts, 35 STAs, 3 gateways and 1 mesh routers. work area (crcles n Fg. 15) and the presented results are averaged over 2 STAs deployments. Each STA has, on average, 2.5 access pont to assocate to. Fg. 16 provdes the numercal results. The graphs confrm the exstng gap n terms of assocaton cost among the dfferent assocaton strateges. As expected, nterference-agnostc assocaton strateges can ncrease the assocaton cost up to 13% (N =75 and assocaton polcy). 7. Brngng network operators nto the competton The focus of prevous sectons was on the assocaton gven statc WMN topologes. Namely, routng paths were fxed between the access ponts and the gateways; thus, the STAs, by choosng the access pont to assocate, nherently choose also the routng path towards the gateways. However, n practcal settngs, the specfc route from each access pont to a gateway s usually determned by the routng protocol/polcy mplemented n the wreless backbone by the network operator. As an example, the network operator may choose the gateway to act as a traffc snk, or may eventually favor specfc mult-hop paths. 7 To ths extent, t s reasonable to assume that multple paths are avalable from each access pont towards a gateway, or, smlarly, that the network operator may choose a specfc gateway among multple opportuntes to route the traffc collected by a gven access pont. Such decson may have the purpose to balance the load among the gateways of the WMN, or, n a compettve scenaro wth multple WMN operators n the same arena, to capture the larger number of customers (STAs) by offerng paths wth hgher qualty. Hereafter, we focus on the latter compettve scenaro, and drop the assumpton of a fxed routng pattern n the wreless mesh network. Each WMN operator owns a set of access ponts, mesh rou- 7 e.g., to mplement load balancng technques at the gateways and/or at the wreless mesh routers.

10 55 A. Argento et al. / Computer Communcatons 35 (212) Average Cardnalty-Based Interference Average Artme [ms] N=5 N=75 N=5 N=75 Fg. 16. Average assocaton cost under dfferent assocaton polces for both the cardnalty-based (left) and the artme-based (rght) cost functon. ters, and a gven number of gateways n the wreless backbone. Multple operators compete wth the target to maxmze ther own revenues (number of customers) by properly settng the best path from the operated AP to one of the avalable gateways. As done n Secton 3 for the assocaton problem, we can model the WMN operators competton as a game. Let O be the set of operators. For each operator j 2 O, let S j be the strategy set, that s the set of mult-hop paths whch can be offered by operator j. The strategy space S s the set of all the possble combnatons of strateges played by the WMN operators,.e., S ¼ S 1 S 2 S n. An element S 2 S; S ¼ðS 1 ; S 2 ;...; S n Þ wth S j 2 S j,sa strategy profle. We can formally defne the B-Level Assocaton Game (BAG) as: BAG ¼hO; S; fp j ðs; WMNAGÞg j2o ð16þ It s further reasonable to decouple the decson tme of end STAs and network operator, by assumng that WMN operators play ther strateges frst, whereas end STAs play the assocaton game rght after, beng the WMN operators actons fully observable by the end STAs. 8 The BAG belongs to the class of mult-stage games: namely, t s a mult-leader, mult-follower game where leaders are the network operators choosng ther routng pattern, whereas followers are the STAs, playng the underlyng assocaton game. The payoffs for the STAs are the ones defned n Secton 3, whereas the payoffs for the generc WMN operator j assocated to the strategy profle S, p j (S,WMNAG), can be defned as the number of STAs whch decde to assocate to the network of operator j under the operators strategy profle S. Consequently, the payoffs of the game among the network operators depend on the underlyng WMNAG,.e., the assocaton game played by the STAs. Fg. 17 depcts a scenaro wth two operators ownng one access pont each provdng connectvty to STAs U 1 and U 2. Two multhop paths (dashed lne) to a couple of gateways are avalable from each access pont (operator). Operator 1 s strategy set s composed of paths P 11 towards gateway 1 and P 13 towards gateway 3. Smlarly, the strategy set of operator 2 s composed of paths P 22 and P 24. The paths composng the strategy sets of the operators may be generally composed of multple wreless network devces whch can nterfere one another. Therefore, referrng to the example reported n Fg. 17, the two operators have to choose frst whch path to actvate from ther 8 The assumpton s reasonable gven the dfferent dynamcs nvolved n the two processes of network confguraton and STA assocaton. Usually, network confguraton has longer dynamcs than user assocaton. Fg. 17. Sample WMN scenaro wth two operators (access ponts) each one wth two avalable paths. operated access ponts, and then the two STAs have to play the assocaton game defned n Secton 3. The equlbrum concept adopted for mult-stage games s the subgame perfect Nash Equlbrum, whch can be qualtatvely defned as the strategy profle (for leader(s) and follower(s)) whch s a Nash Equlbrum of every subgame of the orgnal game [5]. 9 It can be easly shown that: Theorem 7.1. Any subgame perfect equlbrum for BAG s Pareto optmal for the network operators. Proof. The proof comes from the observaton that for every strategy profle of the operators the payoffs of the operators sums up to the total number of STAs. Consequently, gven a NE pont for the BAG, t cannot exst another strategy profle where the payoff of some operators mproves wthout decreasng the payoff of any other operator. By contradcton, f such strategy profle exsted, some operators would get a hgher number of assgned STAs and the others would get the same number of STAs. Ths leads to a greater sum of payoffs, that corresponds to a greater number of STAs, whch contradcts the startng assumpton. h To fnd and characterze the equlbra of the BAG for small sze nstances, we develop a soluton method based on the enumeraton of all the operators strateges. Namely, for each pont of the strat- 9 Note that n our case a subgame s a WMNAG.

11 A. Argento et al. / Computer Communcatons 35 (212) egy space S, we fnd the WMNAG equlbra usng the technque proposed n Secton 5. Algorthm 1 formalzes the soluton procedure n a pseudo-code. Each teraton of the for cycle n Algorthm 1 requres the soluton of the WMNAG whch leads to assocaton relatons among STAs and access ponts. The qualty of such assgnment s then evaluated wth respect to the operators utlty, thus determnng the NE of the overall BAG. Table 4 Characterstcs of the BAG Nash Equlbra. Cost rato at the equlbrum versus the local optmum. N =15 N =25 N =35 N = Algorthm 1. Solve BAG 1: for S 2 S do 2: Solve WMNAG (S) 3: end for 4: Search for Operators Equlbra From the operators pont of vew, a strategy profle s a Nash Equlbrum f there s no player (.e., operator) whch has ncentve n modfyng ts strategy unlaterally. Therefore, a strategy profle S =(S j,s j ) s a Nash Equlbrum for the operators f and only f: p j ððs j ; S j Þ; WMNAGÞ P p j ððs j ; S jþ; WMNAGÞ 8S j 2 S j ; 8j 2 O; where S j s the strategy played by operator j, and S j ndcates the set of strateges played by all the other operators but j. From the consderatons above, the procedure to determne the Nash Equlbra of the BAG can be formalzed n the followng Algorthm 2. We have appled the soluton method to sample network scenaros featurng 2 and 3 operators ownng one access pont each, and beng able to actvate 2 and 3 paths towards gateways. The number of mesh routers n all the topologes has been fxed to 5 for each operator, whereas the number of STAs has been vared from 15 to 45. All the STAs can access all the access ponts n the scenaro. Algorthm 2. Search for Operators Equlbra 1: S eq ={;} 2: for S ¼ðS j ; S j Þ2S do 3: flag = 1 4: for j 2 O do 5: for S j 2 S j do 6: f!ðp j ððs j ; S j Þ; WMNAGÞ P p j ððs j ; S jþ; WMNAGÞÞ then 7: flag = 8: end f 9: end for 1: end for 11: f flag = 1 then 12: S eq +=S 13: end f 14: end for 15: Return S eq Table 4 reports the cost rato among the equlbrum and the local optmum. The local optmum represents the soluton n whch, fxed the strateges of the access ponts, the cost perceved by the users s optmzed. The rato s very close to one. Ths s consstent wth the fact that PoS and PoA for the sngle stage game (WMNAG) are very close to one. Table 5 reports the cost rato among the equlbrum and the global optmum where all the choces (routng patterns and assocaton) are globally optmzed to reduce the STAs perceved cost. Ths measure reflects the cost ncrease perceved by the STAs due Table 5 Characterstcs of the BAG Nash Equlbra. Cost rato at the equlbrum versus the global optmum. N =15 N =25 N =35 N = to the double competton among operators and among STAs themselves. Smlar to the WMNAG, the qualty of the subgame perfect equlbra s close to the global optmum. However, the network confguraton wth 3 operators (access ponts) and 3 paths for each operator features a slghtly hgher cost ncrease for the STA whch can go up to 3%. 8. Concludng remarks The dynamcs of network assocaton n wreless mesh networks have been analyzed n ths work by resortng to game theoretc tools. Namely, we have formalzed the assocaton problem as a non-cooperatve game n whch STAs selfshly play to mnmze a perceved assocaton cost accountng for the path length and the path nterference to reach the gateway. We have further proposed two metrcs to measure the path nterference based on the number of nterferng devces along the path and on the artme parameters. A quanttatve framework has been proposed to determne and characterze the Nash Equlbra of the aforementoned game under the two nterference metrcs. The numercal results derved for sample network topologes suggest that the game features equlbra whch are pretty close to the optmal (centralzed) assocaton pattern. Fnally, the game scenaro has been extended to the network operators whch have been allowed to set/modfy the routng patterns n the WMN backbone to capture the hghest number of customers (assocatng users). Such scenaro has been modeled through a mult-leader, mult-follower game where the network operators choose frst ther strateges (routng patterns), and then the STAs play the assocaton game. As n the case of the plan assocaton game, the Nash subgame perfect equlbra of the multleader, mult-follower game have been proven to be very close to the global optmal soluton. Acknowledgment Ths work s partally supported by MIUR-FIRB Integrated System for Emergency (InSyEme) project under the grant RBIP63BPH. Appendx A. Optmzaton model for mult-channel, artmebased cost functon In ths secton, we present the optmzaton model adopted to characterzed the Nash Equlbra and the optmal soluton of the mult-channel artme-based game. In order to derve the mathematcal formulaton, we assume that:

12 552 A. Argento et al. / Computer Communcatons 35 (212) the wreless lnks n the backbone share the same bearer and can further be tuned to dfferent orthogonal channels n the set C; all the access ponts share the same channel for the access; the backbone and the access network do not nterfere. The parameters used n the model are: ( 1 f devce l s recevng on channel f along path j s jlf ¼ otherwse ( 1 f devces l and m nterfere a lm ¼ otherwse ( 1 f user can choose path j d j ¼ otherwse Ar_access j s the artme (n ls) for user and the access pont j; Ar_back jflk s the artme of the wreless lnk between devce l and devce k operatng on channel f and belongng to path j; M s a bg number used for lnearzng the objectve functon. The decson varables are: 1 f user chooses path j y j ¼ otherwse I Ch s the total path contenton cost perceved by user. To derve the optmal soluton, the followng objectve functon s used: mn!! Ar back jfnm þ Ar access j y j þ I Ch ða:1þ 2U j2p n2n m2n f 2C The constrants of the model are the followng: I Ch P s jlf a lm y ru Ar back ufmk r2u u2p l2ðnnapþ m2ðnngwþ k2n f 2C r þ Ar access ru d rj y ru þ Mðy j 1Þ 8 2 U; 8j 2 P Table B.6 Artme values for IEEE 82.11g at 12 Mbps (QPSK). max 2U and I Ch SNR [db] d [m] BER Artme [ms] [,1] [167,178] [1,2] [156,167] [2,3] [147,156] [3,4] [138,147] [4,5] [129,138] [5,6] [121,129] [6,7] [114,121] [7,8] [17,114] [8,9] [1,17] [9,1] [94,1] Table B.7 Artme value for IEEE 82.11a at 54 Mbps (64 QAM). SNR [db] d [m] BER Artme [ms] [,1] [113,121] [1,2] [16,113] [2,3] [1,16] [3,4] [94,1] [4,5] [88,94] [5,6] [82,88] [6,7] [77,82] [7,8] [73,77] [8,9] [68,73] [9,1] [64,68] r2u r j2p u2p!! Ar back jfnm þ Ar access j y j þ I Ch n2n m2n f 2C l2ðnnapþ m2ðnngwþ k2n f 2C s jlf a lm y ru Ar back ufmk þar access ru d rj y ru þ Mð1 y j Þ 8 2 U; 8j 2 P ða:7þ I Ch P 8 2 U ða:3þ y j ¼ U ða:4þ j2p y j 6 d j 8 2 U; 8j 2 P ða:5þ To obtan the best Nash Equlbrum the followng equlbrum constrants should be added to (A.1) (A.5): Mðy j þ d h 2Þþ Ar back jfnm þ Ar access j n2n m2n þ r2u u2p r f 2C l2ðnnapþ m2ðnngwþ k2n f 2C s jlf a lm Ar back ufmk! þ Ar access ru d rj y ru 6 Ar back hfnm þ Ar access h n2n m2n þ r2u r u2p f 2C l2ðnnapþ m2ðnngwþ k2n f 2C s hlf a lm Ar back ufmk þ Ar access ru d rh y ru!8 2 U; 8j; h 2 P : h j ða:6þ Fnally, to calculate the worst Nash Equlbrum Eqs. (A.1) and (A.2) should be replaced respectvely by the followng two equatons: Appendx B. Artme calculaton Tables B.6 and B.7 report the artme values used n settng up the game envronment. References [1] I.F. Akyldz,. Wang, W. Wang, Wreless mesh networks: a survey, Computer Networks 47 (25) [2] R. Bruno, M. Cont, E. Gregor, Mesh networks: commodty multhop ad hoc networks, IEEE Communcatons Magazne 43 (25) [3] J.D. Camp, E.W. Knghtly, W.S. Reed, Developng and deployng multhop wreless networks for low-ncome communtes, n: Proceedngs of Dgtal Communtes, 25. [4] D. Wu, P. Mohapatra, Qurnet: a wde-area wreless mesh testbed for research and expermental evaluatons, n: IEEE COMSNETS, 21, pp [5] D. Fudenberg, J. Trole, Game Theory, The MIT Press, Cambrdge, USA, [6] R.W. Rosenthal, A class of games possessng pure-strategy Nash equlbra, Internatonal Journal of Game Theory 2 (1973) [7] IEEE, Wreless LAN medum access control (MAC) and physcal layer (PHY) specfcatons, IEEE std , [8] G. Athanasou, T. Koraks, O. Ercetn, L. Tassulas, A cross-layer framework for assocaton control n wreless mesh networks, IEEE Transactons on Moble Computng 8 (28) [9] A.J. Ncholson, Y. Chawathe, M.Y. Chen, B.D. Noble, D. Wetherall, Improved access pont selecton, n: ACM MobSys, 26, pp [1] V. Mhatre, K. Papagannak, Usng smart trggers for mproved user performance n wreless networks, n: ACM MobSys, 26, pp