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3610 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 10, OCTOBER 2014

Utility-Based Resource Allocation for Multi-ChannelDecentralized Networks

Min Sheng, Member, IEEE, Chao Xu, Xijun Wang, Member, IEEE, Yan Zhang, Member, IEEE,Weijia Han, Member, IEEE, and Jiandong Li, Senior Member, IEEE

AbstractThe architecture of decentralization makes futurewireless networks more flexible and scalable. However, due to thelack of the central authority (e.g., BS or AP), the limitation of spec-trum resource, and the coupling among different users, designingefficient resource allocation strategies for decentralized networksfaces a great challenge. In this paper, we address the distributedchannel selection and power control problem for a decentralizednetwork consisting of multiple users, i.e., transmit-receiver pairs.Particularly, we first take the users interactions into account andformulate the distributed resource allocation problem as a noncooperative transmission control game (NTCG). Then, a utility-based transmission control algorithm (UTC) is developed basedon the formulated game. Our proposed algorithm is completelydistributed as there is no information exchange among differentusers and hence, is especially appropriate for this decentralizednetwork. Furthermore, we prove that the global optimal solutioncan be asymptotically obtained with the devised algorithm, andmore importantly, in contrast to existing utility-based algorithms,our method does not require that the converging point is oneNash equilibrium (NE) of the formulated game. In this light, ouralgorithm can be adopted to achieve efficient resource allocationin more general use cases.

Index TermsDecentralized networks, distributed resourceallocation, learning, game theory.

I. INTRODUCTION

D ECENTRALIZED networks are the infrastructure-lesswireless networks consisting of multiple transmit-receivepairs, where each transmitter could dynamically adjust its trans-mission parameters and transmit data to its receiver [1][4].Compared to the conventional networks with the control of cen-tral authorities, e.g., BSs or APs, decentralized networks havemore flexibility and scalability, and hence, span a large numberof real-world implementations, e.g., military communications,disaster relief or sensor networking [2], [4], [5].

Manuscript received January 27, 2014; revised June 18, 2014; acceptedAugust 24, 2014. Date of publication September 11, 2014; date of current ver-sion October 17, 2014. This work was supported in part by the National NaturalScience Foundation of China under Grants 61231008, 61172079, 61201141,61301176, and 91338114, by the 863 Project under Grant 2014AA01A701,and by the 111 Project under Grant B08038. The associate editor coordinatingthe review of this paper and approving it for publication was Y. J. Zhang.

The authors are with the State Key Laboratory of ISN, Xidian University,Xian 710071, China (e-mail: msheng@mail.xidian.edu.cn; cxu@mail.xidian.edu.cn; xijunwang@xidian.edu.cn; yanzhang@xidian.edu.cn; alfret@gmail.com; jdli@mail.xidian.edu.cn).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2014.2357028

The main characteristics of a decentralized network can besummarized as follows.

1) The lack of central controller. In such an infrastructure-less network, each transmitter is responsible for tuningits transmission strategy, e.g., transmission frequency,bandwidth, power, modulation, etc., based on its localobservation. Therefore, self-organization is one funda-mental capability for a decentralized network [6], [7].

2) The limitation of spectrum resource. The available chan-nels are limited in a decentralized network, and hence,users should compete for this precious resource to im-prove their individual performance, e.g., transmission rateor energy efficiency, thereby satisfying their individualQoS requirement.

3) The coupling among different users. Interference occurswhen different users transmit on the same channel simul-taneously. Therefore, each users performance could betuned by properly adjusting the operational parameters ofother users. In other words, the users are coupled.

According to the above three characteristics, there existtwo kinds of conflicts in a decentralized network. One is theconflict between different users which is caused by the lasttwo characteristics, i.e., the limitation of spectrum resource andcoupling among different users. The other one is the conflictbetween system performance and individual requirement whichis mainly introduced by the lack of a central controller. In fact,these two conflicts always make a decentralized network oper-ate at an inefficient point, which is termed as price of anarchy(PoA). For instance, considering some users who operate onthe same channel, if all of them want to maximize their owntransmission rate through power control, then the maximumtransmit power will be adopted by everyone. Obviously, thisis not an efficient power control scheme for this system [8], [9].

In this light, to exploit the benefits promised by the decen-tralized networks, it is essential to design distributed resourceallocation strategies which should fully consider these twoconflicts. Fortunately, game theory which provides a suitableparadigm to analyze the interrelationship between decisionmakers, can be naturally adopted to deal with the first conflict[2], [6][8], [10], [11]. However, designing globally optimalor even Pareto-efficient (Pareto-optimal)1 distributed resourceallocation algorithms for a decentralized network is still an openproblem [2], [6], [7].

1Generally speaking, it is easy to prove that the global optimal solution isalso Pareto-optimal, but not vice versa.

0090-6778 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

SHENG et al.: UTILITY-BASED RESOURCE ALLOCATION FOR MULTI-CHANNEL DECENTRALIZED NETWORKS 3611

In this paper, we consider a multi-user multi-channel decen-tralized network, where each user (consisting of a transmitterand receiver pair) is capable of performing channel selectionand power allocation to satisfy its transmission rate require-ment. In addition, to avoid the high communication overhead,we focus on the network where there is no information ex-change among different users, i.e., no common control channel(CCC) is introduced. We note that this consideration makes thescenario more practical but on the other hand, brings us moredifficulties in designing efficient resource allocation strategies[2][4], [12][14].

Because of the limitation of spectrum resource and couplingamong different users, not all the rate requirements of users(i.e., transmit-receive pairs) can be simultaneously satisfied[4]. Furthermore, recalling that there is no central controllerbeing responsible for scheduling users transmission, it is agreat challenge to provide hard rate guarantee to every userin this decentralized network. For this reason, as studied inprevious work [15][18], we consider softening users require-ments and use a sigmoid function to measure their satisfaction.Specifically, one user has very limited satisfaction when itstransmission rate is below the requirement, but the satisfactionrapidly reaches an asymptotic value when its transmissionrate is above the requirement. Based on this, we formulatethe distributed channel selection and power control problemas a non-cooperative transmission control game (NTCG). Toovercome the lack of communication between different users,a utility-based learning approach is adopted2 and a Utility-based Transmission Control algorithm (UTC) is developed,with which each user can configure its operational parametersjust by measuring local interference. More importantly, al-though there is no guarantee that the Nash equilibrium (NE) forNTCG always exists, it is proved that the decentralized networkcould operate at a global optimal point by implementing UTC.Finally, simulation results verify the validity of our analysisand demonstrate that the performance of our algorithm (e.g.,convergence speed, achieved overall utility, etc.) are better thanthat of the existing distributed algorithms.

The remainder of this paper is organized as follows. InSection II, the related work is presented. Section III describesthe system model and formulates the distributed channel se-lection and power control problem. In Section IV, we developa utility-based transmission control algorithm and analyze itscomplexity as well as efficiency. Finally, numerical and sim-ulation results are presented and analyzed in Section V, andconclusions are drawn in Section VI.

II. RELATED WORK

The game theoretic approach has been applied extensively todesign distributed resource allocation schemes in wireless com-munication systems from both the perspective of transmissionrate as well as energy efficiency [9], [19][22]. In [19][21], theconcerning problem has been formulated as a potential game[23], and then a best response dynamic (BRD) was adopted to

2The definition of utility-based learning approaches will be formally givenin next section.

achieve a pure-strategy NE. However, as discussed in the sem-inal work [23], a potential game always admits multiple pure-strategy NE solutions. Hence, for such a game the operatingpoint achieved by BRD totally depends on the starting pointand may be inefficient. To improve the efficiency of the devisedstrategy, pricing technique was introduced in [9], [22] and thePareto efficiency of the achieved NE is proved.

We note that all of the above schemes require CCC for infor-mation exchange among different agents. Hence, they are notsuitable for the decentralized network, and developing the so-called utility-based or payoff-based learning algorithms is nec-essary. Specifically, when implementing this type of algorithms,each user only needs to access the history of its own actions andutilities, and would make its decision with the local information[24]. To this end, some distributed schemes based on stochasticlearning, no-regret learning and reinforcement learning havebeen proposed in [12][14], respectively. It should be notedthat all the algorithms devised in [12][14] are utility-based,but the converging solution is a probability distribution over theset of available strategies. Therefore, the performance can onlybe evaluated from a statistical perspective in [12][14], i.e., theperformance of each implementation is unpredictable [4].

Recently, some studies begin to focus on developing theutility-based resource allocation strategy which can asymp-totically converge to a fixed configuration (e.g., pure-strategyNE) instead of a probability distribution [3], [4]. In [3], thedistributed channel selection problem was formulated as apotential game, and then a utility-based learning algorithm wasproposed, which could converge to a pure NE. Furthermore, notonly channel selection but also power control was consideredin [4], and another utility-based strategy was designed for oneclass of non-cooperative games. To be more specific, underthe assumption that the set of NE for the proposed game isnot empty and there is at least one NE maximizing the socialwelfare (i.e., sum of the utilities of all users), the proposeddistributed channel selection and power control scheme canasymptotically converge to the global optimal solution [4].

Actually, the above assumption is less plausible in manygeneral cases. The reason lies in two folds: 1. For a non-cooperative game there is no guarantee that the pure-strategyNE always exists [25].3 Particularly, one example can be foundin [21], which is termed as the signal-to-interference-plus-noiseratio (SINR) maximization game. 2. Even if the formulatednon-cooperative game admits a NE, the Pareto-efficiency of itsNE is hard to guarantee [19][21], [25]. Obviously, it is moredifficult to satisfy the more severe requirement that there existssome NE which can maximize the social welfare. In this work, anovel utility-based resource allocation algorithm is developed.More importantly, it has been proved that, even if there is noNE for the formulated game, we can also asymptotically obtainthe globally optimal solution with our proposed algorithm.

III. SYSTEM MODEL AND PROBLEM FORMULATION

As depicted in Fig. 1, we consider a decentralized net-work featuring N communicating users, each consisting of a

3Since the mixed-strategy NE will not be considered in this work, we use NEto denote the pure-strategy NE hereafter for brevity.

3612 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 10, OCTOBER 2014

Fig. 1. Illustration of a decentralized network, where each user consists of onetransmitting node and one receiving node.

transmit-receive pair. Particularly, to transmit data, every userwill choose one channel from the K orthogonal channels, eachof which has bandwidth B0. We consider that each channel canbe assigned to multiple users and meanwhile, the interferenceoccurs when each channel is simultaneously utilized by morethan one user. Without loss of generality, we suppose N K.For notational simplicity, let vectors N and K denote the setof users and channels, respectively, i.e., N = {1, 2, , N}and K = {1, 2, ,K}. Additionally, we denote the channelselected by user n by cn K. In this paper, we consider thatthere is no CCC or central authority for coordination amongusers. That is, all users are autonomous.

Let G RNNK be the channel power gain matrix, wheregkn,m represents the channel gain between transmitter n andreceiver m on channel k. We assume the channel condition isstatic during the underlying operational period, e.g., the quasi-static scenario. The additive noise is modelled as a zero-meanGaussian random variable, and then, for user n, its signal-to-interference-plus-noise ratio (SINR) can be expressed as

n =png

cnn,n

Icnn +B0N0

=png

cnn,n

mN ,m =n(cm, cn)pmg

cnm,n +B0N0

, (1)

where In represents the interference caused to user n, pn is thetransmit power of user n, and N0 is the noise power density.Besides that, the indicative function (cm, cn) is adopted toindicate whether the same channel is used by user m and nsimultaneously or not: if cm = cn, (cm, cn) = 1; otherwise(cm, cn) = 0. In this paper, we consider that each user ncan choose the transmit power pn from a finite set Pn ={p1n, p2n, , pmaxn } [4], [12].

Based on the above, the achievable transmission rate of usern can be expressed as

Rn = B0 log2(1 + n). (2)

Adopting different channels and power levels, one user willobtain different achievable rates. According to (1) and (2), ifuser n transmits on channel cn, Rn can be maximized withpower pmaxn when there is no interference. Therefore, the upperbound of the rate Rn for user n can be defined as

Rmaxn = max

{B0 log2

(1 +

pmaxn gcnn,n

B0N0

)|cn K

}. (3)

Moreover, we consider that each user n has rate requirementRmin...