6928 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014

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1 6928 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014 Optma Power Aocaton and User Schedung n Mutce Networks: Base Staton Cooperaton Usng a Game-Theoretc Approach Janchao Zheng, Student Member, IEEE, Yuemng Ca, Senor Member, IEEE, Yongkang Lu, Yuhua Xu, Student Member, IEEE, Bowen Duan, and Xuemn (Sherman Shen, Feow, IEEE Abstract Ths paper proposes a nove base staton (BS coordnaton approach for nterce nterference mtgaton n the orthogona frequency-dvson mutpe access based ceuar networks. Specfcay, we frst propose a new performance metrc for evauatng end user s quaty of experence (QoE, whch jonty consders spectrum effcency, user farness, and servce satsfacton. Interference graph s apped here to capture and anayze the nteractons between BSs. Then, a QoE-orented resource aocaton probem s formuated among BSs as a oca cooperaton game, where BSs are encouraged to cooperate wth ther peer nodes n the adjacent ces n user schedung and power aocaton. The exstence of the jont-strategy Nash equbrum (NE has been proved, n whch no BS payer woud unateray change ts own strategy n user schedung or power aocaton. Furthermore, the NE n the formuated game s proved to ead to the goba optmaty of the network utty. Accordngy, we desgn an teratve searchng agorthm to obtan the goba optmum (.e., the best NE wth an arbtrary hgh probabty n a decentrazed manner, n whch ony oca nformaton exchange s needed. Theoretca anayss and smuaton resuts both vadate the convergence and optmaty of the proposed agorthm wth farness mprovement. Index Terms OFDMA, nter-ce nterference mtgaton, QoE, BS cooperaton, potenta game, Nash equbrum, decentrazed agorthm, goba optmaty. I. INTRODUCTION INTER-CELL nterference s a fundamenta probem whch mts the performance mprovement n the orthogona frequency dvson mutpe access (OFDMA-based ceuar networks wth reuse of the spectra resource. Tradtonay ths probem s majory addressed by carefuy pannng the spectra resource [1], ncudng conventona frequency pannng Manuscrpt receved December 11, 2013; revsed Apr 6, 2014 and June 16, 2014; accepted June 16, Date of pubcaton Juy 2, 2014; date of current verson December 8, Ths work was supported n part by the project of the Natura Scence Foundatons of Chna under Grants and and n part by the Jangsu Provnca Natura Scence Foundaton of Chna under Grant BK The assocate edtor coordnatng the revew of ths paper and approvng t for pubcaton was D. Nyato. J. Zheng, Y. Ca, Y. Xu, and B. Duan are wth the Coege of Communcatons Engneerng, PLA Unversty of Scence and Technoogy, Nanjng , Chna (e-ma: ongxngren.zjc.s@163.com; caym@vp.sna.com; yuhuaenator@gma.com; bowen @163.com. Y. Lu and X. Shen are wth the Department of Eectrca and Computer Engneerng, Unversty of Wateroo, Wateroo, ON N2L 3G1, Canada (e-ma: yongkang.u.phd@gma.com; sshen@uwateroo.ca. Coor versons of one or more of the fgures n ths paper are avaabe onne at Dgta Object Identfer /TWC (Reuse-1 and Reuse-3, fractona frequency reuse, parta frequency reuse, and soft frequency reuse. However, these approaches reduce nter-ce nterference at the cost of decreasng the spectra effcency. Future network evoutons are envsoned to empoy a fu (or an aggressve frequency reuse to meet the rapdy growng demand of broadband mobe access. Therefore, effcent nterference mtgaton technques are urgenty requred. Recenty, BS cooperaton has emerged as a promsng approach to mtgate nter-ce nterference. Snce any change of resource aocaton n a snge ce w affect the performance of the nearby ces, jont resource aocaton over a custer of neghborng ces va BS coordnaton proves to be effectve [2] [5]. These soutons usuay requre neghborng BSs coordnate ther resource aocaton for the jont network utty optmzaton, whch usuay resut n hgh cost n backhau communcatons wth huge contro overhead. Ths paper treats the coordnaton probem n an aternatve way where user transmsson strateges and resource aocaton schemes, rather than data fows, are coordnated across the BSs [5], [6]; hence, much ess backhau coordnaton bandwdth s needed. Most state-of-the-art work concentrates on the (weghted sum-rate maxmzaton [2], [4], [7], whe the achevabe soutons are generay far from goba optmum. Moreover, the exstng soutons ntroduce unfarness to edge users [2], [7], because edge users usuay experence more path oss whe the network manager tends to prvege the users coser to the BS wth better channe condtons n the resource aocaton. Most exstng work addresses farness ssue ony by usng network-eve crtera ke max-mn but negects the specfc requrements of ndvdua users. In ths paper, we take spectrum effcency, farness and user s requrements nto consderaton jonty to mprove QoE n the formuated optmzaton probem. QoE-drven technques adaptvey aocate the mted resources to enhance end user experence so that they reduce the waste of rado resources compared wth other technques adoptng the objectve metrcs, e.g., sum-rate, whch negect ndvdua users satsfacton of servces. For exampe, f the users wth better channe condtons have been aocated wth adequate resources, QoEdrven optmzer woud then prvege users wth poor channe condtons who coud experence substanta mprovement of satsfacton. Therefore, QoE-drven technques w brng farness whe ncreasng effcency. Ths paper adopts the mean IEEE. Persona use s permtted, but repubcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 ZHENG et a.: OPTIMAL POWER ALLOCATION AND USER SCHEDULING IN MULTICELL NETWORKS 6929 opnon score (MOS [9] to be the utty metrc, whch s wdey used to provde a generc measure of the user s QoE [10] [12], [33]. A. Chaenges and Contrbutons In ths paper, we empoy BS cooperaton to sove the QoEorented resource aocaton probem n the mutce OFDMA networks, whch conssts of user schedung and power aocaton as a jont optmzaton decson by BSs. Furthermore, n the aggressve frequency reuse depoyment, the co-channe nterference makes the resource aocaton among ces couped and correated. Moreover, the non-convexty of the utty metrc (.e., MOS makes the probem more compcated. In ths case, centrazed agorthms cannot guarantee the goba optmaty over the network gven that the demand on backhau sgnang and computatona resources grows rapdy wth the number of ces, subchannes and end users. In practca systems, the nterference sources to ndvdua users usuay come from a sma number of neghborng ces (whch makes t possbe to mt the backhau sgnang and compexty [3]. Therefore, how to desgn an effcent dstrbuted agorthm to fnd the gobay optma souton wth ony oca nformaton exchange throws a great chaenge. We study ths probem by appyng game theory to anayze the dstrbuted decsons made at ndvdua BSs consderng the mutua nterference and coupng among ther strateges [13] [16]. The man contrbutons of our work are summarzed beow: An nterference graph s generated to capture and anayze the nteracton between BSs. Then, based on the nterference graph, the network sum-utty maxmzaton probem s formuated as a oca cooperaton game, where each BS acts as a ratona payer. Furthermore, we prove t to be an exact potenta game, whose best NE pont concdes wth the optma souton wth the sum-utty maxmzaton. We desgn a decentrazed teratve agorthm to obtan the best NE (.e., the goba optmum wth an arbtrary hgh probabty, where ony oca nformaton exchange s needed between neghborng BSs. The convergence, optmaty and computatona compexty are nvestgated. Moreover, the farness mprovement by adoptng QoE as the optmzaton goa s theoretcay anayzed. B. Reated Work In recent years, resource aocaton for ceuar networks has stepped nto the focus of extensve studes, because coordnated resource aocaton brngs sgnfcant performance mprovement by effectvey mtgatng the nter-ce nterference, coordnaton across mutpe ces poses a great chaenge not ony n mpementaton, but aso n fndng the optma soutons, snce the nter-ce nterference eads to nherent nonconvexty n the probem structure [6]. One promsng way s to use heurstc-based agorthms to obtan satsfactory soutons [17] [20]. Another way s to decompose the orgna probem nto mutpe subprobems and sove them teratvey [6], [21] [24]. Besdes, there are some dscussons [25] [29] concentratng on respectve studes (e.g., subchanne aocaton, power contro due to the ntractabty of jont optmzaton. Furthermore, many researchers have referred to game theory to seek for a satsfactory souton. In [27] [29], potenta game based subchanne aocaton agorthms for nterference mnmzaton are proposed. In [26], [30], takng the nter-ce nterference nto account, the authors study the transmt power contro n mutce OFDMA systems by usng non-cooperatve game. However, amost a consder a smpfed system mode and just concentrate on ony one aspect (ether power contro or subchanne aocaton. In [18], jont subchanne and power aocaton s nvestgated by usng game theory. The exstence and unqueness of equbrum are theoretcay proved. However, ths work s decomposed nto two subgames, n whch the subchanne assgnment and power contro are performed teratvey. Therefore, the obtaned souton s suboptma. Buzz et a. [31] use potenta game to make a comprehensve anayss on the jont subchanne and power aocaton n a very genera system mode, but ony suboptma souton s obtaned as we. Moreover, the exstng game theoretc approaches many make an nvestgaton on the exstence and unqueness of the NE pont, but pay ess attenton to the reatonshp between the NE and the goba optmum. In addton, t shoud be noted that the coordnated resource aocaton n the terature many concentrates on QoS optmzaton, whch does not consder the user s satsfacton of servces. QoE has recenty been under the spotght n wreess networks. Genera wreess mutmeda transmsson schemes have been we studed n [12], [32], [33], n whch resource aocaton and mutmeda schedung are the focus. Hassan et a. [34] nvestgate the QoE-drven resource aocaton from the perspectve of the nteracton between the provder and the VoIP user, whch s naturay formuated as a non-cooperatve game. In [10], [11], the authors study the QoE-orented mutuser resource aocaton n the OFDMA systems. To our knowedge, most QoE-drven resource aocaton work s mted n sngece optmzaton. Sheen et a. [35] dscuss the performance evauaton and optmzaton of a genera reay-asssted mutce network and a genetc agorthm s proposed to sove the probem. However, ths work ams at the jont optmzaton of the system parameters, ncudng reay s poston, reuse pattern, path seecton, etc., whch s out of the scope of our work. To sum up, QoE-orented resource aocaton n mutce networks has not been we studed, and the exstng soutons are generay far from goba optmum. Therefore, n ths paper, we empoy BS cooperaton to mprove the effcency of the souton from a game theoretc perspectve. Accordngy, a decentrazed teratve agorthm s desgned to acheve the goba optmum wth an arbtrary hgh probabty. C. Paper Organzaton The remander of the paper s organzed as foows. In Secton II, we present the system mode foowed by the probem formuaton for the QoE-drven resource aocaton. In Secton III, we formuate the oca BS cooperaton game and

3 6930 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014 nvestgate the propertes of ts NE ponts. In Secton IV, a QoEdrven jont user schedung and power aocaton agorthm s proposed to fnd the goba optmum of our probem. Secton V presents smuaton resuts and dscusson. Concusons are drawn n Secton VI. II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Mode We consder an OFDMA-based ceuar network whch conssts of a set of L BSs, denoted by L = {1, 2,...,,...L}. We assume each ce s served by a BS and BSs communcate wth the users n a snge-hop fashon. We aso assume BSs are temporay synchronzed. BSs and users are equpped wth one transmt and one receve antenna, respectvey. N = {1, 2,...,n,...N} s the network user set. The set of users served by BS Ls denoted by N, N N, and N = N. Each user s connected to ony one base staton that s seected based on ong-term channe quaty measurement. Thus, N N =, for. We consder the unversa frequency reuse depoyment n whch every ce shares the whoe bandwdth. The avaabe spectrum s dvded nto K subchannes 1 and the ndex set of a subchannes s denoted by K = {1, 2,...,k,...K}. N = N and K = K are the cardnates of N and K, respectvey. In ths paper we focus on the study of downnk communcatons from BSs to the users. Our anayss can be easy extended to the upnk dscusson. 1 MAC and Physca Layer: In the network, the spectra resource sots are shared by a ces, eadng to an nterference and nose mpared system. Let s k N denote the user connected to base staton n spectra sot k. When perfect synchronzaton s assumed, the dscrete-tme baseband sgna receved by user s k n sot k s gven by r s k = H,s kx s k + }{{} usefu data L =1, H,s k x s k }{{} nterce nterference + Z s k }{{} nose, (1 where x s k and H,s k are the transmtted compex symbo and the compex channe response from BS to user s k, respectvey. Z s k s the addtona nose, whch s modeed as a whte Gaussan varabe wth power E Z s k 2 = σ 2. Suppose user n s connected wth BS,.e., n N.Let δ,n k be the spectra sot aocaton ndcator to denote whether sot k s aocated to the user n n ce : δ,n k =1f the sot s aocated to the user; otherwse, δ,n k =0. Then, the sgna-tonterference-pus-nose rato (SINR of user n wthn ce n sot k, s wrtten as γ k,n = δ,n k pk Gk,n L, (2 =1, δk,n pk Gk,n + σ2 where p k s the transmt power of BS n sot k, Gk,n = Hk,n 2 s the channe power gan from BS to user n n sot k. 1 We w use spectra sot and subchanne nterchangeaby n ths paper. Fg. 1. Generc appcaton mode (The subscrpton of data rate denotng the specfc user s omtted. Wthout oss of generaty, we assume that the bandwdth of each subchanne s ess than the coherence bandwdth of the channe so that fat fadng s consdered n each subchanne. The correspondng achevabe nformaton rate s gven by the foowng Shannon s formua: ( R,n k = B K og 2 1+ γk,n, (3 Γ where Γ= n(5ber/1.5 s BER gap. Then, the aggregate rate of user n s R,n = k K n R,n k, where K n s the set of sots occuped by user n. 2 Appcaton Layer: MOS s used as a measure of the user s QoE for the servces ke vdeo streamng, fe downoad, and web browsng. The vaue of MOS s dstrbuted between 1 and 4.5. MOS =1refects an unacceptabe appcaton quaty, and MOS =4.5corresponds to an exceent quaty experenced by the user. The consdered generc appcaton characterstc [10] resembes a bounded ogarthmc reatonshp between perceved quaty and data rate as ustrated n Fg. 1, descrbed by the MOS as a functon of the data rate, 4.5, R,n R,n 4.5, MOS,n (R,n = a og R,n b, R,n 1.0 <R,n <R,n 4.5, 1, R,n R 1.0 wth,n, (4a 3.5 a = (, (4b og R,n 4.5/R1.0,n b = R 1.0,n ( R 1.0 1/3.5,n R,n 4.5, (4c 0 R 1.0,n <R 4.5,n, n N. (4d The semogarthmc pot of Fg. 1 vsuazes the reated parameters: the parameter a determnes the sope of MOS,n (R,n whe b shfts the curve aong the X-axs. Each user s appcaton characterstc can be parameterzed by ony two parameters, {R 1.0 }, or aternatvey {a, b}.,n,r4.5,n B. Probem Formuaton Snce the system s based on OFDMA, ntra-ce mut-user access s orthogona, whe nter-ce mut-user access s smpy

4 ZHENG et a.: OPTIMAL POWER ALLOCATION AND USER SCHEDULING IN MULTICELL NETWORKS 6931 superposed due to fu reuse of spectrum. It s the superposton of the sots that resuts n severe co-channe nterference, whch majory mts the system performance. Therefore, t s ntutve to decoupe the optmzaton of resources n varous spectra resource sots (.e., frequency bands, or sub-channes, and we may study the user schedung and power aocaton whch maxmze the system performance n a partcuar sot [8]. We w suppress the sot ndex hereafter, concentratng n one arbtrary sot. In any gven spectra resource sot shared by a ces, we denote the user that s granted access to the sot (.e., schedued n ce by s N. Defnton 1: A schedung vector characterzes the set of users smutaneousy schedued across a ces n the same sot: s =(s 1,s 2,...,s,...,s L, where [s] = s. Notng that s N, the constrant set of schedung vectors (.e. the schedung strategy space s gven by S = N 1 N 2 N L, where denotes the Cartesan product. Defnton 2: A transmt power vector characterzes the transmt power vaues used by each BS to communcate wth ts respectve schedued user: p =(p 1,p 2,...,p,...,p L, where [p] = p. Note that n rea practce, the ceuar standards ke the 3GPP LTE standard ony support dscrete power contro 2 n the downnk. We assume each BS can use M 2 dfferent power eves for transmsson, namey {λ 1 P,max,λ 2 P,max,...,λ M P,max }, where 0=λ 1 <λ 2 <...<λ M =1. Then, the constrant set of transmt power vectors s gven by P = {p p {λ 1 P,max,λ 2 P,max,..., λ M P,max }, =1,...,L}. Base statons are coordnated to jonty determne the optma schedung vector and transmt power vector whch maxmze the system utty (.e., MOS. From the system optmzaton pont of vew, the sum-utty optma resource aocaton probem can now be formazed smpy as (s opt, p opt = arg max U 0, (5 s S,p P where U 0 = L MOS = L n N ω,n MOS,n s the network utty functon, ω,n s the weght of the schedued user n the th ce. Remark 1: The sum-mos optma jont user schedung and power aocaton probem for a mutce wreess network beongs to a cass of combnatora optmzaton probems; fndng the gobay optma souton (s opt, p opt s NP-hard. Hence, standard optmzaton technques cannot be apped drecty and even centrazed agorthms cannot guarantee the gobay optma souton. III. INTERFERENCE GAME FOR QOE-ORIENTED BS COORDINATION In ths secton, we dscuss on the dstrbuted souton of the above probem (5 by usng game theory. The abty to mode ndvdua, ndependent decson makers, whose strateges are 2 It s worth notng n [2] that dscrete power contro whch offers two man benefts over contnuous power contro: the transmtter desgn s smpfed, the overhead of nformaton exchange among network nodes s sgnfcanty reduced. Fg. 2. A unatera nterference graph wth 10 BSs (Each node represents a BS, and each drectona edge represents an nterference nk from one end to another end. nteractona, makes game theory partcuary attractve to anayze the performance of decentrazed network/framework. A. Interference Graph In order to quantfy the nter-ce nterference, we empoy the nterference metrc (IM n [2], whch s defned by IM = 1 N G,n, (6 G,n n N where G,n s the channe gan from BS to user n N, and N s the number of eements n N. Notce that the rato G,n /G,n ndcates the amount of nterference caused by BS to user n, and IM s smpy cacuated by averagng the rato G,n /G,n over a users served by BS. The nterference reatonshp s now characterzed by a drectona graph G d =(L,ε. The graph G d conssts of the BS set L, and a set of edges ε L 2. Denote each edge as an ordered par (,, obvousy, (, ε. In ceuar networks, the transmsson s severey nterfered wth ony by BSs ocated n a few surroundng ces, and the nterference from the remote BSs s trva. To capture the near-far effect, decdng the edge of the nterference graph s based on (6. Ony f the vaue of IM s arger than a predefned threshod IM0, there s an edge from BS to, whch means BS causes non-gnorabe nterference to ce. Moreover, snce the mutua nterference s not symmetrca (IM IM n the ceuar system, the produced nterference graph s unatera wth drectona edges, as shown n Fg. 2. Then for each BS, the foowng two neghbor sets can be defned: the n-neghbor set B n : B n = { L:(, ε}. the out-neghbor set B out : B out = {j L:(, j ε}. We use p B n and s B n to denote the power aocaton and user schedung strategy profe of BS s n-neghbors, respectvey. Then, ce s performance s denoted by MOS (p, p B n, s, s B n, snce t s affected by the cochanne nterference from the neghborng BSs. It s worth notng that other metrcs can aso be used to decde the nterference graph (e.g., smpy based on the geographc ocaton of BSs and users [36], or further consderng the

5 6932 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014 traffc oad, etc.. However, there woud be tte varaton n the foowng game mode and the man concusons. Snce the optma constructon of the nterference graph s not the focus of ths work, we adopt the metrc n [2] n order to perform convctve agorthm comparson. B. Game Mode Wth Loca BS Cooperaton Based on the nterference graph, the game s formay denoted by G =[L, {S P } L, G d, {U } L ], where L = {1, 2,...,L} s the set of payers (.e., BSs, S P s the set of avaabe jont power and channe aocaton strategy for payer, and U s the utty functon of payer. To mprove the effcency of the game and obtan the gobay optma souton, the utty functon of each payer s defned as U (p, p D,s, s D =MOS (p, p B n,s, s B n + B out MOS (p, p B n,s, s B n, (7 where D represents the nteractng neghbor set of payer, p D P D and s D S D denote the power aocaton and user schedung strategy profe of payer s nteractng neghbors (excudng, respectvey. P D P, S D S, D, are the sets of acton profes. Accordng to (7, we can get D = B out B n B n B out. (8 Then, referrng to the defntons of B n and B out, the nteractng neghbor set D s further decded by D = B n B out { j : j, B out j B out }. (9 If 1 D 2, we say that two BSs 1 and 2 are nteractng neghbors. Obvousy, 1 D 2 2 D 1. Note that the above defned utty functon s comprsed of two parts: the ndvdua utty of payer and the aggregate utty of ts nterfered neghbors. In other words, when a payer makes a decson, t not ony consders tsef but aso consders ts nterfered neghbors. Then, the oca cooperaton game s expressed as foows: (G : max U (p, p D,s, s D, L. (10 p P,s S Defnton 3 (Nash Equbrum: A resource aocaton profe (p, s =(p 1,p 2,...,p L,s 1,s 2,...,s L s a pure strategy NE pont of G f and ony f no payer can mprove ts utty by devatng unateray,.e., U ( p, p D,s, s D U ( p, p D,s, s D, L, p P \{p }, s S \{s }, (11 where A 1 \ A 2 means that A 2 s excuded from A 1. C. Anayss of NE Theorem 1: The QoE-orented resource aocaton game G s an exact potenta game whch has at east one pure strategy NE. Proof: The foowng proof foows the dea of proof gven n [27] [29]. Frst we construct a potenta functon as Φ(p, p,s, s = L MOS (p, p,s, s, (12 where p and s represents the power aocaton and user schedung strategy profe of a the BSs excudng BS, respectvey. Snce MOS (p, p,s, s =MOS (p, p B n,s, s B n based on the nterference graph, we have Φ(p, p,s, s = MOS (p, p B n,s, s B n L =MOS (p, p B n,s, s B n + MOS (p, p B n,s, s B n + {N\B out }, B out MOS (p, p B n,s, s B n. (13 1 Suppose that an arbtrary payer, say, unateray changes ts transmt power from p to p, then the change n potenta functon s gven by (14, shown at the bottom of the page. For B out,wehave B n ; thus, when changes ts transmt power from p to p, p p B n B n. However, f {N\B out \{}}, p = p B n B n when changes ts transmt power. Thus, the foowng equaton hods: MOS (p, p B,s n, s B n = MOS ( p, p B n,s, s B n, { N\B out }, (15 Φ(p, p,s, s Φ(p, p,s, s = L MOS (p, p B n,s, s B n = MOS (p, p B n,s, s B n + {N\B out }, (MOS ( L ( MOS p, p B n MOS (p, p B n,s, s B n p, p B n,s, s B n,s, s B n + B out (MOS ( p, p B n,s, s B n MOS (p, p B n,s, s B n MOS (p, p B n,s, s B n. (14

6 ZHENG et a.: OPTIMAL POWER ALLOCATION AND USER SCHEDULING IN MULTICELL NETWORKS 6933 On the other hand, the change of ndvdua utty functon caused by ths unateray change s gven by U (p, p D,s, s D U (p, p D,s, s D ( = MOS (p, p B n,s, s B n MOS p, p B n + ( (MOS p,p B,s n,s B n MOS (p,p B n B out,s, s B n,s,s B n. (16 Then, based on (14 (16, we can get Φ(p, p,s, s Φ(p, p,s, s = U (p, p D,s, s D U (p, p D,s, s D. (17 2 Gven transmt power vector p =(p 1,p 2,...,p L,the user seecton probem decoupes across BSs. It s a partcuar property of the downnk n the ceuar network, snce the receved nterference as we as the MOS vaue does not change wth the varaton of user seecton strateges of other ces when the transmt power vector gven. In other words, each BS s MOS vaue s ndependent of the user schedung strateges of other BSs, but ony depends on ts own user schedung strategy,.e., MOS (p, p B n,s, s B n = MOS (p, p B n,s, L. (18 Therefore, when payer unateray changes ts user seecton strategy from s to s,, MOS keeps unchanged. Then, t s easy to get Φ(p, p,s, s Φ(p, p,s, s ( = MOS (p, p B n,s, s B n MOS p, p B n,s, s B n = U (p, p D,s, s D U (p, p D,s, s D. (19 It s shown from (17 and (19 that the change n ndvdua utty functon caused by any payer s unatera devaton equas to the change n the potenta functon. Thus, accordng to the defnton gven n [37], G s an exact potenta game wth network utty U 0 servng as the potenta functon. Exact potenta game s a speca knd of game snce t admts severa promsng propertes, among whch the most mportant one s that every exact potenta game has at east one pure strategy NE pont. Therefore, Theorem 1 s proved. The payers n the game focus on maxmzng ther ndvdua utty functons, as specfed by (10. Ths may resut n neffcency and demma, whch s known as tragedy of commons [38]. Athough Theorem 1 demonstrates that ths game has at east one pure NE pont, anayzng the achevabe performance of NE ponts of a genera exact potenta game s nterestng and mportant. Fg. 3. The schematc dagram of the proposed decentrazed teratve agorthm (Once the power strategy s updated, the user schedung updatng based on Eq. (20 foows, whch s omtted for brevty n ths fgure. Theorem 2: The gobay optma souton of the network sum-mos maxmzaton probem consttutes a pure strategy NE of G. Proof: It s proved by D. Monderer and L. S. Shapey [37] that a Nash equbra are the maxmzers of the potenta functon Φ, ether ocay or gobay. Furthermore, accordng to (12 and the defnton of network utty U 0, we know that the potenta functon concdes wth the network utty U 0. Therefore, a Nash equbra maxmze the network utty U 0 ether ocay or gobay, and the best NE serves as the goba optmum of the network utty. Hence, Theorem 2 s proved. Accordng to Theorem 2, n order to acheve the goba optmum, we ony need to deveop an effectve agorthm to obtan the best NE. IV. DECENTRALIZED ITERATIVE ALGORITHM FOR ACHIEVING GLOBALLY OPTIMAL SOLUTION Wth the jont power aocaton and user schedung probem now formuated as an exact potenta game, there are severa earnng agorthms avaabe n the terature to acheve the pure strategy NE, e.g., best response dynamc [37], fcttous pay [39], [40], and no-regret earnng [41]. However, a of them am at achevng an equbrum souton, and are easy trapped n an undesrabe equbrum. Recenty, a γ-ogt approach has attracted sgnfcant attenton n potenta game theory, e.g., [13], [27], [42], [43], due to ts favorabe property of equbrum seecton and exporng goba optmum. In ths secton, we propose a γ-ogt based decentrazed teratve agorthm to obtan the optma souton to the probem n (5 wth an arbtrary hgh probabty. The agorthm runs at the begnnng of each schedung nterva and has mutpe teratons n whch the user schedung and the transmt power are updated. A. Agorthm Descrpton Let p (t, s (t be the transmt power eve and the user assgnment of BS at teraton t, respectvey, for =1,...,L and t 0. The proposed procedure s descrbed n Agorthm 1 and the schematc dagram s shown n Fg. 3.

7 6934 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014 γ-ogt based decentrazed teratve agorthm Intazaton: Set the teraton t =0, and et each BS, L, seect the maxmum power eve P,max. Then, each BS randomy seects a user for ts communcaton. Loop for t =0, 1, 2,... 1 Payer seecton: A set of non-nteractng BSs, say C(t, s randomy seected n an autonomous and dstrbuted manner. 1, 2 C(t, 1 D 2. Then, each seected BS computes ts current utty vaue U (t by (7 through the communcaton 3 wth neghborng BSs. 2 Exporaton: Each seected BS C(t randomy chooses a power eve ˆp {λ 1 P,max,λ 2 P,max,..., λ M P,max } wth equa probabty 1/M. Then, based on the new transmt power eves, BS as we as ts neghbors ndependenty decdes ts best user assgnment ŝ (t as 4 ŝ (t = arg max ω,n MOS,n, { B out {} }, (20 n N The BSs adhere to ther seectons n an estmaton perod and cacuate ther respectve MOS vaue. Then, the seected BS C(t computes ts exporng utty vaue Û(t by (7 through the communcaton wth ts neghborng BSs. 3 Strategy Updatng: Each seected BS updates ts power eve accordng to the foowng rue: { Pr (p (t +1= ˆp = exp{βû(t} Ψ (21 Pr (p (t +1=p (t = exp{βu (t} Ψ, where Ψ=exp{βÛ(t} +exp{βu (t} and β s a postve parameter. Meanwhe, a the other BSs keep ther seectons unchanged,.e., p (t +1=p (t, L\ C(t. Then, based on the updated power eves, each BS recomputes the best user assgnment s (t +1by (20. End oop unt the stoppng crteron s met. In order to fnd the gobay optma souton, neghborng BSs cooperate to exchange nformaton drecty 5 and ony oca nformaton s nvoved, nteractng neghbors are not aowed to smutaneousy change the transmt power eve n the proposed agorthm. In the payer seecton step of Agorthm 1, the seecton of the non-nteractng BSs set can be mpemented through contenton mechansms over a common contro channe or a prorty-based method n [2]. The stop crteron can be one of the foowng: the maxmum number of teratons s reached, the varaton of the network utty durng a perod s ess than a predefned threshod. 3 Necessary communcaton s used to obtan ts neghbors MOS. 4 In the downnk of the ceuar network, each BS s MOS vaue s ndependent of the user schedung strateges of other BSs, but ony depends on ts own user schedung strategy, as shown n (18. Therefore, the optma user assgnment probem decoupes across BSs when the power vector p s gven. 5 Neghborng BSs are connected though hgh-speed wrene, thus ther nformaton exchange s very easy. The γ-ogt based decentrazed agorthm s nspred by the work n [13], [42], [43], where the dea of probabstc decson makng s proposed and deveoped. The probabstc decson makng rue n step 3 s referred to as Botzmann exporaton strategy [13], [45], [46], and the parameter β s anaogous to the concept of temperature n smuated anneang. We ntroduce such a probabstc strategy seecton nto our agorthm for the coordnated resource aocaton probem n order to escape from oca optma ponts and fnay converge to the optma NE (.e., goba optmum. In addton, the same resource aocaton probem s aso addressed n [2], whe the desgned agorthm there s essentay the best response (BR n whch each payer expores ts whoe strategy space and seects the best strategy. It shoud be noted that the best response may easy get trapped at an undesrabe NE [43]. The basc requrement for the convergence of the exstng ogt agorthm s that ony one payer updates ts acton at one tme [27], [42], [43]. However, n a arge-scae mut-ce network, the scheme of ony one payer s strategy updatng woud sow the convergence of the agorthm. To acceerate the convergence, we mprove the agorthm by aowng mutpe (non-neghborng payers to update ther respectve actons smutaneousy. Secondy, n the typca ogt agorthm, each actve payer s strategy updatng s based on the exporaton n the whoe strategy space. Our probem s jont power aocaton and user schedung. Thus, the strategy space s twodmensona, and each actve payer shoud expore the acton from the two-dmensona strategy space. In ths case, the compexty s hgh, and the convergence speed sows down. To decrease the compexty and aso acceerate the convergence, we modfy the agorthm by decomposng the jont probem nto two steps (.e., reducng the scae of the expored strategy space whe keepng the optmaty of the souton. The convergence of the proposed modfed agorthm needs to be revsted, whch w be proved n Subsecton C. B. Convergence and Optmaty Anayss Theorem 3: If a payers adhere to the proposed decentrazed teratve agorthm, the unque statonary dstrbuton π(p, s of any jont user schedung and power aocaton strategy profe (p, s, s gven by: π(p, s = exp {βφ(p, s} (22 exp {βφ(p, s}, p P where P s the space of transmt power profe for a BSs, Φ s the potenta functon gven n (12 and s s the user schedung vector whch s unquey determned by p,.e., s = g(p. Proof: Gven transmt power vector p, the user seecton probem decoupes across BSs, thus each BS can decde ts own user schedung strategy ndependenty n our proposed agorthm. Hence, the user schedung vector s s unquey determned by p, and we use functon g( to denote ths reatonshp. Foowng smar proof gven n [27], [42], [47], we denote the power aocaton state n the t-th teraton by p(t = (p 1 (t,p 2 (t,...,p L (t. Notaby, p(t s a dscrete tme

8 ZHENG et a.: OPTIMAL POWER ALLOCATION AND USER SCHEDULING IN MULTICELL NETWORKS 6935 Markov process, whch s rreducbe and aperodc. Therefore, t has an unque statonary dstrbuton. Denote any two arbtrary network states by X and Y, X, Y P, and the transton probabty from X to Y by Pr(Y X. In the foowng part, we w show that the unque dstrbuton must be (22 by verfyng that the dstrbuton (22 satsfes the foowng baanced equaton: π(xpr(y X =π(y Pr(X Y. (23 If X = Y, (23 obvousy hods. Then, we focus on the case of X Y. Note that ony non-neghbor BSs are aowed to update ther strateges smutaneousy n each teraton, whch resuts n the change of correspondng eements n X. For cear presentaton, we denote X by (p 1,p 2,...,p L, where the teraton ndex t and the subchanne superscrpt k are omtted. Wthout oss of generaty, suppose that the set of non-nteractng BSs who smutaneousy update ther strateges s C = {1, 2,..., C }, where C denotes the number of C s eements. Therefore, Y =(p 1,p 2,...,p C,p C +1,p C +2,...,p L. Addtonay, we assume the probabty of C to be chosen as the set of updatng payers s η. Snce any power eve has probabty 1/M of beng chosen n the proposed agorthm, we can get (24, shown at the bottom of the page. By defnng α as (25, shown at the bottom of the page, we have π(xpr(y X =α exp{βφ(x, g(x} Cexp{βU (p, p D,g(p, p D } { =α exp βφ(x, g(x+β } U (p, p D,g(p, p D. C Due to the symmetry property, we aso have (26 π(y Pr(X Y { = α exp βφ(y,g(y + β } U (p, p D,g(p, p D. C (27 Construct a sequence as X 0,X 1,X 2,...,X C, where X 0 =X and X =(p 1,p 2,...,p,p +1,p +2,...,p L, C. Obvousy, Y = X C. We obtan Φ(Y,g(Y Φ(X, g(x =Φ ( X C,g ( X C Φ(X0,g(X 0 = (Φ (X,g(X Φ(X 1,g(X 1 C = (U (X,g(X U (X 1,g(X 1. (28 C Because a payers n C are not mutuay nteractng neghbors,.e., 1, 2 C, 1 D 2. Therefore, U (X,g(X U (X 1,g(X 1 = U (p, p D,g(p, p D U (p, p D,g(p, p D. (29 Accordng to (28 and (29, we can get Φ(Y,g(Y Φ(X, g(x = (U (p, p D,g(p, p D U (p, p D,g(p, p D. (30 C Then, (26 and (27 mmedatey yed the baanced (23. Thus, we have π(xpr(y X = π(y Pr(X Y X P X P = π(y Pr(X Y =π(y, (31 X P whch s exacty the baanced statonary equaton of the Markov process p(t. Snce the proposed agorthm has an unque statonary dstrbuton and the dstrbuton gven by (22 satsfes the baanced equatons of ts Markov process, we can concude that ts statonary dstrbuton must be (22. Therefore, Theorem 2 s proved. Theorem 4: Wth a suffcenty arge β, the proposed agorthm acheves the gobay optma souton to the sum-mos maxmzaton probem wth an arbtrary hgh probabty. Proof: Let p opt and s opt = g(p opt be the gobay optma power aocaton vector and user schedung vector, respectvey. Furthermore, Theorem 2 has demonstrated that the goba optmum s exacty the best pure strategy NE of G, whch maxmzes the potenta functon gobay. Thus, we have (p opt, s opt = arg max Φ(p, s. (32 p P,s S π(xpr(y X= X P exp {βφ(x, g(x} k exp {βφ(x, g(x} η M C exp {βu (p, p D,g(p, p D } exp {βu (p, p D,g(p, p D } +exp{βu (p, p D,g(p, p D } (24 η α = M 1 X Pk exp {βφ(x, g(x} exp {βu (p, p D,g(p, p D } +exp{βu (p, p D,g(p, p D } C (25

9 6936 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014 In addton, we have proved n Theorem 3 that the agorthm converges to a unque statonary dstrbuton π(p, s gven by (22, whch rees on the parameter β. When the parameter β s suffcenty arge (.e., β, exp{βφ(p opt, s opt } exp{βφ(p, s }, (p, s A\(p opt, s opt, where A s the jont strategy space. In ths case, accordng to (22, the unque statonary dstrbuton w be (0,...,0, 1, 0,...,0. The probabty 1 s gven to the gobay optma souton (p opt, s opt whch maxmzes the potenta functon, whe other soutons (p, s A\(p opt, s opt are n probabty 0. That s, m β π(popt, s opt =1, (33 whch substantates that the proposed agorthm converges to the goba optmum wth an arbtrary hgh probabty. Thus, the proof s competed. Remark 2: The proposed approach eads to optma network sum-utty wth an arbtrary hgh probabty no matter whch metrc (e.g., MOS, nformaton rate s defned as the utty functon. For nstance, f the nformaton rate s desgned as the utty functon, the proposed agorthm can acheve the sumrate optma souton. Overa, the proposed approach s generc to sove ths cass of NP-hard probems. C. Computatona Compexty Anayss In each teraton, each seected BS needs a random number to choose a power eve wth a computatona compexty of O(1, and then decdes the best user assgnment wth a compexty of O( N. As for the computaton of the MOS vaue, t needs B n +1addtons and Bn +5mutpcatons (dvsons to frst compute the nformaton rate, and then two comparsons and no more than 2 mutpcatons and one ogarthmc operaton to cacuate the MOS vaue. Then, t needs B out 1 addtons to compute the utty U accordng to (22. Thus, the tota compexty for computng the utty s O( B n Bout. In addton, the procedure of strategy updatng nvoves the operatons of 2 exponents, 1 addtons and 4 mutpcatons, and hence the compexty s O(1. Therefore, n tota, the computatona compexty for each seected BS 6 s O( B n B out + N. The compexty depends on the number of served users as we as the scae of the neghbor set. The scae of the neghbor set then rees on the predefned threshod IM 0 for the nterference metrc (IM. If the threshod IM 0 ssettobeow,the nterference graph w capture more nterference nks (even weak nterference nks, whch s coser to the rea nterference envronment. In ths case, the scae of the neghbor set w be arger, and more nterferng BSs w be ncorporated nto the coordnaton to further mprove the performance. However, t ntroduces hgher computatona compexty. In the extreme case (IM 0 =0,BS s neghbor set w ncude a the other BSs,.e., B n = Bout = L 1, thus, the computatona compexty s O( L 2 + N. 6 Snce the non-seected BSs do not have any operaton, the computatona compexty s 0. The above anayss provdes the computatona compexty for each teraton of the proposed agorthm. Furthermore, the whoe computatona compexty aso rees on the number of teratons needed for convergence (.e., convergence speed. However, there s a tradeoff between the performance and convergence speed of the proposed agorthm. On one hand, we have proved n Theorem 4 that the probabty of achevng the goba optmum by our proposed agorthm woud be cose to 1 when β s suffcenty arge, however, t cannot be obtaned n fnte number of teratons [44]. On the other hand, f β s not suffcenty arge for practca appcaton, there may exst performance oss whch w be shown n the smuaton part. D. Farness Anayss Note that sum-rate optma resource aocaton schemes [2], [7] tend to prvege users wth good channe condtons, whe the good users may not need such a ot of spectra bands, whch resuts n the waste of resources. In contrast, f a user cannot contrbute enough capacty gan to the system to outwegh the generated nterference, t may not be schedued n the spectra sots. Thus, a user may be aocated a number of spectra sots over ts need or none at a. To sove ths probem, we empoy sum-mos as the optmzaton goa whch not ony depends on the user s channe condton, but aso consders the user s requrement. In the foowng, we w prove that our proposed agorthm amng at sum-mos maxmzaton w sove the farness ssue effectvey. A snge typca ce s consdered. Suppose that there are N dentca vdeo-stream users n ce, and the number of subchannes (.e., spectra sots s K and K = N. For smpcty, we assume the K subchannes are a dentca, whch brng the same rate gan for the same user. That s, Rn 1 = Rn 2 = = Rn K = Rn, 0 n. Moreover, we assume R1 0 >R2 0 > >RN 0. In the foowng, we anayze the farness by usng Jan s farness ndex (JFI [48], whch transates a resource aocaton vector {R 1,R 2,...,R N } nto a score n the nterva of [1/N, 1] and hgher JFI means the resource aocaton s farer. The foowng theorem characterzes the acheved farness for dfferent optmzaton goa. Theorem 5: Suppose R 4.5 >R1 0 >R2 0 > >RN 0 >R1.0 and RN 0 > 2b, the JFI acheved by sum-rate maxmzng agorthm s 1/N, whe that acheved by sum-mos maxmzng agorthm s ower bounded by 1+2λ(N 1/N, where λ s constraned by { R λ 2 0 λ 1 2 2λ R 0 N (34 Proof: Snce R1 k >R2 k > >RN k, k, a subchannes w be aocated to user 1 by sum-rate maxmzng agorthm. Thus, R 1 = k K Rk 1 = KR1, 0 whe R 2 = = R N =0.In ths case, JFI s obvousy 1/N. As for the sum-mos optma scheme, a og(r1/b k > a og(r2/b k > >aog(rn k /b due to the monotony property of the ogarthmc functon. Therefore, the frst subchanne w be schedued to the frst user. When t comes to the schedung of the second subchanne, the ncrement of

10 ZHENG et a.: OPTIMAL POWER ALLOCATION AND USER SCHEDULING IN MULTICELL NETWORKS 6937 the frst user s MOS, say ΔMOS 1,s(aog(R1 1 + R1/b 2 a og(r1/b. 1 Note that Rn 1 = Rn 2 =...= Rn K = Rn, 0 n, we have ΔMOS 1 = a og 2. When R2 2 > 2b, a og(r2/b 2 > ΔMOS 1. Therefore, the second subchanne w be assgned to user 2. Foowng ths ne of anayss, each user w be aocated exact one subchanne to. In ths case, each user s rate can be expressed as R n = Rn, 0 n. Note that ( N 2 N R n = (R n 2 +2 R R j, (35 <j N we am to acheve the JFI bound by provng the foowng nequaty: R R j λ ( (R 2 +(R j 2, j. (36 For anayss, we rewrte the above nequaty as λ(r 2 + λ(r j 2 R R j 0. (37 Now, t s easy to get the constraned condton for the above nequaty beng rght as λ =0,or λ>0 1 4λ R 1 4λ 2 1+ j 2λ R R (38 1 4λ 2 j 2λ. Fg. 4. Smuated network confguraton wth 49 ces. TABLE I TRANSMITTED VIDEO STREAMS Snce R n = Rn, 0 n and R1 0 >R2 0 >...>RN 0, we can get the requrement as (34. In ths case, the bound of the JFI farness s gven by ( N 2 N R n (R n 2 +2 R R j <j N J = = N N (R n 2 N N (R n 2 = N (R n 2 +2λ <j N N N (R n 2 ( (R 2 +(R j 2 N (R n 2 +2λ(N 1 N (R 2 N N (R n 2 1+2λ(N 1 =. (39 N Accordng to Theorem 5. when RN 0 /R0 1 ncreases, λ can take a arger vaue. Then, the JFI for the sum-mos maxmzaton goa gets arger, snce t ncreases wth λ. When R1 0 =...= RN 0, λ can take 1/2, thus the JFI reaches 1. =1 V. S IMULATION RESULTS In ths secton, we evauate the performance of the proposed agorthm by Matab smuatons. A. System Descrpton and Parameters Settng Smar to [2], [23], we consder a 49-ce OFDMA network confguraton, as shown n Fg. 4. Each hexagona ce has a radus of 500 meters, and each BS s ocated at the center of ts servng ce, and adjacent BSs are separated by 3/2 km from each other. To nvestgate the case of severe nter-ce nterference, 8 remote users (each wth a separate vdeo stream are generated as a unform dstrbuton wthn the edge-regon of each ce (at east 400 meters away from the BS. Parameters of the vdeos [10] are summarzed n Tabe I and the weght of each user s set to be 1 for smpcty. The maxma transmt power of each BS s set to be 46 dbm, and s equay spt across subchannes. The BER gap Γ s set to be 1. The tota bandwdth B s dvded nto M =16 subchannes smar to [23] and the bandwdth of each subchannes s set to be 200 KHz. G k,n = Hk,n 2 s the channe power gan from BS to user n on subchanne k, whch s expressed as G k,n =(d,n θ ε k,n, where d,n s the dstance between BS and user n, θ s the path oss exponent and ε k,n s the fadng coeffcent. Rayegh fadng mode s consdered n the smuaton, and the channe gans are exponentay dstrbuted wth unt-mean. The pass oss exponent θ s set to be 3.7 and the nose power experenced at each recever s assumed dentca and has a power of 130 dbm. In the proposed agorthm, the obtaned souton s coser to optmum gven a arger β, at the cost of convergence tme [43]. To acheve a tradeoff between optmaty and convergence speed, we choose β = t/10 n the smuaton, where t s the teraton step.

11 6938 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014 Fg. 6. The evouton of power aocaton and user schedung strategy versus the number of teratons n a snge tra. Fg. 5. Convergence behavor n a snge tra. B. Convergence and Optmaty The convergence curve of the proposed agorthm s shown n Fg. 5, and the convergence curve of the best response (BR agorthm n [2] s presented for comparson. In order to capture the convergence behavor, the resuts are acheved by snge smuaton tra. Moreover, the gobay optma souton s potted by exhaustve search to evauate the optmaty of our proposed agorthm. Because the goba optmum cannot be found by exstng computng technques n arge scae networks, ths fgure studes a 7-ce sma network (ce 1 7 n Fg. 4. The number of power eves s set to be 4. As shown n Fg. 5, the network uttes by the two agorthms are updated n each teraton and both greaty mproved at the convergence tme. Furthermore, our proposed agorthm can acheve the goba optmum wth an arbtrary hgh probabty, whe the BR agorthm n [2] ony obtans a oca optmum. It shoud be noted that the BR agorthm n [2] converges faster than our proposed agorthm, snce the probabstc updatng s empoyed n our agorthm for goba optmum. Addtonay, our proposed agorthm shares the same amount of sgnang exchange and communcaton overhead as the BR agorthm. The detaed sgnang overhead anayss and comparson can be found n [2], and the nterested readers can refer to [2] for further readng. Next, Fg. 6(a and (b present the power aocaton and user schedung strategy updatng versus the number of teratons, respectvey. The evouton of number of payers seectng dfferent power eves s shown n Fg. 6(a. It s seen that the number of payers on dfferent power eves remans unchanged n about 80 teratons, whch further vadates the convergence of the proposed agorthm. Addtonay, the user schedung strategy updatng s presented n Fg. 6(b, where ony 3 ces strateges are shown. In fact, the other ces strategy updatng s qute smar, whch s omtted for concson and brevty. In Fg. 7, we compare the proposed agorthm wth stateof-the-art agorthms (best response (BR agorthm [2], [37], fcttous pay [39], [40], no-regret earnng [41]. The resuts are obtaned by smuatng 500 ndependent tras and then takng the average vaue. The stop crteron for each tra Fg. 7. Performance comparson of dfferent agorthms. s that the maxmum number of teratons (200 teratons s reached. It s noted that the average utty acheved by the proposed agorthm may not reach goba optmum wthn fnte number of teratons, as anayzed n Secton IV-C. Among these 500 tras, the gobay optma souton was reached by the proposed agorthm for 69 tras wthn about 170 teratons, and n the remanng 431 tras the goba optmum cannot be found wthn 200 teratons. However, the resut at 200th teraton s cose to the goba, and the margna gan decreases whe the margna cost ncreases sgnfcanty. Therefore, the resut at the 200th teraton s a good approxmaton of the optma one. Besdes, Fg. 7 shows that the average uttes acheved by dfferent agorthms a ncrease wth the number of teratons. In specfc, the BR agorthm and the fcttous pay converge fastest, the proposed agorthm converges reatvey sower, and the convergence speed of the no-regret earnng agorthm s sowest. However, a agorthms converge wthn 200 teratons. In terms of the acheved network utty, a the agorthms present good performance. In more detas, the proposed agorthm s near-optma, the BR agorthm and the fcttous pay foow, whe the performance of the no-regret agorthm s reatvey worse. It s theoretcay proved that the BR agorthm [37] and the fcttous pay [39] can converge to

12 ZHENG et a.: OPTIMAL POWER ALLOCATION AND USER SCHEDULING IN MULTICELL NETWORKS 6939 Fg. 8. Evauaton of performance oss when dfferent β are seected. Fg. 10. Improvement of utty versus number of teratons for dfferent power eves (M =2, 4, 8. Fg. 9. The acheved network utty versus parameter β. Fg. 11. Improvement of utty versus the sze of n-neghbor set. NE n potenta games, whch s ether gobay or ocay optma souton of ths probem. However, the no-regret earnng proves to converge ony to the correated equbrum (CE [41], whch does not show a cear reatonshp wth the goba/oca optmum. Therefore, the no-regret agorthm presents reatvey worse performance. As anayzed n Secton IV-C, there exsts a tradeoff between performance and compexty (convergence speed. In order to get a cear understandng of the performance oss, we present n Fg. 8 the network utty acheved by the proposed agorthm when dfferent vaues of β are seected. Fg. 8 shows that the arger β s, the coser to optmum the proposed agorthm can acheve. However, the smaer β s, BSs are more ncned towards unformy payng a ther actons, whch yeds ower performance gans [13]. Moreover, when β s sma, the convergence curve fuctuates, snce t may oscate around severa good soutons. Besdes, we pot Fg. 9 to further evauate how the seecton of parameter β affects the acheved network utty. As shown n Fg. 9, arger β yeds hgher network utty, but further ncreasng β beyond 140 ony obtans margna gans of network utty. C. Network Utty In Fg. 10, we evauate the performance of our proposed agorthm n terms of dfferent power eves (M =2, 4, 8. Smar to [2], the normazed power eves are set to be {0, 1}, {0,1/4,1/2,1}and{0,( 2 /8,=0,...,6} for M =2, 4, 8, respectvey. As we can see from Fg. 10, ncreasng M from 2 to 4 brngs sgnfcant performance gan, whe further ncreasng the number of power eves beyond 4 ony acheves margna benefts. Moreover, the ncrease of the number of power eves makes the convergence of the agorthm sow down. In Fg. 11, we study the performance of our proposed agorthm under dfferent szes of the n-neghbor set n the whoe 49-ce network, and the agorthm n [2] s aso potted for comparson. By propery seectng the nterference threshod IM 0, the sze of the n-neghbor set, Bn,, s set to be B, whch s vared from 2 to 12 n the smuaton. Increasng B s benefca snce a arger number of nterferng BSs are coordnated; however, ncreasng B nevtaby ncreases sgnang overhead as we as computatona compexty n each teraton of the dstrbuted procedure. Furthermore, when B s arger

13 6940 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014 Fg. 12. Improvement of JFI versus the number of subchannes. Fg. 13. The acheved rate and utty of each user by dfferent agorthms (K = 16. than 6, ncreasng B cannot brng substanta performance mprovement. In practca mpementaton, we shoud make a tradeoff between performance and sgnang overhead. Based on the smuaton resuts, settng the sze of the n-neghbor set to be 6 s approprate n the studed arge-scae network for good performance. Moreover, Fg. 11 aso shows that our proposed agorthm outperforms the exstng agorthm (especay when B s arger than 6. Addtonay, ncreasng the number of transmt power eves beyond 4 w not obtan sgnfcant benefts by both agorthms. D. Farness Evauaton In ths part, we perform the farness comparson of the two dfferent agorthms wth dfferent optmzaton goas. Snce we focus on the farness among users, users n a snge ce are nvestgated. Fg. 12 shows the evouton of Jan s farness ndex (JFI versus the number of avaabe subchannes, K. TheJFIs n terms of achevabe rate and utty are presented. The JFI of utty s hgher than that of rate, because the utty functon reduces the gap between users rates to mnor dfference of MOS vaues (1 to 4.5. Secondy, the JFIs n both subfgures of Fg. 12 get mproved wth the ncreasng number of subchannes due to the mutchanne dversty gan, whch s the advantage of OFDMA n frequency-seectve channes. A user experencng fadng on one subchanne, can be schedued on another when t meets a better channe, f there are enough subchannes. Last but not east, Fg. 12 further vadates the cam n Theorem 5 that the proposed agorthm for sum-mos maxmzaton can acheve sgnfcant farness mprovement aganst the exstng sum-rate optma scheme, snce sum-rate optmzaton tends to prvege users wth better channe condtons, who are generay coser to the base-staton. To better ustrate how the sum-mos optma scheme performs, the rate and utty acheved by each user are shown n Fg. 13, where the users are sorted n descendng order of ther performance.

14 ZHENG et a.: OPTIMAL POWER ALLOCATION AND USER SCHEDULING IN MULTICELL NETWORKS 6941 VI. CONCLUSION In ths paper, we have nvestgated the mutce coordnaton among mutpe BSs for nterference mtgaton n the QoEorented resource aocaton. A game-theoretc approach has been proposed n whch the exstence of the jont-strategy NE has been proved. Then, the gobay optma souton for the network sum-utty maxmzaton has been obtaned usng a decentrazed teratve agorthm wth an arbtrary hgh probabty, where ony oca nformaton exchange s nvoved. The proposed agorthm has been anayzed and proved to converge to the best NE (.e., goba optmum. Moreover, farness among users has been mproved wth theoretca anayss. Smuaton resuts have vadated the effectveness of the proposed agorthm. For our future work, we w extend the presented mode to the case where base statons have mutpe antennas. It s aso nterestng and chaengng to extend the mode to the heterogeneous networks such as a mxture of macroces and sma ces. 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Aarts, Smuated Anneang: Theory and Appcatons. Amsterdam, The Netherands: Rede, [46] R. S. Sutton and A. G. Barto, Renforcement Learnng: An Introducton. Cambrdge, MA, USA: MIT Press, [47] J. Marden, G. Arsan, and J. Shamma, Cooperatve contro and potenta games, IEEE Trans. Syst., Man, Cybern., vo. 39, no. 6, pp , Dec [48] R. Jan, D. Chu, and W. Haws, A Quanttatve Measure of Farness and Dscrmnaton for Resource Aocaton n Shared Computer System, Dgt. Equp. Co., Tech. Rep., Yuhua Xu (S 08 receved the B.S. degree n communcatons engneerng and the Ph.D. degree n communcatons and nformaton systems from PLA Unversty of Scence and Technoogy, Nanjng, Chna, n 2006 and 2014, respectvey. He s currenty an Assstant Professor wth the Coege of Communcatons Engneerng, PLA Unversty of Scence and Technoogy. Hs research nterests focus on opportunstc spectrum access, earnng theory, game theory, and dstrbuted optmzaton technques for wreess communcatons. Mr. Xu was an Exempary Revewer for the IEEE COMMUNICATIONS LETTERS n 2011 and Bowen Duan receved the B.S. degree n communcatons engneerng from Lanzhou Jaotong Unversty, Lanzhou, Chna, n He s currenty workng toward the M.S. degree n communcatons and nformaton systems n the Insttute of Communcatons Engneerng, PLA Unversty of Scence and Technoogy, Nanjng, Chna. Hs current research nterests ncude cooperatve communcatons, resource aocaton, and game theory. Janchao Zheng (S 13 receved the B.S. degree n eectronc engneerng from PLA Unversty of Scence and Technoogy, Nanjng, Chna, n He s currenty workng toward the Ph.D. degree n communcatons and nformaton systems n the Coege of Communcatons Engneerng, PLA Unversty of Scence and Technoogy. Hs research nterests focus on nterference mtgaton technques, earnng theory, game theory, and optmzaton technques. Yuemng Ca (M 05 SM 12 receved the B.S. degree n physcs from Xamen Unversty, Xamen, Chna, n 1982 and the M.S. degree n mcroeectroncs engneerng and the Ph.D. degree n communcatons and nformaton systems from Southeast Unversty, Nanjng, Chna, n 1988 and 1996, respectvey. He s currenty wth the Coege of Communcatons Engneerng, PLA Unversty of Scence and Technoogy, Nanjng. Hs current research nterests ncude cooperatve communcatons, sgna processng n communcatons, wreess sensor networks, and physca ayer securty. Yongkang Lu receved the Ph.D. degree from the Unversty of Wateroo, Wateroo, Canada. He s currenty a Postdoctora Feow wth the Broadband Communcatons Research (BBCR Group, Unversty of Wateroo. Hs research nterests ncude protoco anayss and resource management n wreess communcatons and networkng, wth speca nterest n spectrum and energy-effcent wreess communcaton networks. Xuemn (Sherman Shen (M 97 SM 02 F 09 receved the B.Sc. degree from Daan Martme Unversty, Daan, Chna, n 1982 and the M.Sc. and Ph.D. degrees from Rutgers Unversty, New Jersey, NJ, USA, n 1987 and 1990, respectvey, a n eectrca engneerng. He s currenty a Professor and the Unversty Research Char of the Department of Eectrca and Computer Engneerng, Unversty of Wateroo, Wateroo, Canada, where he was the Assocate Char for Graduate Studes from 2004 to Hs research focuses on resource management n nterconnected wreess/wred networks, wreess network securty, soca networks, smart grd, and vehcuar ad hoc and sensor networks. He has coauthored/edted sx books and has pubshed over 600 papers and book chapters n wreess communcatons and networks, contro, and fterng. Dr. Shen s a Feow of the IEEE, The Engneerng Insttute of Canada, and The Canadan Academy of Engneerng and a Dstngushed Lecturer of the IEEE Vehcuar Technoogy and IEEE Communcatons Socetes. He s an Eected Member of the IEEE ComSoc Board of Governors and the Char of the Dstngushed Lecturers Seecton Commttee. He served as the Technca Program Commttee Char/Cochar for IEEE Infocom 14 and IEEE VTC 10 Fa, the Symposa Char for IEEE ICC 10, the Tutora Char for IEEE VTC 11 Sprng and IEEE ICC 08, the Technca Program Commttee Char for IEEE Gobecom 07, the Genera Cochar for Chnacom 07 and QShne 06, the Char for the IEEE Communcatons Socety Technca Commttee on Wreess Communcatons and P2P Communcatons and Networkng. He aso serves/served as the Edtor-n-Chef of IEEE Network, Peer-to-Peer Networkng and Appcaton, and IET Communcatons; a Foundng Area Edtor of the IEEE TRANSAC- TIONS ON WIRELESS COMMUNICATIONS; an Assocate Edtor of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, Computer Networks, and ACM/Wreess Networks; and a Guest Edtor of IEEE JSAC, IEEE Wreess Communcatons, IEEE Communcatons Magazne, and ACM Mobe Networks and Appcatons. He was a recpent of the Exceent Graduate Supervson Award n 2006 and the Outstandng Performance Award n 2004, 2007, and 2010 from the Unversty of Wateroo; the Premer s Research Exceence Award (PREA n 2003 from the Provnce of Ontaro, Canada; and the Dstngushed Performance Award n 2002 and 2007 from the Facuty of Engneerng, Unversty of Wateroo. He s a Regstered Professona Engneer of Ontaro.