Network-wide energy efficiency in wireless networks with multiple access points

Transcription

1 TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIES Trans. Emerging Tel. Tech. (0) Published online in Wiley Online Library (wileyonlinelibrary.com)..5 RESEARCH ARTICLE Networ-wide energy efficiency in wireless networs with multiple access points Omur Ozel and Elif Uysal-Biyioglu * Department of Electrical and Computer Engineering, University of Maryland, College Par, MD 07, USA Department of Electrical and Electronics Engineering, Middle East Technical University, Anara 06800, Turey ABSTRACT This paper presents a distributed mechanism for improving the overall energy efficiency of a wireless networ where users can control their uplin transmit power targeted to the multiple access points in the networ. This mechanism lets the networ achieve a trade-off between energy efficiency and spectral efficiency through the use of suitably designed utility functions. A user s utility is a function of throughput and average transmission power. Throughput is assumed to be a sigmoidal function of signal-to-interference-plus-noise ratio. Each user, being selfish and rational, acts to maximise its utility in response to signal-to-interference-plus-noise ratio by adjusting its power. The resulting mechanism is a distributed power control scheme that can incline towards energy-efficient or spectrally efficient operating points depending on the choice of utility function. Existence and uniqueness of Nash equilibrium points in this game are shown via convergence of the distributed power iterations. It is shown that, in the best-response strategy, each user selects a single access point. An extension of this result for a multicarrier system is considered, and the corresponding power levels used for various priorities between energy efficiency and spectral efficiency are characterised. Finally, several numerical studies are presented to illustrate the analysis. Copyright 0 John Wiley & Sons, Ltd. KEY WORDS networ energy efficiency; distributed power control; multiple access points; utility function; game theory; target SINR *Correspondence E. Uysal-Biyioglu, Department of Electrical and Electronics Engineering, Middle East Technical University, Anara 06800, Turey. elif@eee.metu.edu.tr Received July 00; Revised 0 January 0; Accepted 8 March 0. INTRODUCTION There is a well-nown trade-off between energy efficiency and transmission rate [,], and in the context of a wireless networ, increasing transmission power only maes sense if, considering the networ s response, it will ultimately lead to an appreciable gain in rate. Because of the broadcast nature of wireless communication, and the interference this tends to cause, the performance of a user in a wireless networ can be highly dependent on other users actions. One of the possible actions is the choice of transmission power. In order to achieve a certain rate, for example, a user may need to increase its transmit power as interference level increases. This, in turn, can increase the interference on others, who respond, and so on. This interaction may culminate at a stable operating point where every user is satisfied with its own level of signal-to-interference ratio. However, this operating point may not always be energy efficient. In principle, a networ can be engineered to converge to a desired operating point by using a suitable power control algorithm. Of course, for implementability, distributed algorithms are attractive. Distributed power control, where wireless nodes mae their own power control decisions (possibly asynchronously), has been the focus of a large body of early studies [ 8]. In recent years, game theory [9] has been used to model the interference-induced interaction in wireless communication [0 ] and to obtain distributed algorithms. In fact, communication networs form an increasingly popular setting for the application of game theory []. For one thing, the terminals (nodes) are quite truly rational and usually selfish players. A power control game arises when users are able to adjust their power in response to the interference they are subject to because of other data transmissions, with the goal of maximising the utility of their communication with their intended receiver. Depending on the constraints and Copyright 0 John Wiley & Sons, Ltd.

2 the utility function, there may be an equilibrium or several equilibria in this game. The networ designer s problem is to design the utility function to give rise to a desirable equilibrium from the perspective of the whole networ. The particular goal of interest in this paper is to maximise the networ energy efficiency, that is, to minimise the total power used per overall throughput in the networ. One approach through which users can be driven to be efficient while also trying to maximise their rate has been pricing []. In this case, users try to maximise a net utility, which is utility minus a price, where price is a function of, say, power. However, pricing is not always natural in settings where there is no centre to collect them or if the centre is not also a player in the game. As also established in the recent wor in [5], in settings such as interference networs where user s actions to maximise their own utilities indirectly affect others, pricing methods may not result in efficient or Pareto optimal allocations. It is argued [5] that different techniques are required for guaranteed convergence to globally optimal power allocations. Furthermore, in settings such as ad hoc or sensor networs where networ-wide energy efficiency is important, pricing is unnatural, as, for one thing, there is typically no price collector. All of the aforementioned reasons motivate us to pose the following question: Can we set up a simple utility model for uplin power control with multiple access points (APs) that does not include an explicit pricing mechanism yet drives the networ to an operating point with tuneable energy efficiency and spectral efficiency? The objective of this wor is to devise a distributed mechanism of uplin power control in an interference networ with multiple base stations (or APs). The main parameters of the problem are the time average power gains of users to the APs, assumed to be valid over a period during which a power allocation decision will be used. The goal is to show the existence of a distributed mechanism for nodes to adjust their transmission powers aimed at each base station, whereby the whole networ can strie a balance between energy efficiency and spectral efficiency. This problem is formulated in a game theoretic setting. The best-response strategies (reaction curves) are found, and the existence and uniqueness of Nash equilibrium (NE) are shown. Iterative methods to reach equilibrium are presented. In addition, the behaviour of equilibria that result from utility functions with varying degrees of priority given to energy efficiency versus throughput maximisation are investigated. The structure of the rest of the paper is as follows. In the next subsection, we review and discuss some of the most related wor from the literature in order to put this contribution in the proper context. The following section presents the system model and the basic definitions to be used in the problem formulation. In Section, the expression and properties of the utility function are provided and the power control game in a single-ap system is analysed. In Section, the multiple-base station vector power control game is analysed. Considering different priorities for different applications, the trade-off between energy efficiency and spectral efficiency is pointed in Section 5. The games and concepts are numerically illustrated in Section 6. Conclusions are presented in Section 7... Related wor One of the earliest studies of a problem formulation similar to the one herein is [6]. In [6], power control games in a single-cell system are considered, with a utility function in the form of the ratio of rate to power. A socially optimum operating point is derived, and prices are introduced to obtain a point closer to the social optimum. In [7], power control in a code division multiple access (CDMA) system is modelled as a noncooperative game with utility proportional to rate. Using linear pricing results in admission control, because users may opt out of the networ as they try to unilaterally optimise net utilities. In [8 0], power control games in a CDMA system are established with an energy efficiency goal. With different types of receivers, adaptive modulation and coding, hybrid games are obtained and equilibrium points are analysed. Analysis of noncooperative power control in a single-cell multicarrier CDMA system is presented in []. The multiple-base station problem addressed in this paper, while carrying similarities to the multicarrier CDMA problem, does not reduce to it, as users that select different base stations still potentially cause interference on each other (whereas users selecting different carriers do not). The gradual removal problem formulated in a recent wor [] is relevant to the scope of this paper in the sense that, at equilibrium, the transmitting users attain their target signal-to-interference ratios by transmitting the minimum overall power. Other approaches to distributed power control have appeared in the literature. In [], sum rate in a multicell system is maximised in a fading environment. Again, with respect to the sum rate criterion, [] and [5] have recently and independently shown the optimality of binary power control in all signal-to-noise ratio regimes. A distributed sum rate maximisation scheme that supports unequal user priorities was proposed in []. Fairness of non-game theoretic as well as game theoretic distributed power control algorithms in a general interference networ has been explored in [6], wherein numerical results showed the advantage of game theoretic modelling with respect to fairness and efficiency. The focus of this paper is distinguished from those of the aforementioned studies in that, vector power control in a networ with multiple choices of receiver stations for each node is addressed in a multichannel system. With somewhat similar motivation and goals as in this paper, the recent wor in [7] considered allocation of base stations and distributed base stations in an LTE * networ. As *LTE stands for Long Term Evolution, a recent cellular communication standard.

3 in some of the previously mentioned wors, overall networ efficiency is targeted in this paper. NEs are obtained and analysed with respect to the trade-off between spectral efficiency and energy efficiency, and the emphasis of the solution on one or the other is controlled by a priority exponent (see [8] for a recent treatment on spectral efficiency versus energy efficiency). In the next section, the system model in consideration is made explicit, and the problem formulation is defined.. SYSTEM MODEL AND DEFINITIONS We consider a wireless networ of K users and M APs (Figure ). Users can transmit with a rate of up to R bps in a common frequency band B Hz. Let the channel power gain between user i and AP b be given by the real constant h ib > 0. Channel gains are assumed constant during operation. More generally, the h ib s can be considered as average channel gains in a fading environment. Not all users have to be heard by each base station. This is captured by setting h ib D 0 for user i and base station b. Messages are sent from nodes to APs, and each AP hears each user s transmission, provided the user s channel gain to that AP is sufficiently large. The user can potentially exploit this to increase its rate, by sending different messages to different APs. Let the message signal of user l to destination AP j be X lj. According to this model, the signal Y b, received at base station b is Y b D KX ld j D User MX p hlb X lj C Z b () h b Access Point b Figure. Wireless networ with several users and access points. Z b is additive noise at AP b. For convenience, we model Z b as white Gaussian with zero mean and EjZ b j D. P ij is the average power of the message X ij : EjX ij j D P ij. Each user is subject to a power constraint: MX P ij 6 P max 8i () j D We consider single-user decoders in the receivers. Depending on the receiver structure, cross-channel gains may be suppressed by additional processing gains. A typical application of this model is the direct-sequence CDMA [9] with specific spreading codes for each possible lin. As user i sends different data to different users, user i s own message to APs other than b are also treated as interference at AP b. Interference will be treated as noise, as it is usually done in practical receivers, and will be modelled as Gaussian, which can be a good assumption as the number of independent interferers grows (see, for example, [5, 0, ]). G p stands for processing gain. The signalto-interference-plus-noise ratio (SINR) of user in the receiver of AP b is h b D G b P b p PKiD P () C Mj D h ib P ij h b P b The model with multiple APs has been motivated by a number of communication scenarios: (i) a local area networ where wireless nodes may be in the range of multiple APs; (ii) an ad hoc wireless networ with multiple gateway stations that enable connection to a larger, wired networ; and (iii) the microdiversity system in [] where multiple APs are considered as a single AP having multiple antennae distributed in space. The understanding in these scenarios is that the connections between APs are wired and the communication among them is straightforward. In each of these settings, the signalling and coding can tae different forms. In addition to the immediate example of a CDMA system given in the previous paragraph, other relevant models include multicarrier signalling and time division; in an orthogonal frequency division multiplexing strategy [], users can allocate different subcarriers to different APs and divide their total instantaneous power among the subcarriers []. In this case, the structure of the problem is somewhat different than the single-carrier version in that users are only subject to interference from users on the same subcarrier. Similarly, users could allocate different time slots to access to different base stations, allocating a long-term average power constraint between time slots. Again, interference is between subsets of users using the same time slot []. Although characterising the equilibrium points may be more complicated in the multicarrier and multislot models, some of the results in this paper continue to hold, as will be argued later in this paper. Note that the effects of strategies of the other users are observed in the denominator of the SINR expression in Equation (). Hence, users are in such an interaction that

4 performance of one user is degraded when another user attempts to increase its power. This interaction is observed not only in single-user decoders but also in multiuser detectors such as minimum mean square error [8] and minimum mean square error successive interference cancellation [0,5]. In order to analyse this interaction among users, we will employ static noncooperative game theory. A static game D ŒU ; fs i g; fu i g is defined using three components [9]: () user set U ; () action or strategy set S i, 8i U ;and () utility u i as a function of elements of S i, 8i U. The user set is the index set of players: U D f; ; :::;Kg. Given the other users actions, users unilaterally maximise their utility in their strategy set. An operating point at which no user can achieve higher utility by unilateral changes in action is called a Nash equilibrium. This captures the noncooperative nature of the problem. Definition. An NE is the vector of strategies se D Œs ;s ;s ;:::;s K such that u i.s i ; Es i / > u i.s i ; Es i / 8s i S i is satisfied for all user i where Es i D.s ;s ;:::;s i ; s ic ;:::;s K /. Note that an NE may not be socially optimal, that is, there may be a point with utilities u 0 i that is feasible and yet u 0 i >u i 8i,whereu i is the value of user i s utility at NE. Actually, it is possible to obtain higher total utility by using a cooperative mechanism such as pricing [7,6]. However, in our setup, users are selfish and are not directly interested in the overall performance of the networ; each user optimises its own utility in its own action space. Hence, we assume noncooperative operation. Given actions of users other than, Es, the best response (in other words, the reaction curve) ofuser, r,is r.es /, arg max s S u Q real numbers, u i W Kj D S j! R. The value of the function u i represents the level of satisfaction of user i with respect to some goal. Usually, in a communication scenario, satisfaction of a node is related to the communication performance such as throughput, outage probability, bit error rate (BER), SINR and power or energy cost. The choice of utility can also depend on external conditions: when spectral resources are scarce, throughput carries high utility, whereas if energy is limited, a utility that decreases with transmit power is appropriate. However, a combination of these parameters must determine the level of satisfaction for mobile data users. Bits successfully sent per joule of energy spent has been a well-nown utility function [6, 8, 6] that appropriately combines throughput and cost terms, encouraging energy-efficient behaviour. The standard definition of throughput, also adopted in this paper, is the long-term average data rate (bits per transmission) achieved. Taing into account lin layer framing and error control mechanisms whereby a data pacet (say, a constant number of bits) is declared unsuccessful if more than a certain number of bit errors occur and accounting for resulting pacet drops, which happen with finite probability, throughput by definition is upper bounded by the long-term average coding rate, R. In previous literature, throughput was often modelled as a sigmoidal function of SINR (Figure ) [7]. The main reason for this is, as a certain threshold in SINR is exceeded, pacet success probability quicly rises towards with many practical as well as optimal modulation and coding schemes. As a very simple example for the occurrence of the sigmoid, consider the following: pacets of length L symbols are sent using binary phase shift eying (BPSK) modulation technique, and the code rate is R bits per symbol. Each bit is decided erroneously with probability BER./. Then, the long-term average throughput T is T D R. BER.// L () Concave Nash equilibrium can also be defined in terms of best responses. Es is NE iff s D r.es / 8. Inotherwords, NE is a fixed point of best responses. Consequently, the concept of NE is well suited to the wireless networ power control problem, and we will analyse stable operating points through examining the existence and properties of NE. Convex Inflection Point. UTILITY FUNCTION AND THE SINGLE-ACCESS POINT SYSTEM In game theoretic terms, utility function u i is a mapping from the Cartesian product of action sets S j of users to 0 Figure. Function f./ versus in normal scale. It is plotted for binary phase shift eying modulation with pacet length L D 00 bits.

5 When BER./ is decreasing with a convex shape with respect to, as it is typically the case, T is a sigmoid (e.g. [7]), that is, there is an inflection point such that T./is convex in Œ0; / and concave in.;/. Note that Equation () is in the form of an effective rate, that is, rate multiplied by an efficiency function f./ as T D Rf. /. The sigmoid model for f./ is valid in many communication scenarios. In a system with fixed coding and modulation and a lin layer error control mechanism such as automatic repeat request (ARQ) with cyclic redundancy chec [8], f./ has sigmoidal shape. The sigmoidal shape still holds [9] even if messages heard by each AP were decoded in a common centre and even when modulation and rate are adapted to changes in SINR. Consistent with observations from many theoretical and practical research results [8,, 5, 8, 9], f./will be assumed to have sigmoidal shape in this paper. Let T b and p b be the throughput and power of user for communication with AP b, respectively. The utility function u is defined as the ratio of total throughput to total dissipated power: P M bd u D T b P M bd p b Note that the motivation for having the power term in the denominator of the utility function is to encourage energy-efficient behaviour of users. To avoid associating a positive utility with no transmission, it is reasonable to have u! 0 when P M bd p b D 0 for all. This will automatically hold when the throughput function tends to zero as SINR vanishes, which is the case in almost all practical lin layer mechanisms [8,, 5, 8, 9]: (5) lim f./d 0 (6)!0 definition). It is observed and can be verified that, given P, u i.p ;P / 8i are quasiconcave with respect to p (Figure ). Given p j, j, changes linearly with p. Letting Qh be an effective channel gain of user and the noise variance in its receiver, SINR expression in Equation () is Qh D D Q h p (9) G p h Pi h i p i C (0) p that optimises u over the compact set S D Œ0; P max is such that either it is on the boundary or D 0; p S () Proceeding by taing the derivative and using the linearity of SINR with transmit power, we find the best response r.p / of user as r.p / D min ( ) ;P max Qh () is a unique positive solution of the following equation [0]: f./d f 0./ () The value of depends on the sigmoidal function f./ such that the horizontal component of the intersection point in Figure is strictly greater than the inflection point of the sigmoid [0]. Note that the shape of the sigmoidal function is determined by the modulation and coding scheme. Before approaching the general problem, we will first consider the single-ap system. Let M D. The strategy set of each user i is S i D Œ0; P max,wherep max is the maximum power level allowed for each user. Utility function of user, with power level p,is u D T p (7) T is the long-term average rate as in Equation (). Let D ŒU ; fs i g; fu i g be the one-shot game in which each user unilaterally performs the following optimisation: Utility max u.p ;P / 8 U (8) p S where P / stands for the vector of powers of all users except the th. An important property possessed by the utility functions u i that plays a ey role in the existence and uniqueness of equilibrium is quasiconcavity (see [6] for Power Figure. Typical variation of utility function u with power p given other users powers. It is quasiconcave, monotone increasing up to some value of power and monotone decreasing afterward. Note that, depending on P max, the decreasing regime may not be observed, but this does not violate quasiconcavity.

6 γ f (γ) f(γ) update algorithm O I./ diverges, and for some of the users, p i D P max and i <, whereas other users achieve at NE. On the other hand, the feasibility condition is necessary (but not sufficient) for all users to achieve. Now that the power control game in a single-ap system has been analysed, we turn our attention to the general multiple-ap system in the next section.. THE POWER CONTROL GAME γ * Figure. is a unique positive-valued solution of Equation (). By definition, solution(s) of the fixed-point equations p D r.p / is (are) the NE(s). Consider the corresponding fixed-point iteration p.t C / D r.p.t// where the power of user i at iteration t is p i.t/. The iterations converge to the unique fixed-point iff NE is unique. To investigate the convergence of the fixed-point iterations, it is useful to view the iterations as a power update algorithm I./ such that p.t C / D I.p.t//,wherep.t/ D Œp.t/; p.t/;:::;p K.t/. In our problem, the explicit form of I./ is such that I i.p.t// D minf Op i.t/; P max g, where Op i.t/ D P j i h j p j.t/ C ;id ;;:::;K G p h i () It is evident from the aforementioned expression that our I./ satisfies the standard power update algorithm definition of Yates [] and if algorithm I./ O with IO i.p.t// D Op i.t/ is a standard algorithm, then I i.p.t// D minf Op i.t/; P max g has a unique fixed point. Hence, we conclude that our power update algorithm I./ has a unique fixed point; consequently, has a unique NE. In general, p D r.p / 8 form a system of K nonlinear equations. In our particular problem in Equation (), the nonlinearity of r is due to clipping with P max.ifp max is assumed sufficiently large, NE is a solution to the following system of K linear equations []: h p Pi h i p i C D 8 D ;;:::;K (5) The aforementioned linear system may have a unique solution, infinitely many solutions or no solution. If is feasible, then the system has a unique solution. The feasibility of can be determined using Perron Frobenius theory []. By analysing the problem in terms of received powers, one can show that the feasibility condition is < G p K. If this condition is not satisfied, power Consider the general model with M APs (Figure ). As before, users are subject to power constraint P max.however, now, they are allowed to transmit to more than one AP at a time. In other words, users can divide their power budget and transmit (different) data to different APs in order to (possibly) obtain a multiplexing gain. In this case, the strategy set of a user is 8 9 < M S D : Œp p ::: p M R M C W X = p j 6 P max ; j D (6) The utility function is as in Equation (5): P M bd u D T b P M bd p b (7) T b is the long-term average rate of user in AP b. We will analyse D ŒU ; fs g; fu g, and the corresponding user optimisation is as follows: max u.p ; P / (8) p S where p D Œp ;p ;:::;p M and P D Œp ; p ;:::;p ; p C ;:::;p K. The main result of the paper is stated in the following theorem, which stipulates the special form of the best-response strategy in which each user transmits to a single AP. Theorem. The utility maximising strategy of user, p,givenp in game is such that p p b D ; if b D b 0; otherwise b n o D arg max bhb b (9) (0) 0 p D max ; A () b hb b G p h hb D b b C P K id i h P Mj () ib D p ij

7 Proof. The proof relies on the results obtained for single- AP system. First, the set over which the optimisation is performed is extended to Œ0; P max M. The result for the optimum in single-ap system is used, and a componentwise summation yields the desired conclusion. The details are given in Appendix A. Theorem suggests that each user should just transmit to the AP that requires minimum power in order to maximise its utility unilaterally. Although obtained from different contexts, the similarity of best-response strategy to the sum-rate optimum strategy of transmitting to a single user in each channel state in the fading broadcast channel model [] is notable. The aim of the resource allocation formulation in [] is to maximise long-term average rate, given an average power budget. In contrast, in our formulation, the power budget is optimally divided among base stations to maximise the utility. Put in a different way, optimising energy efficiency requires achieving a target SINR by choosing the best AP, whereas rate maximisation allocates all power resource to the best channel... Equivalence with base station selection and power control game In conclusion of Theorem, the problem reduces to wellnown joint AP assignment and power control problem [6]. Therefore, the game, in which users strategies are power vectors, can be analysed by another game, in which the strategies are one of APs and (scalar) transmit power for that AP. Consider a game D ŒU ; fs g; fu g for which the strategy set S is S D A P 8 D ;;:::;K () where A DfA ;A ;:::;A M g and P D Œ0; P max, with A i being the ith AP. Let a and p be the AP assignment and transmit power of user, respectively. a D b h a p is the SINR of user. Each user has the following utility function: u D R f. a / p () The joint AP assignment and power control game was originally proposed in [6]. In order to find the best-response strategy, optimisation is performed in two stages [6]. First, the base station for which user s SINR is maximum is chosen. Then, the power is adjusted to the level that optimises utility function in Equation () for the chosen base station and given other power levels. Note that the best-response strategies of and are equivalent. In [6], is proved to have a unique NE. Similar to the single-ap problem, the existence proof is based on compactness, convexity of Œ0; P max and quasiconcavity of f./; the uniqueness is proved by direct verification that the best-response strategy defines a standard power update algorithm [6]. This way, the equilibrium of the vector power control game is characterised in terms of base station selection and (scalar) power control game, which is emphasised once more in the following theorem. Theorem. and have unique NEs. In the NEof,user only transmits to the APthat is assigned in the NE of with nonzero power and transmit power p in theneof, and are equal for all. An earlier wor of Yates [5] posed a non-game theoretic integrated power control and AP assignment problem. Although the formulation was not that of a game, it applies to the problem at hand. In a K user and M -AP system, minimum total transmit power vector is found under SINR constraints i 0 where each user is assigned to only one AP. i > 0 i 8i D ;;:::;K (5) If 0 i are feasible, then there exists a unique solution for minimum total transmit power vector problem. Perron Frobenius theory [] is again deployed for analysing feasibility. Assuming that each user is assigned to a fixed AP, SINR constraints and channel gains are combined in one matrix. The feasibility condition is that resultant Perron Frobenius eigenvalue PF <for some assignment among M K possible assignments. In particular, for a single-ap system with K users, the feasibility condition reduces to a simple inequality on the number of users and processing gain: < G p K. Set 0 D. Provided that P max is sufficiently large and feasibility is satisfied, the NE points of and are equivalent to the unique solution of minimum total transmit power problem with 0 D... Extension to multicarrier multiple-access channel systems The system model can be extended to a multicarrier multiple-ap system. The availability of multiple carriers introduces an extra dimension to the strategy sets: now, users, by picing which subcarriers to use to access which base station (and how much of their total power to allocate to it), are picing a subset of interferers. It is important to note that the results in [] and Theorem straightforwardly combine to conclude that the best-response strategy would be to transmit to only one AP by putting the total power on a single carrier. However, the game may not have unique NE in this case. Because of the orthogonality among the carriers, monotonicity and thus the standardness property cease to hold, as different users can transmit in different carriers, one user may not respond to an increase in another user s power. Therefore, the uniqueness of NE is not guaranteed, and in fact, as observed in [], for some values of channel gains, multiple NEs may exist.

8 5. ENERGY AND SPECTRAL EFFICIENCY The analysis in Section was based on a utility function that emphasises energy efficiency and has units of bits per joule. Unilateral optimisation of utilities led users to reach atargetsinr, which is the unique solution of the equation f./ D f 0./. However, the spectrum resource is inefficiently used in case has a low value. This observation points to the trade-off between energy efficiency and spectral efficiency. In order to explicitly address this trade-off in the game setting, we introduce a priority exponent >0so that the cost of transmitting with power p isassumedtobep. Then, the utility function of user for single-ap system is as follows: u D T p (6) The priority exponent brings a variable degree of energy efficiency to the utility function. For D, the utility function in Equation (7) is obtained. <means that users value spectral efficiency more, whereas >drives users to be more energy efficient. The equilibrium SINR (as a function of the exponent ). / is the unique solution of f./d f 0./ The variation of. / for different values of is shown in Figure 5. A similar modification can be made to the utility function for multiple-ap system. If the cost of communication with AP b is p for user, then the total cost is b the P summation of the costs of communication to all APs: M bd p. Similar to the previous utility function, the b modified utility represents total throughput per total cost as follows: u D P M bd T b P M bd p b (7) In this case, although not immediately obvious, the best response is again transmitting to a single AP that requires the lowest power to reach. /. Details about the calculation of best-response strategy for this case is given in Appendix B. 6. NUMERICAL ILLUSTRATIONS In this section, we will provide graphical and numerical illustrations on how NE is reached using the best response strategy and on the variation of target SINR with respect to priority exponent. In particular, we will first dwell on iterative application of best-response strategy by using the utilities in Equation (5) for a practical setting. Then, we show the variation of target SINR with the exponent, and several plots that illustrate the variation for certain practical modulation schemes will be provided. 6.. Iterative application of best response We will illustrate how NE is reached by iterative application of best-response strategy given in Theorem. In particular, assuming a.t/ is the base station selection of user at step t, power is updated to p.t C / with the use of the following synchronous two-step algorithm: () a.t C / D arg max aa a.t/,and ( ) () p.t C / D min Qh ;P a.tc/.tc/ max. γ * =G p /(N ) sigmoid α= α=/ α= α=/ We consider the uplin of a 0-user, four-ap directsequence CDMA wireless networ deployed in a m area. Users are uniformly located in the area. Simulation parameters are given in Table I. For simplicity, we assume that there is no channel coding, although the results can be generalised by assuming a coding gain. Data are sent Table I. Simulation parameters. Energy Efficient Spectrally Efficient Figure 5. is the intersection of two functions f./ and f 0./. The horizontal axis of the intersection point is the equilibrium SINR,. /. For <, as the priority of spectral efficiency is higher, taes higher values. It is not possible for users to mutually reach in case > Gp. K M Number of access points K Number of users 0 R Bit rate 0 bps G p Processing gain 50 AWGN power in receiver W Modulation technique BPSK P max Maximum power W AWGN, additive white Gaussian noise; BPSK, binary phase shift eying.

9 y (meters) x (meters) Figure 6. Access point (AP) and user locations in the simulated area. Numbers in bold are attached to APs, and small-sized numbers are attached to each user, indicating to which AP it communicates in the Nash equilibrium of game. Note that each user communicates to only one AP, although it is allowed to communicate with all APs. as pacets of length 000 bits, and if the pacets cannot be received (i.e. if an error is detected), then pacets are retransmitted with an ARQ mechanism. Accordingly, the BER for communication between user and AP b for an additive white Gaussian noise channel is BER b D Q p b Figure 6 on the border of two quadrants where a user is closer to AP but communicates to AP. For the simulated distribution of users, no users select the same AP; therefore, D 6 db is the SINR in NE. The evolution of power and SINR values in the iterative application of best-response strategy is observed in Figures 7 and 8, respectively. Note that, at each iteration, data are For this case, ARQ throughput expression is T b D R Q p 000, b whereq is the complementary error function. Considering the condition in Equation (6), the efficiency function is chosen as f./d Q p Power (W) For the previous f./, the solution of the equation f./ D f 0./ is calculated as D 6 db. Note that, if every user were to communicate to every AP, then feasibility condition would become < G p K D 50 9,which is not satisfied. However, if there are at most users from which each AP receives data, then 6-dB SINR is feasible. We now that each user communicates to only one AP as a best-response strategy. Hence, unless or more users select the same AP, SINR in the NE is 6 db. Locations of APs and users are given, and APs to which each user communicates in NE is illustrated in Figure 6. Note that users do not necessarily communicate to the closest AP. A counterexample can be readily observed in Iteration Figure 7. Evolution of transmit power of five arbitrarily selected users (users 5, as the label next to each curve indicates) from the simulation experiment. Each curve plots the power versus iteration number for one particular user applying the best-response strategy. At each iteration, users select different access points (APs), but as equilibrium is approached, AP selection is also fixed. Bold numbers attached at the end of each curve indicates the equilibrium AP selection.

10 SIR (normal) γ = Iteration Figure 8. Evolution of the observed signal-to-interference-plusnoise ratio (SINR) of five arbitrarily selected users from the simulation experiment. Each curve plots the SINR versus iteration number for one particular user applying the best-response strategy. The common SINR at Nash equilibrium is. 0 NC FSK BPSK Figure 9. Variation of with respect to the priority exponent for binary phase shift eying (BPSK)-modulated and noncoherent frequency shift eying (NC-FSK)-modulated automatic repeat request transmission with 000-bit length pacets. transmitted to different APs and overshoots are observed. As algorithm approaches to the equilibrium, the AP selection is also fixed. Note that SINR of each user converges to. 6.. Variation of target signal-to-interference-plus-noise with In Figure 5, the intersection points that define the target SINR varies with the priority exponent, andas is increased, the value of transmitting with less power is increased so that the target is decreased. We will illustrate the variation of target SINR with in a described practical setting. In particular, the solution of the equation f./ D f 0./, which yields the target SINR,is calculated for selected efficiency functions and for various values of. Because an explicit expression of is not possible for a given f./, the calculation of the solution of the equation (which is guaranteed to be unique) has been performed numerically using Newton s method with a precision of 0:0%. Two different functions that correspond to BPSK and noncoherent frequency shift eying (NC-FSK) modulation schemes are used, which respectively are as follows: f BPSK./ D Q p f ncfsk./ D e Resultant variation of with for efficiency functions of BPSK and NC-FSK is provided in Figure 9. It is observed that target SINR decreases as is increased. 0.9 NC FSK BPSK Figure 0. Normalised throughput at equilibrium versus for binary phase shift eying (BPSK) and noncoherent frequency shift eying (NC-FSK). Note that for FSK is higher than that for BPSK. This results from the difference between the BER performance of BPSK and that of FSK. In order to achieve the same BER, -db-higher SINR is required in FSK than that in BPSK []. It is interesting to observe that about the same difference in the target SINR results from the efficiency function of the modulation scheme. Note that previous observation does not imply that FSK is more spectrally efficient than BPSK. Actually, equilibrium throughput (assuming that P max is large and is reached) of FSK is lower than that of BPSK as shown in Figure 0. Because FSK has worse BER performance, the concave part of the sigmoid starts at a higher SINR, and that is why of FSK is higher than that of BPSK. In conclusion, BPSK has higher spectral efficiency, and the priority exponent can introduce different degrees of energy efficiency for different modulation schemes.

11 7. CONCLUSIONS AND DISCUSSION This paper studied vector power control in the uplin of a general wireless communication networ with multiple APs. The problem has been formulated within a noncooperative game framewor. Given the other users strategies, each user is allowed to optimise its own utility function by selecting a transmission power level allocation for each AP. The main novelty in the approach of the paper is the choice of utility function, which is an increasing function of throughput and a decreasing function of power. Of course, as the two goals of high throughput and low energy consumption are at odds, a trade-off surface, which depends on the relative weights of throughput and power that are in the utility function, is introduced. The motivation for our choice of utility was to obtain a tuneable degree of networ energy efficiency and networ spectral efficiency and to accomplish this without an explicit pricing mechanism. A vector power control game was proposed using this utility function. Throughput was assumed to be a sigmoidal function of SINR. The best-response strategy of the game was shown to have a special structure; although the users are allowed to spend portions of their power on different APs, they end up choosing to transmit to a single AP. Hence, it was shown that the game decouples into AP selection and power control. The existence and uniqueness of the NE of this game has been established using this special structure. It has been observed that the best-response strategy leads to a target SINR-based power control algorithm. The equilibrium operating point is characterised by a target SINR, whose value is determined by the coding and modulation type. When the parameters of the problem instance such as maximum power levels and time average channel gains deem feasible, NE corresponds to achieve the minimum total transmit power vector under average SINR. Hence, effectively, energy efficiency is optimised while satisfying an average quality of service. After obtaining these basic results about the structure of the equilibrium point, we then studied the variation of the operating point in response to the tuning of the utility function to emphasise energy efficiency versus spectral efficiency, and vice versa. The variation in the target SINR with respect to priority exponent was analysed, and the trade-off between energy efficiency and spectral efficiency was verified. Numerical illustrations that exhibit how the iterative algorithms reach NE and show the variation of with respect to are presented. We observe that convergence occurs after several iterations, and the approach is a convincingly efficient way of arriving at a networ-optimised operating point. Moreover, users are treated reasonably fairly in the sense that they all observe almost equal SINR at the equilibrium operating point. This indicates that, in networs where pricing is not natural, a simple distributed method of letting the networ operate at a point of desired quality of service and energy efficiency is possible. This result is consistent with the related energy efficiency-related results of, for example, Meshati et al. [8 0], although the wor in this paper has addressed vector power control in a networ with multiple choices of receiver stations. We believe that the results motivate further wor involving an in-depth treatment including a more general utility function that captures the essence of the energy spectral efficiency trade-off. This would be of particular value in response to the growing interest in energy-efficient communications and distributed networs. The noncooperative game setup may be developed to include nodes that harvest energy from the environment. Then, not only the time average use of power but also the current state of stored energy would be a parameter determining the users action spaces. Also, although we have discussed lins to pricing and suggested that this approach is an alternative to it, we have not made this lin precise. It was pointed out, for example, that using linear pricing results in an admission control mechanism. It would be interesting to study the lins between the approach here and admission control in future wor. APPENDIX A: PROOF OF THEOREM It will be shown that f bhb p P p > Pb f. b/ b p 8Œp p p M S b (A) Let W D Œ0; P max. Without loss of generality, assume D (i.e. consider the first user) and b D (i.e. for the first user, the maximum b h b parameter is obtained with base station ). Hence, b h > b h b 8b f;;:::;mg There are two cases to consider: () p D b h, () p D P max (A) Assume the first case. Note that b D b.p ;p ;:::; p M / is a function of user s power strategy vector Œp p p M given the other p ij.asforb D, we have 8p W f. / p > fœ.p ;p D 0; p D 0;:::; p M D 0/ p (A) Then, the inequality follows: f. / p > fœ.p ;p ;p ;:::;p M / p 8Œp p p M S (A)

12 Considering for b D, maximisation is at either p D P max or p D b h. For the latter case, the procedure is similar to the previous one. In the former case, at p D P max,then f..p D 0; p D P max ;p D 0;:::;p M D 0/ P max 6 f. / p 0 6 f. / p (A5) for some p 0 > P max such that p 0 D b h ; then, it is obvious that p 0 > P max > p. Hence, it follows 8p W that f. / p > fœ.p D 0; p ;p D 0; :::;p M D 0/ p (A6) We immediately see that f. / p > fœ.p ;p ;p ;:::;p M / p 8Œp p p M S (A7) By proceeding similarly for other base stations, the inequalities obtained in Equations (A) and (A7) can be generalised as follows: f. / p > fœ b.p ;p ;p ;:::;p M / p b 8Œp p p M W M and 8b f;;:::;mg Converting the inequalities to p b p > fœ b.p ;p ;p ;:::;p M / f. / (A8) (A9) In the second case, p D P max. Using the maximality assumption of b h, we observe that p b D P max 8b f; ; : : : ; M g. f bhb p b f bhb P max p D 8b f;;:::;mg: P b max (A) Again, using the maximality of b h, we reach Equation (A8). Hence, the desired result follows for the second case. APPENDIX B: THE BEST RESPONSE WITH PRIORITY EXPONENT Let W D Œ0; P max. Assuming b D and without loss of generality, letting D, we claim that f bh p P.p b / > f. b/ Pb p 8Œp p p M S b (B) where b h > b h b 8b f;;:::;mg Again, there are two cases to consider: () p D b h, () p D P max (B) We will prove the claim for the first case as the proof for the second case can be shown by similar arguments to the proof in Appendix A. As for b D, wehave8p W f.. //.p / > fœ.p ;p D 0; p D 0;:::;p M D 0/ p (B) Then, the inequality follows: f.. //.p / > fœ.p ;p ;p ;:::;p M / p (B) 8Œp p p M S and summing over b, we obtain P b p P b p b > fœ b.p ;p ;p ;:::;p M / f. / (A0) 8Œp p p M S Converting once more, we obtain the desired result. 8Œp p p M S Considering for b D, maximisation is at either p D P max or p D. / b h. Similar to the proof in Appendix A, we can show the following result: f.. //.p / > fœ b.p ;p ;p ;:::;p M / p b 8Œp p p M W M and 8b f;;:::;mg (B5)

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