Energy-Efficient Power and Rate Control withQoS Constraints: A Game-Theoretic ApproachFarhad Meshkati, H. Vincent Poor, Stuart C. SchwartzDepartment of Electrical EngineeringPrinceton University, Princeton, NJ 08544E-mail: {meshkati,poor,stuart}@princeton.eduRadu V. BalanSiemens Corporate Research755 College Road East, Princeton, NJ 08540E-mail: [email protected]—A game-theoretic model is proposed to study thecross-layer problem of joint power and rate control with qualityof service (QoS) constraints in multiple-access networks. In theproposed game, each user seeks to choose its transmit power andrate in a distributed and selfish manner in order to maximize itsown utility and at the same time satisfy its QoS requirements. Theuser’s QoS constraints are specified in terms of the average sourcerate and the upper bound on the average delay. The utility functionconsidered here measures energy efficiency and the delay includesboth transmission and queueing delays. The Nash equilibriumsolution for the proposed non-cooperative game is derived anda closed-form expression for the utility achieved at equilibriumis obtained. It is shown that the QoS requirements of a usertranslated into a “size” for the user which is an indication ofthe amount of network resources consumed by the user. Usingthis framework, the tradeoffs among throughput, delay, networkcapacity and energy efficiency are also studied.I. INTRODUCTIONFuture wireless networks are expected to support a varietyof services with diverse quality of service (QoS) requirements.Because of the hostile characteristics of wireless channels andscarcity of radio resources such as power and bandwidth,efficient resource allocation schemes are necessary for design ofhigh-performance wireless networks. The objective is to use theradio resources as efficiently as possible and at the same timesatisfy the QoS requirements of the users in the network. QoSis expressed in terms of constraints on rate, delay or fidelity.Since in most practical scenarios, the users’ terminals arebattery-powered, energy efficient resource allocation is crucialto prolonging the battery life of the terminals.In this work, we study the cross-layer problem of QoS-constrained joint power and rate control in wireless networksusing a game-theoretic framework. We consider a multiple-access network and propose a non-cooperative game in whicheach user seeks to choose its transmit power and rate in sucha way as to maximize its energy-efficiency (measured in bitsper Joule) and at the same time satisfy its QoS requirements.The QoS constraints are in terms of the average source rate andthe upper bound on the average total delay (transmission plusqueueing delay). We derive the Nash equilibrium solution forthe proposed game and use this framework to study trade-offsamong throughput, delay, network capacity and energy effi-ciency. Network capacity here refers to the maximum numberof users that can be accommodated by the network.Joint power and rate control with QoS constraints havebeen studied extensively for multiple-access networks (see forexample [1] and [2]. In [1], the authors study joint powerand rate control under bit-error rate (BER) and average delayconstraints. [2] considers the problem of globally optimizingthe transmit power and rate to maximize throughput of non-real-time users and protect the QoS of real-time users. Neitherwork takes into account energy-efficiency. Recently tradeoffsbetween energy efficiency and delay have gained more at-tention. The tradeoffs in the single-user case are studied in[3]–[6]. The multiuser problem in turn is considered in [7]and [8]. In [7], the authors present a centralized schedulingscheme to transmit the arriving packets within a specific timeinterval such that the total energy consumed is minimizedwhereas in [8], a distributed ALOHA-type scheme is proposedfor achieving energy-delay tradeoffs. Joint power and ratecontrol for maximizing goodput in delay-constrained networksis studied in [9].This work is the first one that attempts to study QoS-constrained power and rate control in multiple-access networksusing a game-theoretic framework. In our proposed game-theoretic model, users choose their transmit powers and rates ina competitive and distributed manner in order to maximize theirenergy efficiency and at the same time satisfy their delay andrate QoS requirements. Using this framework, we also analyzethe the tradeoffs among throughput, delay, network capacity andenergy efficiency. It should be noted that power control gameshave previously been studied in [10]–[17]. However, [10]–[16]do not take into account the effect of delay, and [17] onlyconsiders transmission delay and does not perform any ratecontrol.The remainder of this paper is organized as follows. In Sec-tion II, we describe the system model. The proposed joint powerand rate control game is discussed in Section III and its Nashequilibrium solution is derived in Section IV. We then describean admission control scheme in Section V. Tradeoffs amongthroughput, delay, network capacity and energy efficiency arestudied in Section VI using numerical results. Finally, we giveconclusions in Section VII.II. SYSTEM MODELWe consider a direct-sequence code-division multiple-access(DS-CDMA) network and propose a non-cooperative (distrib-uted) game in which each user seeks to choose its transmitpower and rate to maximize its energy efficiency (measuredin bits per joule) while satisfying its QoS requirements. Wespecify the QoS constraints of user k by (rk, Dk) where rkisthe average source rate and Dkis the upper bound on averagedelay. The delay includes both queueing and transmissiondelays. The incoming traffic is assumed to have a PoissonFig. 1. System model based on an M/G/1 queue.distribution with parameter λkwhich represents the averagepacket arrival rate with each packet consisting of M bits. Thesource rate (in bit per second), rk, is hence given byrk= M λk. (1)The user transmits the arriving packets at a rate Rk(bps)and with a transmit power equal to pkWatts. We consider anautomatic-repeat-request (ARQ) mechanism in which the userkeeps retransmitting a packet until the packet is received atthe access point without any errors. The incoming packets areassumed to be stored in a queue and transmitted in a first-in-first-out (FIFO) fashion. The packet transmission time for userk is defined asτk=MRk+ ²k'MRk, (2)where ²krepresents the time taken for the user to