Framework for resource allocation in heterogeneous wireless networks using game theory

Transcription

1 Framework for resource allocatio i heterogeeous wireless etworks usig game theory Mariaa Dirai ad Tijai Chahed GET/Istitut Natioal des Telecommuicatios UMR CNRS rue C. Fourier EVRY CEDEX - Frace {mariaa.dirai,tijai.chahed}@it-evry.fr Abstract. This is a framework for resource allocatio i a heterogeeous system composed of various access etworks, for istace Third Geeratio wireless etworks (3G) ad WLAN, i the presece of multimedia traffic, amely voice ad data. Our aim is to preset a game theoretical modelig of routig ad loadbalacig strategies alog with admissio cotrol ad pricig i cooperative ad o-cooperative settigs. 1 Itroductio I a heterogeeous eviromet composed of more tha oe access etwork, say third geeratio (3G) [1] [2] ad IEEE based WLAN [3] etworks, the questio is that of how to allocate resources i a way that is optimal both to the user ad operator (see Figure 1). UMTS/HSDPA voice or data? Iteret WLAN Fig. 1. Referece model. The aswer is multi-fold. O the user side, the issue is that of routig. Optimality i this case refers to the best QoS, accordig to the type of traffic, at the best price. O the operator side, the issue is that of admissio cotrol ad pricig so as to maximize reveue which is proportioal to resource utilizatio itself subject to grated QoS ad the prices assiged to each type of resources as well as their cost.

2 Game theory [4] [5] studies iteractios ad wiig strategies for parties ivolved i situatios where their iterest coflict with each other. It has applicatios to real games, ecoomics, commerce, politics ad recetly telecommuicatios. The research area of etworkig games ad their applicatio i telecommuicatios has kow rapid developmets through the last years [6]. Sice optimizatio theory is uable to take ito accout iteractios betwee differet actors, be they users, protocols, ature or other, game theory has show to be a useful tool allowig to study behavior ad evetual equilibrium of complicated ad iteractig systems. Ispired from ecoomical eviromets, game theoretical modelig proved to be a powerful tool to study resources allocatio i homogeeous as well as heterogeeous systems presetig cotedig users or classes for those resources. More specifically, it has bee used i power cotrol [7] ad access to a commo shared lik, but also to study flow cotrol problems ad to fid structural properties of equilibrias i systems ivolvig routig ito liks with differet capacities ([8], [9]). For a survey o applicatios but also o methodologies ad challeges of etworkig games i telecommuicatio, the reader is refered to Referece [6]. Our aim i this work is to model iteractio betwee differet, o-homogeeous types of traffic, amely voice ad data, with evetually differet radio coditios, competig for the use of resources i heterogeeous etworks offerig differet QoS. The dyamic sharig ature of wireless protocols i recet etworks, particularly Medium Access Cotrol (MAC) protocols, makes the game theoretical tool suitable to study iteractio ad resultig strategies of ivolved players, be they the ed users askig for resources ad subsequetly a QoS level, ad the etwork strategies for admittig those calls ad pricig them after admissio. At this poit, we have to distiguish betwee cooperative ad o-cooperative settigs, both o the user ad operator side. O the user side, users may cooperate so as to achieve the best global utility fuctio. This is ot very probable. Ofte, users are o-cooperative wherei oly maximizig the idividual utility fuctio is at sake. O the operator side, cooperatio is typically give whe the heterogeeous access etworks belog to oe operator ad the objective is agai to maximize some global utility fuctio. If differet players ow the differet etworks, the problem becomes a game where each oe tries to maximize its ow utility fuctio. Now, takig the problem as a whole, with both users ad operator(s), the resource allocatio problem may be solved through some classical Nash equilibrium [5], defied as beig the players actios from which o player has a icetive to uilaterally deviate, or may be brought to some optimal operatio poit through for example icetives by the operator to guide users to choices that simultaeously maximize their utility fuctio as well as his. The remaider of this work is orgaized as follows. I sectio II, we give a basic itroductio for the tool of game theory. I Sectio III, we ivestigate the routig ad show the distictio betwee cooperative versus the o-cooperative cases. The same distictio pertais for admissio cotrol ad pricig too ad is preseted i Sectio IV. Sectio V presets a cross-layer modelig of the two systems uder cosideratio. Evetually, Sectio VI cocludes the paper.

3 2 Game theory primer A strategic game has three compoets: A set of players Pl A set of possible actios A pl for each player pl Pl A set of utilities U, where for each player pl Pl, the utility for each player is a fuctio of the actio profile a = (a g, a g ), a g beig the actio of player pl ad a g the vector of all other players strategies. A actio profile belogs to the set of actios profiles deotes by A = A pl. I other words a utility fuctio is a pl Pl mappig from the set of all actios profiles A to the set of real umbers R. Two settigs may arise i a system ivolvig several players. Players may cooperate ad the problem reduces to a optimizatio problem where a sigle player drives the system to a social equilibrium. A stadard criterio used i game theor to express efficiecy of such equilibrium is Pareto efficiecy. A strategy profile a is called Pareto efficiet if there is o other strategy a for which: 1) all users do at least as well 2) at least oe user does strictly better. Aother importat settig is that of o-cooperative players where each decisio maker selfishly chooses its strategy. I this case, the equilibrium reached, whe it exists, is called a Nash equilibrium ad is defied as the poit from which o player fids it beeficial to uilaterally deviate. Pareto efficiecy is a desirable operatig poit for a give system i geeral, however o-cooperative equilibria are i geeral Pareto iefficiet.a importat questio that arises is how to drive a system where decisio makers have a o-cooperative behavior to the system s optimal poit. This questio has bee addressed i [10]. 3 User side : routig/load-balacig We cosider a QoS-based routig, where each type of traffic, broadly classified ito streamig versus elastic ad each further decomposed ito classes accordig to various radio coditios, chooses the proportio of flows to be set to each subetwork. At this poit, two situatios arise. Oe ca imagie some cooperative behavior of differet classes, where the routig cotroller wats to achieve i a social maer, the best global utility, QoS divided by a give price i this case, for differet classes i a fair maer. Aother case arises whe o-cooperative classes behave as selfish players, each player tryig to choose idividually its routig strategy so as to optimize its idividual utility, i.e., its ow perceived QoS ormalized to the price. Recall that i the cooperative case, Pareto-optimal poits are defied as poits correspodig to equilibria from which ay deviatio will lead to a degradatio i the performace of at least oe player i the cooperative game. It should be oted that i a multi-class eviromet, there is a ifiite umber of solutios, so-called Paretooptimal strategies. The otio of fairess is the itroduced to select a uique operatig poit. I the theory of cooperative games this is kow as the Nash arbitratio scheme.

4 I the o cooperative case, however, every class acts selfishly to optimize its ow performace measure regardless of others performace. Such games are characterized by the Nash equilibrium poit, whe it exists, defied as the (routig) strategy profile from which o player fids it beeficial to uilaterally deviate. This ca arise i situatios where a decetralized routig decisio is adopted ad where ed users choose their subetwork i a selfish maer so as to optimize their idividual performace measure. From a operator poit of view, it is more beeficial to operate o Pareto-optimal poits sice Nash equilibrium poits are i some cases iefficiet compared to Paretooptimal solutios. I what follows, we first preset utility fuctios, the measure that assesses the user degree of satisfactio from a give settig. 3.1 User utilities We cosider a set of voice ad data users, with idex j J = {v, d} deotig their respective types ad a idex k K = {1,..., K} deotig each type s radio coditios. Let these users share resources i a set of N = {1,..., N} possible parallel subetworks. We cosider probabilistic routig of calls accordig to their type of traffic ad radio coditios. A type-(j, k) selects the -th subetwork with a probability r j,k. Let λ j,k = N =1 λj,k be the total flow demad of class-(j, k) users, where λj,k = rj,k λj,k is the mea rate of class-(j, k) flow that is routed through subetwork. The utility fuctio J j,k of class-(j, k) is the utility achieved by that class ad depeds o its ow strategy give by the rate vector Λ j,k = (λ j,k ) N, but it also depeds o other classes routig decisios, deoted by Λ (j,k). Or: J j,k = J j,k (Λ j,k, Λ (j,k) ) For a QoS-based routig, the atural cadidate for utility fuctios is some performace measure see by the call, blockig probability for voice B v,k, ad the mea trasfer time for data M d,k, ormalized to the uitary price p of resources of each subetwork (p correspods to the price per uit time for the case of voice ad to a price per uit volume for the case of data). I this work, we defie the utility fuctio of a idividual (j, k)-class by : where X j,k time M d,k for data. J j,k (Λ j,k, Λ (j,k) ) = N =1 λ j,k is equal to the blockig probability Bv,k (1 Bj,k )Xj,k p for voice traffic ad mea trasfer Remark 1. Other utility fuctios are possible. For istace, the oe where a subclass tries to maximize some utility related oly to the throughput or mea trasfer time, while maitaiig its blockig probability below a give acceptable limit. I this case, we are i the presece of a costraied optimizatio/game problem where the weights o the blockig probabilities are the Lagrage multipliers.

5 Remark 2. Stability coditios are required i the case where o admissio cotrol is implemeted i the etworks. I this case the admissible regio should be specified. The remaider depeds o whether strategies are cooperative or ot. 3.2 No-cooperative routig I a o-cooperative settig, differet classes are cosidered as selfish players where each class implemets a routig strategy so as to maximize its ow et utility fuctio as a respose to others strategies without ay cocer about others utilities. For a (j, k)- class user, the set of all possible strategies is give by: F j,k = {(Λ j,k ) R N : λ j,k 0 for N; N =1 λ j,k = λ j,k } I this case, optimality caot be well defied. The Nash equilibrium is cosidered as a specific form of optimality [5]. Whe it exists, the Nash equilibrium is a routig strategy profile from which o class fids it beeficial to uilaterally deviate, i.e., o class fids it beeficial for its perceived QoS to uilaterally chage the amout of load it is sedig to each subetwork. More precisely, a (Λ j,k ) vector is a Nash equilibrium if for all (j, k), j J, k K: Λ j,k argmax f j,k F j,k J j,k (f j,k, Λ (j,k) ) meaig that Λ j,k is the best strategy of class-(j, k) player while other players strategies are fixed. If the above-metioed utility fuctios are covex i the routig strategy Λ j,k, the Kuh-Tucker optimality coditios are applicable ad imply that the respose of users of class-(j, k) give by Λ j,k is the optimal respose to other classes strategies give by Λ (j,k) if ad oly if there exist Lagrage multipliers l j,k ad (s j,k ) N = (s j,k 1,..., sj,k N )) such that [9][8] J j,k λ j,k (Λ j,k, Λ (j,k) ) l j,k s j,k N l j,k 0, s j,k 0, λj,k λ j,k = λj,k =1 s j,k λj,k = 0 = 0 = 1,..., N 0 = 1,..., N (1) 3.3 Cooperative routig We ow tur to the cooperative case where for istace a cetral operator assigs calls to each subetwork i a probabilistic maer esurig fairess i terms of the QoS perceived by differet classes. I this case, user classes are cosidered as cooperative

6 players tryig to share resources so as to optimize a overall utility fuctio. The JKdimesioed cooperative game reduces the to a optimizatio problem where the cetral decisio maker (the router) maximizes a global utility fuctio built of idividual oes. The routig strategy is a routig vector (λ j,k ) N,j J,k K. The set of all possible routig vectors is give by: F = {Λ = (λ j,k ) N,j J,k K : λ j,k 0 for N, j J, k K; N =1 λ j,k = λ j,k } We are iterested i Pareto optimality. I this settig, the solutio provides that o player ca icrease its utility without adversely affectig the others [11]. Pareto optimility leads to a set of P 1 equatios for P players, therefore a ifiite umber of operatig poits called Pareto boudary. To choose oe operatig poit, the otio of fairess is itroduced. The cooperative game ca be formulated as follows: max J(Λ) Λ F J.J 1 = γ where γ ( γ = 1) is a JK s dimesioed vector defiig the directio i which the Pareto poit is required. The Pareto boudary ca be foud by evaluatig Pareto poits i all possible directios γ, i other words γ refers to the fairess degree that a cetralized decisio maker might give to differet classes. 4 Network side : Admissio Cotrol ad pricig While ed users, if give the right to decide o their routig strategy, are oly iterested i the QoS they perceive regardless of the good use of resources, the etwork operator(s) do care about the way resources are utilized. I other words, supposig that a call of type (j, k) J K has a reveue p j, the operator should choose its prices as well as its Call Admissio Cotrol (CAC) strategy so as to maximize its total reveues R give by: R(Λ, CAC, p) = (p I)λ j,k (1 B j,k ) j J,k K where I represets the cost of the ivestmet made by the operator for the give techology. Please ote that i the above expressio the reveues of the etwork are a result of both the offered load, the price ad the implemeted admissio strategy. For every routig strategy, be it cooperative or ot, ad i order to maximize its reveue, the operator must offer a attractive price ad implemet some itelliget admissio cotrol to make the best profit of his resources give that load is dictated by ed users. Remark 3. If o fairess cosideratios towards differet classes of users are take ito accout from the operator side, maximizig reveues oly may lead to very ufair situatios where for istace users experiecig bad radio coditios are costatly blocked. (2)

7 We ow cosider cooperative versus o-cooperative cofiguratios. For the sake of simplicity, we adopt a threshold-based admissio cotrol leadig to closed-form expressios. Nevertheless, our framework is geeral ad ca be used for other families of admissio cotrol strategies such as truk reservatio. 4.1 Cooperative case To make the best use out of the etwork resources, the strategy of the operator is to choose a threshold parameter T j,k for each class (j, k). The set of all possible strategies for admissio cotrol is give by : where T = T j,k T j,k N j J k K j,k = {0,..., N ad N j,k is the maximum umber of admitted users of class-(j, k) assumig that this class is the oly oe served i etwork. The dimesio of this set is give by T = N j,k. j J k K } I the case of a wireless etwork where capacity is shared i a oliear maer, determiig the set T of all possible strategies for admissio cotrol is more complex. The set of threshold strategies is a subset of the above-metioed T cotaiig elemets correspodig to feasible states, i.e., the set of all possible strategies where all admitted users obtai sufficiet resources (at the MAC layer) so as to satisfy their QoS. Similarly, the same aalysis holds for the pricig strategy. I the cooperative case, a cetralized decisio maker chooses a vector of prices (p) N i a fiite set of possible prices P. I the case of a sigle operator implemetig admissio cotrol, the objective fuctio is as follows 4.2 No-cooperative case max R(Λ, T, p) (3) T T,p P I this case, each etwork has its ow utility fuctio ad each etwork optimizes its CAC parameters ad chooses its prices idepedetly of other etworks. Formally, each etwork N solves selfishly the followig maximizatio problem, cosiderig other etwork admissio strategies fixed to T ad prices to p : max T T,p P R (Λ, T, T, p, p ) (4) where (T) is the set of all possible etwork strategies give by: T = T j,k j J k K ad each subetwork chooses its ow price p from a fiite set of possible prices P.

8 4.3 Optimal strategies The set of all possible strategies is a fiite set limited to the threshold values guarateeig some QoS to the admitted users as well as prices. A extesive search algorithm is used to fid the optimal threshold parameters [12]. Some algorithms acceleratig the search for the optimal threshold parameters ca be ru by orderig traffic classes accordig to their reveues. Admissio as well as pricig strategies eed ot be ru o the same time scales. The operator may well fix the price first ad the optimize accordig to admissio cotrol. 5 Performace metrics The above-metioed utility fuctios, both for etwork ad user, have bee formulated i terms of performace metrics: mea trasfer time ad blockig probabilities. These performace metrics are derived as follows. Cosider a subetwork where the arrival of (j, k)-class users is Poissoia with mea rate λ j,k for fresh users ad hj,k for hadoff users. The service is expoetial with mea rate µ j,k. The mea service time of voice users is costat; it depeds o the share of resources for data trasfers. I what follows, idex will be suppressed for clarity. I 3G etworks, voice calls shall be assiged to costat rate dedicated liks whereas data oes shall share the leftover power o shared liks implemetig High Speed Dowlik Packet Access (HSDPA). At the MAC layer, HSDPA implemets, amog other mechaisms, opportuistic schedulig, typically through the use of the Proportioal Fair Schedulig (PFS) algorithm. The service rate of a class-(k) data call i such a system is give by: µ j,k (x) = ψk (C K k=1 xv,k φ v,k ) k G(x d ) x d where ψ k is a atteuatio factor related to radio coditios of class-k users, C is the overall system capacity, φ v,k is the share of resources for voice users out of the total resources ad G(.) is the schedulig gai [13]. I IEEE WLAN, the MAC layer is based o CSMA/CA, ad all flows, voice ad data, are subject to competitio. Voice frames are however severely affected by aggressive data sources as the latter are typically saturated oes, i.e., always with a frame to sed. The share of voice ad data users i this case is a oliear fuctio of the umber of users of each type i the system ad is explicitly give i Refereces [14] ad [15]. The overall system ca be described by a Markov process. It is however irreversible which makes product form expressios for the steady-state distributio impossible. As proposed i [16], we cosider a quasi-statioary regime, where data calls would reach steady states betwee voice calls arrivals ad departures. Accordigly, the margial distributio of voice calls i the system is give by a M/M/c/c queueig system with steady state distributio of the umber of ogoig voice calls give by [17]:

9 where π(x v ) = 1 K q k (x v,k ) (5) G k=1 q k (x v,k )= ( λv,k +h v,k µ v,k ) xv,k x v,k! ( λv,k +h v,k µ v,k ) T v,k v,k T v,k ( hv,k µ v,k )x x v,k! if x v,k T v,k if T v,k < x v,k H v,k ad G = x v X v k=1 K q k (x v,k ) is the ormalizatio costat. X v is the state space of voice users for which QoS is guarateed o the packet level, T v,k is the threshold value above which o ew voice arrival is admitted ad H v,k is the threshold value above which o voice call i hadover is admitted. As of data, it ca be modeled as a M/G/1-Processor Sharig (PS) queue with steady-state probabilities give by: where f k (x d,k x v ) = π(x d x v ) = 1 H ( λd,k +h d,k µ d,k ) xd,k (x) x d,k! K f k (x d,k ) (6) k=1 ( λd,k +h d,k µ d,k ) T d,k )xd,k T d,k ( hd,k (x) µ d,k (x) x d,k! if x d,k T d,k if T d,k < x d,k H d,k where H is the ormalizatio costat obtaied by settig the sum of all joit probabilities to oe, T d,k ad H d,k are defied similarly to T v,k ad H v,k. These joit steady-state probabilities π(x v, x d ) are give by: π(x v, x d ) = π(x d x v ) π(x v ) (7) Now, the performace measures are give as follows. The blockig probabilities of a class-k fresh arrival voice or data user is give by: ad for calls i hadover B j,k = x (j,k) H j,k x j,k =T j,k π(x (j,k), x j,k )

10 B j,k h = x (j,k) π(x (j,k), H j,k ) The mea file trasfer time W k of a class-k data call is give by the a phase-2 type distributio takig ito accout the service received i both subsystems i case of a hadover. The sojour time S i each subsystem is give by the miimum betwee the dwell time V i the subsystem ad W ( 1 S = 1 V + 1 W ) [18]: S k = x d,k λ d,k (1 B d,k ) + h d,k (1 B d,k h ) where x d,k is the mea umber of data flows of class k i the subetwork. 6 Coclusio We preseted i this work a framework for modelig the relatioships betwee users, operators as well as the relatioship betwee them i a heterogeeous eviromets where several wireless etworks share the access of some Iteret cloud. We covered the cases of cooperative optimizatio ad o-cooperative games betwee the differet players as these cases arise i real. Our ew step shall be devoted to the umeric aalysis of such strategies i a attempt to quatify etwork-orieted issues, such as what the best optios for voice ad data users are, whether it is better for etwork operators to cooperate or ot ad do users iterest correspod to the etwork s oe. Refereces 1. 3GPP TS High Speed Dowlik Packet Access (HSDPA); Overall UTRAN descriptio. 2. 3GPP TS High Speed Dowlik Packet Access (HSDPA); Layer 2 ad 3 aspects. 3. IEEE stadard for iformatio techology- telecommuicatios ad iformatio exchage betwee systems- local ad metropolita area etworks- specific requiremets Part II: wireless LAN medium access cotrol (MAC) ad physical layer (PHY) specificatios. 4. Drew Fudeberg, Jea Tirole. Game Theory T. Basar, G. J. Olsder, Dyamic Nocooperative Game Theory, Academic Press, Lodo/New York, Jauary E. Altma, T. Bouloge, R. El Azouzi, T. Jimeez ad L. Wyter, A survey o etworkig games, Computers ad Operatios Research, T. Alpca, T. Basar, R. Srikat, ad E. Altma, CDMA uplik power cotrol as a ocooperative game, Wireless Networks, vol. 8, pp , November Y. A. Korilis, A. A. Lazar, ad A. Orda, Capacity allocatio uder o-cooperative routig, IEEE Trasactios o Automatic Cotrol, 42(3): , March E. Altma, R. El-Azouzi, ad V. Abramov, No-Cooperative Routig i Loss Networks, Performace Evaluatio 49(1 4), 43-55, 2002.

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