Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.2543
RESEARCH ARTICLE
Network-wide energy efficiency in wireless networks
with multiple access points
Omur Ozel1and Elif Uysal-Biyikoglu2*
1Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA 2Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara 06800, Turkey
ABSTRACT
This paper presents a distributed mechanism for improving the overall energy efficiency of a wireless network where users can control their uplink transmit power targeted to the multiple access points in the network. This mechanism lets the network achieve a trade-off between energy efficiency and spectral efficiency through the use of suitably designed util-ity functions. A user’s utilutil-ity 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 priori-ties between energy efficiency and spectral efficiency are characterised. Finally, several numerical studies are presented to illustrate the analysis. Copyright © 2012 John Wiley & Sons, Ltd.
KEY WORDS
network energy efficiency; distributed power control; multiple access points; utility function; game theory; target SINR
*Correspondence
E. Uysal-Biyikoglu, Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara 06800, Turkey. E-mail: elif@eee.metu.edu.tr
Received 13 July 2010; Revised 20 January 2012; Accepted 28 March 2012
1. INTRODUCTION
There is a well-known trade-off between energy efficiency and transmission rate [1, 2], and in the context of a wireless network, increasing transmission power only makes sense if, considering the network’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 network 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 interfer-ence level increases. This, in turn, can increase the inter-ference 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 network can be engineered to converge to a desired operating point by using a suitable power con-trol algorithm. Of course, for implementability, distributed algorithms are attractive. Distributed power control, where wireless nodes make their own power control decisions (possibly asynchronously), has been the focus of a large body of early studies [3–8].
In recent years, game theory [9] has been used to model the interference-induced interaction in wireless communi-cation [10–12] and to obtain distributed algorithms. In fact, communication networks form an increasingly popular set-ting for the application of game theory [13]. 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 theutilityof their communication with their intended receiver. Depending on the constraints and
the utility function, there may be an equilibrium or several equilibria in this game. The network designer’s problem is to design the utility function to give rise to a desirable equilibrium from the perspective of the whole network. The particular goal of interest in this paper is to maximise thenetwork energy efficiency, that is, to minimise the total power used per overall throughput in the network.
One approach through which users can be driven to be efficient while also trying to maximise their rate has been
pricing[14]. In this case, users try to maximise a net util-ity, 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 work in [15], in set-tings such as interference networks where user’s actions to maximise their own utilities indirectly affect others, pric-ing methods may not result in efficient or Pareto optimal allocations. It is argued [15] that different techniques are required for guaranteed convergence to globally optimal power allocations. Furthermore, in settings such as ad hoc or sensor networks where network-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 uplink power con-trol with multiple access points (APs) that does not include an explicit pricing mechanism yet drives the network to an operating point with ‘tuneable’ energy efficiency and spectral efficiency?
The objective of this work is to devise a distributed mechanism of uplink power control in an interference network 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 network can strike a balance between energy efficiency and spec-tral efficiency. This problem is formulated in a game theo-retic 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 vary-ing 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 work from the literature in order to put this contribution in the proper context. The following sec-tion presents the system model and the basic definisec-tions to be used in the problem formulation. In Section 3, the expression and properties of the utility function are pro-vided and the power control game in a single-AP system is analysed. In Section 4, the multiple-base station vec-tor 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 illus-trated in Section 6. Conclusions are presented in Section 7. 1.1. Related work
One of the earliest studies of a problem formulation similar to the one herein is [16]. In [16], 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 opti-mum’ operating point is derived, and prices are introduced to obtain a point closer to the social optimum. In [17], 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 admis-sion control, because users may opt out of the network as they try to unilaterally optimise net utilities.
In [18–20], power control games in a CDMA system are established with an energy efficiency goal. With dif-ferent types of receivers, adaptive modulation and cod-ing, hybrid games are obtained and equilibrium points are analysed. Analysis of noncooperative power control in a single-cell multicarrier CDMA system is presented in [21]. The multiple-base station problem addressed in this paper, while carrying similarities to the multicarrier CDMA prob-lem, 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 work [22] 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 [23], sum rate in a multicell system is maximised in a fading environment. Again, with respect to the sum rate criterion, [24] and [25] 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 [24]. Fairness of non-game the-oretic as well as game thethe-oretic distributed power con-trol algorithms in a general interference network has been explored in [26], 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 network 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 work in [27] considered allocation of base sta-tions and distributed base stasta-tions in an LTE*network. As
*LTE stands for ‘Long Term Evolution’, a recent cellular communica-tion standard.
in some of the previously mentioned works, overall net-work efficiency is targeted in this paper. NEs are obtained and analysed with respect to the trade-off between spec-tral efficiency and energy efficiency, and the emphasis of the solution on one or the other is controlled by a ‘prior-ity exponent’ (see [28] 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.
2. SYSTEM MODEL
AND DEFINITIONS
We consider a wireless network ofKusers and M APs (Figure 1). Users can transmit with a rate of up toRbps in a common frequency bandBHz. Let the channel power gain between useri and APbbe given by the real con-stanthi b>0. Channel gains are assumed constant during operation. More generally, thehi b’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 settinghi bD0for useriand base stationb.
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 mes-sages to different APs. Let the message signal of userl to destination APj beXlj. According to this model, the signalYb, received at base stationbis
YbD K X lD1 M X jD1 p hlbXljCZb (1) User k Access Point b kb h
Figure 1.Wireless network with several users and access points.
Zbis additive noise at APb. For convenience, we model Zbas white Gaussian with zero mean andEjZbj2D2. Pij is the average power of the messageXij:EjXijj2D Pij. Each user is subject to a power constraint:
M
X
jD1
Pij 6Pmax8i (2)
We consider single-user decoders in the receivers. Depending on the receiver structure, cross-channel gains may be suppressed by additional processing gains. A typi-cal application of this model is the direct-sequence CDMA [29] with specific spreading codes for each possible link. As useri sends different data to different users, useri’s own message to APs other thanbare also treated as inter-ference at APb. Interference will be treated as noise, as it is usually done in practical receivers, and will be mod-elled as Gaussian, which can be a good assumption as the number of independent interferers grows (see, for example, [5, 30, 31]). Gp stands for processing gain. The signal-to-interference-plus-noise ratio (SINR) of user k in the receiver of APbis kbDGp hkbPkb 2CPKiD1PMjD1hi bPij hkbPkb (3)
The model with multiple APs has been motivated by a number of communication scenarios: (i) a local area net-work where wireless nodes may be in the range of multiple APs; (ii) an ad hoc wireless network with multiple gateway stations that enable connection to a larger, wired network; and (iii) the microdiversity system in [32] where multi-ple APs are considered as a single AP having multimulti-ple 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 take 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 multiplex-ing strategy [33], users can allocate different subcarriers to different APs and divide their total instantaneous power among the subcarriers [31]. 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 allo-cate 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 [34]. Although characterising the equilibrium points may be more complicated in the mul-ticarrier 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 (3). Hence, users are in such an interaction that
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 [18] and minimum mean square error successive interference can-cellation [20,35]. In order to analyse this interaction among users, we will employ static noncooperative game theory.
A static gameDŒU ;fSig;fuigis defined using three components [9]:
(1) user setU;
(2) action or strategy setSi,8i2U; and
(3) utilityuias a function of elements ofSi,8i2U.
The user set is the index set of players:U D f1; 2; : : : ; 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 aNashequilibrium. This captures the noncooperative nature of the problem. Definition 1. An NE is the vector of strategies sE D Œs1; s2; s3; : : : ; sKsuchthat
ui.si;sEi/>ui.si;sEi/8si2Si
is satisfied for all useri whereEsi D .s1; s2; : : : ; si1; siC1; : : : ; sK/.
Note that an NE may not be socially optimal, that is, there may be a point with utilitiesu0ithat is feasible and yet u0i> ui 8i, whereui is the value of useri’s utility at NE. Actually, it is possible to obtain higher total utility by using a cooperative mechanism such as pricing [7, 16]. However, in our setup, users are selfish and are not directly interested in the overall performance of the network; each user opti-mises itsownutility in itsown action space. Hence, we assume noncooperative operation.
Given actions of users other than k, Esk, the best response (in other words, thereaction curve) of user k, rk, is
rk.Esk/,arg max sk2Sk
uk
Nash equilibrium can also be defined in terms of best responses.Esis NE iffsk Drk.Esk /8k. In other words, NE is a fixed point of best responses. Consequently, the concept of NE is well suited to the wireless network power control problem, and we will analyse stable operat-ing points through examinoperat-ing the existence and properties of NE.
3. UTILITY FUNCTION AND THE
SINGLE-ACCESS POINT SYSTEM
In game theoretic terms, utility functionui is a mapping from the Cartesian product of action setsSj of users to
real numbers, ui W QKjD1Sj ! R. The value of the functionui represents the level of satisfaction of useri with respect to some goal. Usually, in a communication scenario, satisfaction of a node is related to the communi-cation 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 combina-tion of these parameters must determine the level of sat-isfaction for mobile data users.Bits successfully sent per joule of energy spenthas been a well-known utility func-tion [16, 18, 36] 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 trans-mission) achieved. Taking into account link layer framing and error control mechanisms whereby a data packet (say, a constant number of bits) is declared unsuccessful if more than a certain number of bit errors occur and accounting for resulting packet drops, which happen with finite prob-ability, 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 2) [37]. The main reason for this is, as a certain threshold in SINR is exceeded, packet success probability quickly rises towards 1 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: packets of lengthLsymbols are sent using binary phase shift keying (BPSK) modulation technique, and the code rate isRbits per symbol. Each bit is decided erroneously with probabilityBER. /. Then, the long-term average throughputT is TDR.1BER. //L (4) 0 1
Convex
Concave
Inflection PointFigure 2.Function f. /versus in normal scale. It is plot-ted for binary phase shift keying modulation with packet length
When BER. / is decreasing with a convex shape with respect to , as it is typically the case, T is a sigmoid (e.g. [37]), that is, there is an inflection point such that T . /is convex inŒ0; /and concave in. ;1/. Note that Equation (4) is in the form of an effective rate, that is, rate multiplied by anefficiency functionf ./asTDRf . /.
The sigmoid model for f . / is valid in many com-munication scenarios. In a system with fixed coding and modulation and a link layer error control mechanism such as automatic repeat request (ARQ) with cyclic redundancy check [38], f . / has sigmoidal shape. The sigmoidal shape still holds [19] even if messages heard by each AP were decoded in a common centre and even when mod-ulation and rate are adapted to changes in SINR. Consis-tent with observations from many theoretical and practical research results [18, 21, 35, 38, 39],f . /will be assumed to have sigmoidal shape in this paper.
LetTkbandpkbbe the throughput and power of userk for communication with AP b, respectively. The utility functionuk is defined as the ratio oftotalthroughput to total dissipated power:
ukD PM bD1Tkb PM bD1pkb (5)
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 transmis-sion, it is reasonable to haveuk!0whenPMbD1pkbD0 for allk. This will automatically hold when the through-put function tends to zero as SINR vanishes, which is the case in almost all practical link layer mechanisms [18, 21, 35, 38, 39]:
lim
!0f . /D0 (6)
Before approaching the general problem, we will first consider the single-AP system. LetM D1. The strategy set of each user i is S1i D Œ0; Pmax, where Pmax is the maximum power level allowed for each user. Utility function of userk, with power levelpk, is
u1kD Tk pk
(7) Tk is the long-term average rate as in Equation (4). Let 1 D ŒU ;fS1ig;fu1igbe the one-shot game in which each user unilaterally performs the following optimisation:
max pk2S1k
u1k.pk; Pk/8k2 U (8)
wherePk/stands for the vector of powers of all users except thekth.
An important property possessed by the utility functions ui that plays a key role in the existence and uniqueness of equilibrium is quasiconcavity (see [16] for
definition). It is observed and can be verified that, given Pk, ui.pk; Pk/ 8i are quasiconcave with respect to pk(Figure 3).
Givenpj,j ¤k,kchanges linearly withpk. Letting
Q
hkbe aneffective channel gainof userkand2the noise variance in its receiver, SINR expression in Equation (3) is
kD Qhkpk (9) Q hkD Gphk P i¤khipiC2 (10)
pk that optimises uk over the compact set Sk D Œ0; Pmax is such that either it is on the boundary or it satisfies
@uk @pk
D0 ; pk2Sk (11)
Proceeding by taking the derivative and using the linear-ity of SINR with transmit power, we find the best response rk.Pk/of userkas rk.Pk/Dmin ( Q hk ; Pmax ) (12)
is a unique positive solution of the following equation [40]:
f . /Df0. / (13)
The value ofdepends on the sigmoidal functionf . / such that the horizontal component of the intersection point in Figure 4 is strictly greater than the inflection point of the sigmoid [40]. Note that the shape of the sigmoidal function is determined by the modulation and coding scheme.
Utility
Power
Figure 3.Typical variation of utility function uk with power pk given other users’ powers. It is quasiconcave, monotone increasing up to some value of power and monotone decreasing afterward. Note that, depending onPmax, the decreasing regime
γ f’(γ) f(γ)
γ*
Figure 4. is a unique positive-valued solution of
Equation (13).
By definition, solution(s) of the fixed-point equations pk D rk.Pk/is (are) the NE(s). Consider the corre-sponding fixed-point iterationpk.t C1/ D rk.Pk.t // where the power of user i at iteration t is pi.t /. The iterations converge to the unique fixed-point iff NE is unique.
To investigate the convergence of the fixed-point itera-tions, it is useful to view the iterations as a power update algorithmI ./such thatp.tC1/DI .p.t //, wherep.t /D Œp1.t /; p2.t /; : : : ; pK.t /. In our problem, the explicit form of I ./ is such thatIi.p.t // D minf Opi.t /; Pmaxg, where O pi.t /D Pj¤ihjpj.t /C2 Gphi ; iD1; 2; : : : ; K (14) It is evident from the aforementioned expression that our I ./ satisfies the standard power update algorithm definition of Yates [41] and if algorithm I .O / with
O
Ii.p.t //D Opi.t /is a standard algorithm, thenIi.p.t //D minf Opi.t /; Pmaxghas a unique fixed point. Hence, we con-clude that our power update algorithmI ./has a unique fixed point; consequently,1has a unique NE.
In general, pk D rk.Pk/ 8k form a system of K nonlinear equations. In our particular problem in Equation (12), the nonlinearity of rk is due to clip-ping with Pmax. If Pmax is assumed sufficiently large, NE is a solution to the following system of K linear
equations [41]: hkpk
P
i¤khipiC2
D8kD1; 2; : : : ; K (15)
The aforementioned linear system may have a unique solu-tion, infinitely many solutions or no solution. If is feasible, then the system has a unique solution. The fea-sibility ofcan be determined using Perron–Frobenius theory [42]. By analysing the problem in terms of received powers, one can show that the feasibility condition is < Gp
K1. If this condition is not satisfied, power
update algorithmI .O /diverges, and for some of the users, pi D Pmax 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 sys-tem has been analysed, we turn our attention to the general multiple-AP system in the next section.
4.
THE POWER CONTROL GAME
Consider the general model withM APs (Figure 1). As before, users are subject to power constraintPmax. How-ever, now, they are allowed to transmit to more than one AP at a time. In other words, users can divide their power bud-get and transmit (different) data to different APs in order to (possibly) obtain a multiplexing gain.
In this case, the strategy set of a userkis
S2kD 8 < :Œpk1pk2 : : : pkM2R M C W M X jD1 pkj6Pmax 9 = ; (16) The utility function is as in Equation (5):
u2kD PM bD1Tkb PM bD1pkb (17)
Tkbis the long-term average rate of userkin APb. We will analyse 2 D ŒU ;fS2kg;fu2kg, and the corresponding user optimisation is as follows:
max pk2S2k
u2k.pk;Pk/ (18)
where pk D Œpk1; pk2; : : : ; pkM and Pk D Œp1; p2; : : : ;pk1;pkC1; : : : ;pK.
The main result of the paper is stated in the follow-ing theorem, which stipulates the special form of the best-response strategy in which each user transmits to a single AP.
Theorem 1. The utility maximising strategy of user k, pk, givenPkin game2is suchthat
pkbD pk; ifbDbk 0; otherwise (19) bkDarg max b n bhkb o (20) pkDmin 0 @Pmax; bhkb k 1 A (21) bhkbD Gphkb b2CPKiD1 i¤khi bPMjD1pij (22)
Proof. The proof relies on the results obtained for single-AP system. First, the set over which the optimisation is performed is extended toŒ0; PmaxM. The result for the optimum in single-AP system is used, and a component-wise summation yields the desired conclusion. The details
are given in Appendix A.
Theorem 1 suggests that each user should just trans-mit 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 [43] is notable. The aim of the resource allocation formulation in [43] is to maximise long-term average rate, given an average power budget. In contrast, in our formu-lation, the power budget is optimally divided among base stations to maximise the utility. Put in a different way, opti-misingenergy efficiencyrequires achieving a target SINR by choosing the best AP, whereas rate maximisation allocates all power resource to the best channel.
4.1. Equivalence with base station selection and power control game
In conclusion of Theorem 1, the problem reduces to well-known joint AP assignment and power control problem [36]. Therefore, the game2, 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 game3DŒU ;fS3kg;fu3kgfor which the strategy setS3kis
S3kDAP 8kD1; 2; : : : ; K (23)
whereAD fA1; A2; : : : ; AMgandP DŒ0; Pmax, withAi being theith AP. Letakandpkbe the AP assignment and transmit power of userk, respectively.kak Dbhkakpk
is the SINR of user k. Each user has the following utility function:
u3kDR f .kak/
pk
(24) The joint AP assignment and power control game3 was originally proposed in [36]. In order to find the best-response strategy, optimisation is performed in two stages [36]. 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 (24) for the chosen base station and given other power levels.
Note that the best-response strategies of2and3are equivalent. In [36],3is proved to have a unique NE. Sim-ilar to the single-AP problem, the existence proof is based on compactness, convexity ofŒ0; Pmaxand quasiconcav-ity off . /; the uniqueness is proved by direct verification that the best-response strategy defines a standard power update algorithm [36]. This way, the equilibrium of the
vector power control game2is characterised in terms of base station selection and (scalar) power control game3, which is emphasised once more in the following theorem. Theorem 2. 2and3have unique NEs. In the NE of 2, userkonly transmits to the AP that is assigned in the
NE of3withnonzero power and transmit powerpk in
the NE of2, and3are equal for allk.
An earlier work of Yates [5] posed a non-game theo-retic integrated power control and AP assignment problem. Although the formulation was not that of a game, it applies to the problem at hand. In aK user andM-AP system, minimum total transmit power vector is found under SINR constraintsi0where each user is assigned to only one AP.
i> 0
i8iD1; 2; : : : ; K (25)
Ifi0 are feasible, then there exists a unique solution for minimum total transmit power vector problem. Perron– Frobenius theory [42] is again deployed for analysing fea-sibility. 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 eigenvaluePF< 1for some assignment among MK possible assignments. In particular, for a single-AP system withKusers, the feasibility condition reduces to a simple inequality on the number of users and processing gain:< Gp
K1.
Setk0 D . Provided thatPmax is sufficiently large and feasibility is satisfied, the NE points of2and3are equivalent to the unique solution of minimum total transmit power problem withk0 D.
4.2. 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 picking which subcarriers to use to access which base station (and how much of their total power to allocate to it), are picking asubsetof interferers. It is important to note that the results in [21] and Theorem 1 straightfor-wardly 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 haveuniqueNE in this case. Because of the orthogonality among the carriers, mono-tonicity 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 [21], for some values of channel gains, multiple NEs may exist.
5. ENERGY AND
SPECTRAL EFFICIENCY
The analysis in Section 4 was based on a utility function that emphasises energy efficiency and has units ofbits per joule. Unilateral optimisation of utilities led users to reach a target SINR, which is the unique solution of the equa-tionf . /Df0. /. However, the spectrum resource is inefficiently used in casehas a low value. This observa-tion 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 powerp is assumed to be p˛. Then, the utility function of userkfor single-AP system is as follows:
ukD Tk
p˛k (26)
The priority exponent˛brings a variable degree of energy efficiency to the utility function. For˛D1, the utility func-tion in Equafunc-tion (7) is obtained.˛ < 1means that users value spectral efficiency more, whereas˛ > 1drives users to be more energy efficient.
The equilibrium SINR (as a function of the exponent˛) .˛/is theuniquesolution of
f . /D 1 ˛f
0. /
The variation of.˛/for different values of˛is shown in Figure 5.
A similar modification can be made to the utility func-tion for multiple-AP system. If the cost of communica-tion with APb is p˛kb for userk, then the total cost is
sigmoid α=1 α=1/2 α=3 α=1/4 γ*=G p/(N−1)
Energy Efficient Spectrally Efficient
Figure 5. is the intersection of two functions f. / and 1
˛f
0. /
. The horizontal axis of the intersection point is the equilibrium SINR,.˛/. For˛ <1, as the priority of spectral
efficiency is higher,takes higher values. It is not possible for
users to mutually reach
in case> Gp
K1.
the summation of the costs of communication to all APs:
PM
bD1p˛kb. Similar to the previous utility function, the modified utility represents total throughput per total cost as follows: ukD PM bD1Tkb PM bD1p˛kb (27)
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 cal-culation 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.
6.1. Iterative application of best response We will illustrate how NE is reached by iterative appli-cation of best-response strategy given in Theorem 1. In particular, assumingak.t /is the base station selection of userkat stept, power is updated topk.tC1/with the use of the following synchronous two-step algorithm:
(1) ak.tC1/Darg maxa2Aka.t /, and
(2) pk.tC1/Dmin ( Q hka k.tC1/.tC1/ ; Pmax ) .
We consider the uplink of a 30-user, four-AP direct-sequence CDMA wireless network deployed in a 4 km2 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. M Number of access points 4
K Number of users 30
R Bit rate 104bps
Gp Processing gain 50
2 AWGN power in receiver 51015W Modulation technique BPSK Pmax Maximum power 1 W
AWGN, additive white Gaussian noise; BPSK, binary phase shift keying.
0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 x (meters) y (meters) 1 2 4 3 4 4 4 4 4 4 4 4 1 1 1 1 1 2 2 2 2 4 4 1 2 3 3 3 3 3 3 3 3 2
Figure 6.Access point (AP) and user locations in the simulated area. Numbers in bold are attached to APs, and small-sized num-bers are attached to each user, indicating to which AP it communicates in the Nash equilibrium of game2. Note that each user
communicates to only one AP, although it is allowed to communicate with all APs.
as packets of length 1000 bits, and if the packets cannot be received (i.e. if an error is detected), then packets are retransmitted with an ARQ mechanism. Accordingly, the BER for communication between userkand APbfor an additive white Gaussian noise channel is
BERkbDQpkb
For this case, ARQ throughput expression is Tkb D R1Qpkb1000, where Q is the complementary error function. Considering the condition in Equation (6), the efficiency function is chosen as
f . /D1Qp1000 1
2
1000
For the previous f . /, the solution of the equation f . / D f0. /is calculated as D 6dB. Note that, if every user were to communicate to every AP, then fea-sibility condition would become< Gp
K1 D
50 29, which is not satisfied. However, if there are at most12users from which each AP receives data, then6-dB SINR is feasible. We know that each user communicates to only one AP as a best-response strategy. Hence, unless13or more users select the same AP, SINR in the NE is6dB.
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
Figure 6 on the border of two quadrants where a user is closer to AP 4 but communicates to AP 1.
For the simulated distribution of users, no 13 users select the same AP; therefore,D6dB is the SINR in NE. The evolution of power and SINR values in the iterative appli-cation of best-response strategy is observed in Figures 7 and 8, respectively. Note that, at each iteration, data are
1 2 3 4 5 6 7 8 9 10 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Iteration Power (W) 2 4 4 3 1
Figure 7.Evolution of transmit power of five arbitrarily selected users (users 1–5, 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 selec-tion is also fixed. Bold numbers attached at the end of each
1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 Iteration SIR (normal) γ=
Figure 8.Evolution of the observed signal-to-interference-plus-noise ratio (SINR) of five arbitrarily selected users from the simulation experiment. Each curve plots the SINR versus iter-ation number for one particular user applying the best-response
strategy. The common SINR at Nash equilibrium is
.
transmitted to different APs and overshoots are observed. As algorithm approaches to the equilibrium, the AP selec-tion is also fixed. Note that SINR of each user converges to.
6.2. 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˛, and as˛ is increased, thevalueof transmitting with less power is increased so that the targetis decreased. We will illus-trate the variation of target SINR with˛ in a described practical setting. In particular, the solution of the equation f . / D 1˛f0. /, 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 givenf . /, the calculation of the solution of the equation (which is guaranteed to be unique) has been performed numerically using Newton’s method with a pre-cision of0:01%. Two different functions that correspond to BPSK and noncoherent frequency shift keying (NC-FSK) modulation schemes are used, which respectively are as follows: fBPSK. /D1Q p 1000 1 2 1000 fncFSK. /D 11 2e 10001 2 1000
Resultant variation of with ˛ for efficiency func-tions of BPSK and NC-FSK is provided in Figure 9. It is observed that target SINRdecreases as˛is increased.
0 1 2 3 4 5 6 7 8 9 10 4 5 6 7 8 9 10 NC−FSK BPSK
Figure 9.Variation ofwith respect to the priority exponent
˛ for binary phase shift keying (BPSK)-modulated and nonco-herent frequency shift keying (NC-FSK)-modulated automatic
repeat request transmission with 1000-bit length packets.
0 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 NC − FSK BPSK 1
Figure 10.Normalised throughput at equilibrium versus˛for binary phase shift keying (BPSK) and noncoherent frequency
shift keying (NC-FSK).
Note thatfor 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, 3-dB-higher SINR is required in FSK than that in BPSK [44]. It is interesting to observe that about the same difference in the target SINRresults from the efficiency function of the modulation scheme.
Note that previous observation does not imply that FSK is more spectrally efficient than BPSK. Actually, equilib-rium throughput (assuming thatPmax is large and is reached) of FSK is lower than that of BPSK as shown in Figure 10. Because FSK has worse BER performance, the concave part of the sigmoid starts at a higher SINR, and that is whyof 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.
7. CONCLUSIONS
AND DISCUSSION
This paper studied vector power control in the uplink of a general wireless communication network with multiple APs. The problem has been formulated within a noncoop-erative game framework. 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 func-tion of throughput and a decreasing funcfunc-tion 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 moti-vation for our choice of utility was to obtain a tuneable degree of network energy efficiency and network spec-tral 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 differ-ent 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 unique-ness 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 equi-librium operating point is characterised by a target SINR , whose value is determined by the coding and modu-lation type. When the parameters of the problem instance such as maximum power levels and time average chan-nel gains deemfeasible, 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 util-ity 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 effi-ciency was verified. Numerical illustrations that exhibit how the iterative algorithms reach NE and show the vari-ation 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 network-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 networks where pricing is not natural, a simple distributed method of letting the network 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, Meshkatiet al.
[18–20], although the work in this paper has addressed vec-tor power control in a network with multiple choices of receiver stations.
We believe that the results motivate further work involv-ing an in-depth treatment includinvolv-ing 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 com-munications and distributed networks. The noncooperative game setup may be developed to include nodes that harvest energy from the environment. Then, not only the time aver-age 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 links to pricing and sug-gested that this approach is an alternative to it, we have not made this link precise. It was pointed out, for example, that using linear pricing results in an admission control mecha-nism. It would be interesting to study the links between the approach here and admission control in future work.
APPENDIX A: PROOF OF
THEOREM 1
It will be shown that f bhkb kp k pk > P bf .kb/ P bpkb 8Œpk1pk2 pkM2Sk (A1) Let W D Œ0; Pmax. Without loss of generality, assume k D 1(i.e. consider the first user) andbk D 1(i.e. for the first user, the maximumbhbkparameter is obtained with base station 1). Hence,
bh11>bh1b8b2 f1; 2; : : : ; Mg (A2)
There are two cases to consider: (1) p11 D
bh11
, (2) p11 DPmax
Assume the first case. Note that1bD1b.p11; p12; : : : ; p1M/ is a function of user 1’s power strategy vector Œp11 p12 p1Mgiven the otherpij. As forbD1, we have8p112W f./ p11 > f Œ11.p11; p12D0; p13D0; : : : ; p1M D0/ p11 (A3) Then, the inequality follows:
f ./ p11 > f Œ11.p11; p12; p13; : : : ; p1M/ p11 (A4) 8Œp11p12 p1M2S1
Considering forb D2, maximisation is at eitherp12 D Pmax or p12 D
bh12
. For the latter case, the proce-dure is similar to the previous one. In the former case, at p12DPmax, then f .12.p11D0; p12DPmax; p13D0; : : : ; p1M D0/ Pmax 6f . / p120 6 f ./ p11 (A5)
for somep120 >Pmaxsuch thatp0
12D
bh21
; then, it is obvi-ous thatp012>P
max>p11. Hence, it follows8p122W that f./ p11 > f Œ12.p11D0; p12; p13D0; : : : ; p1MD0/ p12 (A6) We immediately see that
f ./ p11 > f Œ12.p11; p12; p13; : : : ; p1M/ p12 (A7) 8Œp11p12 p1M2S1
By proceeding similarly for other base stations, the inequalities obtained in Equations (A4) and (A7) can be generalised as follows: f ./ p11 > f Œ1b.p11; p12; p13; : : : ; p1M/ p1b (A8) 8Œp11p12 p1M2WMand8b2 f1; 2; : : : ; Mg
Converting the inequalities to p1b
p11
>f Œ1b.p11; p12; p13; : : : ; p1M/
f ./ (A9)
8Œp11p12 p1M2S1
and summing overb, we obtain
P bp1b p11 > P bf Œ1b.p11; p12; p13; : : : ; p1M/ f ./ (A10) 8Œp11p12 p1M2S1
Converting once more, we obtain the desired result.
In the second case,p11 D Pmax. Using the maximal-ity assumption ofbh11, we observe thatp1bDPmax8b2
f1; 2; : : : ; Mg. f bh1bp1b p1b D f bh1bPmax Pmax 8b2 f1; 2; : : : ; Mg: (A11) Again, using the maximality of bh11, we reach Equation (A8). Hence, the desired result follows for the second case.
APPENDIX B: THE BEST RESPONSE
WITH PRIORITY EXPONENT
˛
LetW DŒ0; Pmax. AssumingbkD1and without loss of generality, lettingkD1, we claim that
f bh11p1 .p1/˛ > P bf .1b/ P bp˛1b 8Œp11p12 p1M2Sk (B1) where bh11>bh1b8b2 f1; 2; : : : ; Mg (B2)
Again, there are two cases to consider: (1) p11 D
bh11
, (2) p11 DPmax
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 forbD1, we have8p112W
f..˛// .p11/˛ >
f Œ11.p11; p12D0; p13D0; : : : ; p1M D0/ p11˛
(B3) Then, the inequality follows:
f ..˛// .p11/˛ >
f Œ11.p11; p12; p13; : : : ; p1M/
p11˛ (B4)
8Œp11p12 p1M2S1
Considering forb D 2, maximisation is at eitherp12 D Pmaxorp12D
.˛/
bh12
. Similar to the proof in Appendix A, we can show the following result:
f ..˛// .p11/˛
>f Œ1b.p11; p12; p13; : : : ; p1M/
p1b˛ (B5)
Converting the inequalities to p1b˛ .p11/˛ >f Œ1b.p11; p12; p13; : : : ; p1M/ f ..˛// (B6) 8Œp11p12 p1M2S1
and summing overb, we obtain
P bp˛1b .p11/˛ > P bf Œ1b.p11; p12; p13; : : : ; p1M/ f ..˛// (B7) 8Œp11p12 p1M2S1
Converting once more, we obtain the desired result.
ACKNOWLEDGEMENT
This work was supported by TUBITAK under grant 106E119.
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