TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIES
Trans. Emerging Tel. Tech. (2012)
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 Ozel1 and Elif Uysal-Biyikoglu2*
1 Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA
2 Department 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 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 © 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 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.
Copyright © 2012 John Wiley & Sons, Ltd.
In principle, a network 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 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 communication [10–12] and to obtain distributed algorithms. In fact,
communication networks form an increasingly popular setting 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 the utility of their communication with
their intended receiver. Depending on the constraints and
O. Ozel and E. Uysal-Biyikoglu
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
the network 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 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 work in [15], in settings such as interference networks 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 [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 control 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 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 work 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 3, the
expression and properties of the utility function are provided and the power control game in a single-AP system
is analysed. In Section 4, 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.
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 optimum’ 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 admission 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 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 [21].
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 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 theoretic as well as game theoretic distributed power control 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 stations and distributed base stations in an LTE* network. As
* LTE stands for ‘Long Term Evolution’, a recent cellular communication standard.
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
O. Ozel and E. Uysal-Biyikoglu
in some of the previously mentioned works, overall network 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 [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 of K users and M APs
(Figure 1). 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 hib > 0. Channel gains are assumed constant during
operation. More generally, the hib ’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 hib 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 Xlj . According to this model, the
signal Yb , received at base station b is
Yb D
K X
M p
X
hlb Xlj C Zb
(1)
lD1 j D1
User k
hkb
Access Point b
Figure 1. Wireless network with several users and access
points.
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
Zb is additive noise at AP b. For convenience, we model
Zb as white Gaussian with zero mean and EjZb j2 D 2 .
Pij is the average power of the message Xij : EjXij j2 D
Pij . Each user is subject to a power constraint:
M
X
Pij 6 Pmax 8i
(2)
j D1
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
[29] with specific spreading codes for each possible link.
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, 30, 31]). Gp stands for processing gain. The signalto-interference-plus-noise ratio (SINR) of user k in the
receiver of AP b is
kb D Gp
2
C
hkb Pkb
K PM
iD1
j D1 hib Pij hkb Pkb
P
(3)
The model with multiple APs has been motivated by a
number of communication scenarios: (i) a local area network 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 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
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 multiplexing 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 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 [34]. 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 (3). Hence, users are in such an interaction that
O. Ozel and E. Uysal-Biyikoglu
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 cancellation [20,35]. In order to analyse this interaction among
users, we will employ static noncooperative game theory.
A static game D ŒU ; fSi g; fui g is defined using three
components [9]:
(1) user set U ;
(2) action or strategy set Si , 8i 2 U ; and
(3) utility ui as a function of elements of Si , 8i 2 U .
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 a Nash equilibrium.
This captures the noncooperative nature of the problem.
Definition 1. An NE is the vector of strategies sE D
such that
Œs1 ; s2 ; s3 ; : : : ; sK
/ > ui .si ; sEi
/ 8si 2 Si
ui .si ; sEi
is satisfied for all user i where sEi D .s1 ; s2 ; : : : ; si1 ;
siC1 ; : : : ; sK /.
Note that an NE may not be socially optimal, that is,
there may be a point with utilities u0i that is feasible and yet
u0i > ui 8i , where ui 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, 16]. However,
in our setup, users are selfish and are not directly interested
in the overall performance of the network; each user optimises its own utility in its own action space. Hence, we
assume noncooperative operation.
Given actions of users other than k, sEk , the best
response (in other words, the reaction curve) of user k,
rk , is
QK
real numbers, ui W
j D1 Sj ! R. The value of the
function ui 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-known utility function [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 transmission) 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 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 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 length L symbols are sent using
binary phase shift keying (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.1 BER. //L
1
rk .Esk / , arg max uk
Concave
sk 2Sk
Nash equilibrium can also be defined in terms of best
/ 8k. In other words,
responses. sE is NE iff sk D rk .Esk
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 operating points through examining the existence and properties
of NE.
3. UTILITY FUNCTION AND THE
SINGLE-ACCESS POINT SYSTEM
In game theoretic terms, utility function ui is a mapping
from the Cartesian product of action sets Sj of users to
(4)
Inflection Point
Convex
0
Figure 2. Function f ./ versus in normal scale. It is plotted for binary phase shift keying modulation with packet length
L D 400 bits.
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
O. Ozel and E. Uysal-Biyikoglu
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 in Œ0; / and concave in . ; 1/. Note that
Equation (4) 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 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 modulation and rate are adapted to changes in SINR. Consistent 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.
Let Tkb and pkb be the throughput and power of user k
for communication with AP b, respectively. The utility
function uk is defined as the ratio of total throughput to
total dissipated power:
PM
bD1 Tkb
uk D PM
bD1 pkb
lim f . / D 0
k D hQ k pk
hQ k D P
(6)
(9)
Gp hk
2
i¤k hi pi C (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
D 0 ; p k 2 Sk
@pk
(11)
Proceeding by taking the derivative and using the linearity of SINR with transmit power, we find the best response
rk .Pk / of user k as
(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 transmisP
sion, it is reasonable to have uk ! 0 when M
bD1 pkb D 0
for all k. This will automatically hold when the throughput function tends to zero as SINR vanishes, which is
the case in almost all practical link layer mechanisms
[18, 21, 35, 38, 39]:
!0
definition). It is observed and can be verified that, given
Pk , ui .pk ; Pk / 8i are quasiconcave with respect to
pk (Figure 3).
Given pj , j ¤ k, k changes linearly with pk . Letting
hQ k be an effective channel gain of user k and 2 the noise
variance in its receiver, SINR expression in Equation (3) is
(
; Pmax
rk .Pk / D min
hQ k
)
(12)
is a unique positive solution of the following
equation [40]:
f . / D f 0 . /
(13)
The value of depends on the sigmoidal function f . /
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.
u1k D
Tk
pk
(7)
Utility
Before approaching the general problem, we will first
consider the single-AP system. Let M D 1. 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 user k, with power level pk , is
Tk is the long-term average rate as in Equation (4). Let
1 D ŒU ; fS1i g; fu1i g be the one-shot game in which
each user unilaterally performs the following optimisation:
max u1k .pk ; Pk / 8k 2 U
pk 2S1k
(8)
where Pk / stands for the vector of powers of all users
except the kth.
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
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
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 on Pmax , the decreasing regime
may not be observed, but this does not violate quasiconcavity.
O. Ozel and E. Uysal-Biyikoglu
γ f’(γ)
update algorithm IO./ 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 system has been analysed, we turn our attention to the general
multiple-AP system in the next section.
f(γ)
4. THE POWER CONTROL GAME
γ*
Figure
4. is
a
unique positive-valued
Equation (13).
solution
of
By definition, solution(s) of the fixed-point equations
pk D rk .Pk / is (are) the NE(s). Consider the corresponding fixed-point iteration pk .t C 1/ 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 iterations, it is useful to view the iterations as a power update
algorithm I ./ such that p.t C 1/ D I .p.t //, where p.t / D
Œp1 .t /; p2 .t /; : : : ; pK .t /. In our problem, the explicit
form of I ./ is such that Ii .p.t // D minfpOi .t /; Pmax g,
where
P
j ¤i hj pj .t / C 2
; i D 1; 2; : : : ; K
pOi .t / D
Gp hi
(14)
It is evident from the aforementioned expression that
our I ./ satisfies the standard power update algorithm
definition of Yates [41] and if algorithm IO./ with
IOi .p.t // D pOi .t / is a standard algorithm, then Ii .p.t // D
minfpOi .t /; Pmax g has a unique fixed point. Hence, we conclude that our power update algorithm I ./ has a unique
fixed point; consequently, 1 has 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 clipping with Pmax . If Pmax is assumed sufficiently large,
NE is a solution to the following system of K linear
equations [41]:
hk p k
P
D 8k D 1; 2; : : : ; K
2
h
p
C
i
i
i¤k
Consider the general model with M APs (Figure 1). As
before, users are subject to power constraint Pmax . 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 k is
8
9
M
<
=
X
M
pkj 6 Pmax
S2k D Œpk1 pk2 : : : pkM 2 RC W
:
;
j D1
(16)
The utility function is as in Equation (5):
PM
bD1 Tkb
u2k D PM
bD1 pkb
Tkb is the long-term average rate of user k in AP b. We will
analyse 2 D ŒU ; fS2k g; fu2k g, and the corresponding
user optimisation is as follows:
max u2k .pk ; Pk /
(18)
pk 2S2k
where pk D Œpk1 ; pk2 ; : : : ; pkM and Pk D Œp1 ;
p2 ; : : : ; pk1 ; pkC1 ; : : : ; pK .
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 1. The utility maximising strategy of user k,
pk , given Pk in game 2 is such that
D
pkb
pk ;
0;
if b D bk
otherwise
n
o
bk D arg max b
hkb
(19)
(20)
b
(15)
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 [42]. By analysing the problem in terms of received
powers, one can show that the feasibility condition is
Gp
. If this condition is not satisfied, power
< K1
(17)
0
pk
1
A
D min @Pmax ;
b
h (21)
kbk
b
hkb D
b2 C
PK
Gp hkb
iD1 i¤k
hib
PM
j D1 pij
(22)
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
O. Ozel and E. Uysal-Biyikoglu
Proof . The proof relies on the results obtained for singleAP system. First, the set over which the optimisation is
performed is extended to Œ0; Pmax 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 1 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 [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 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.
4.1. Equivalence with base station
selection and power control game
In conclusion of Theorem 1, the problem reduces to wellknown joint AP assignment and power control problem
[36]. Therefore, the game 2 , 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 3 D ŒU ; fS3k g; fu3k g for
which the strategy set S3k is
S3k D A P 8k D 1; 2; : : : ; K
(23)
where A D fA1 ; A2 ; : : : ; AM g and P D Œ0; Pmax , with Ai
being the i th AP. Let ak and pk be the AP assignment and
hkak pk
transmit power of user k, respectively. kak D b
is the SINR of user k. Each user has the following
utility function:
u3k D R
f .kak /
pk
(24)
The joint AP assignment and power control game 3
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 of 2 and 3 are
equivalent. In [36], 3 is proved to have a unique NE. Similar to the single-AP problem, the existence proof is based
on compactness, convexity of Œ0; Pmax and quasiconcavity of f . /; 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
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
vector power control game 2 is characterised in terms of
base station selection and (scalar) power control game 3 ,
which is emphasised once more in the following theorem.
Theorem 2. 2 and 3 have unique NEs. In the NE of
2 , user k only transmits to the AP that is assigned in the
NE of 3 with nonzero power and transmit power pk in
the NE of 2 , and 3 are equal for all k.
An earlier work 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 i0 where each user is assigned to only one AP.
0
i > i 8i D 1; 2; : : : ; K
(25)
0
If i are feasible, then there exists a unique solution for
minimum total transmit power vector problem. Perron–
Frobenius theory [42] 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 < 1 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
Gp
.
gain: < K1
0
Set k D . Provided that Pmax is sufficiently large
and feasibility is satisfied, the NE points of 2 and 3 are
equivalent to the unique solution of minimum total transmit
power problem with k0 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 a subset of interferers. It is important
to note that the results in [21] and Theorem 1 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 [21], for some values of channel gains,
multiple NEs may exist.
O. Ozel and E. Uysal-Biyikoglu
5. ENERGY AND
SPECTRAL EFFICIENCY
The analysis in Section 4 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
a target SINR , 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 ˛ > 0 so that the
cost of transmitting with power p is assumed to be p ˛ .
Then, the utility function of user k for single-AP system is
as follows:
uk D
Tk
pk˛
(26)
The priority exponent ˛ brings a variable degree of energy
efficiency to the utility function. For ˛ D 1, the utility function in Equation (7) is obtained. ˛ < 1 means that users
value spectral efficiency more, whereas ˛ > 1 drives users
to be more energy efficient.
The equilibrium SINR (as a function of the exponent ˛)
.˛/ is the unique solution of
f . / D
the
of the costs of communication to all APs:
PMsummation
˛
bD1 pkb . Similar to the previous utility function, the
modified utility represents total throughput per total cost
as follows:
PM
bD1 Tkb
uk D PM
˛
bD1 pkb
(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 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.
6.1. Iterative application of best response
1 0
f . /
˛
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 communica˛ for user k, then the total cost is
tion with AP b is pkb
We will illustrate how NE is reached by iterative application of best-response strategy given in Theorem 1. In
particular, assuming ak .t / is the base station selection of
user k at step t , power is updated to pk .t C 1/ with the use
of the following synchronous two-step algorithm:
(1) ak .t C 1/ D arg maxa2A ka .t /, and
(
(2) pk .t C 1/ D min
hQ ka .tC1/ .tC1/
)
; Pmax .
k
*
γ =G /(N−1)
p
sigmoid
α=1
α=1/2
α=3
α=1/4
We consider the uplink of a 30-user, four-AP directsequence 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.
Energy Efficient
Spectrally Efficient
Figure 5. is the intersection of two functions f ./ and
1 0
f ./. 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
Gp
users to mutually reach in case > K 1
.
M
K
R
Gp
2
Pmax
Number of access points
Number of users
Bit rate
Processing gain
AWGN power in receiver
Modulation technique
Maximum power
4
30
104 bps
50
5 1015 W
BPSK
1W
AWGN, additive white Gaussian noise; BPSK,
binary phase shift keying.
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
O. Ozel and E. Uysal-Biyikoglu
2000
3
4
1800
3
4
4
1600
4
1400
y (meters)
3
4
4
3
3
3
4
3
4
1200
4
4
3
3
4
1
1000
2
1
2
800
2
2
2
2
600
1
400
2
1
1
1
1
200
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
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 2 . 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 user k and AP b for an
additive white Gaussian noise channel is
BERkb D Q
p
kb
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, 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 Tkb D
1000
p
R 1 Q kb
, where Q is the complementary
error function. Considering the condition in Equation (6),
the efficiency function is chosen as
0.018
2
0.016
0.014
p 1000
f . / D 1 Q 1000
1
2
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 feaGp
D 50
sibility condition would become < K1
29 , which
is not satisfied. However, if there are at most 12 users from
which each AP receives data, then 6-dB SINR is feasible.
We know that each user communicates to only one AP as
a best-response strategy. Hence, unless 13 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
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
Power (W)
4
0.012
0.01
0.008
4
0.006
1
0.004
3
0.002
1
2
3
4
5
6
Iteration
7
8
9
10
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 selection is also fixed. Bold numbers attached at the end of each
curve indicates the equilibrium AP selection.
O. Ozel and E. Uysal-Biyikoglu
14
10
NC−FSK
BPSK
12
9
SIR (normal)
10
8
8
7
6
6
γ =4
5
2
4
0
1
2
3
4
5
Iteration
6
7
8
0
1
2
3
4
5
6
7
8
9
10
9
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 .
Figure 9. Variation of with respect to the priority exponent
˛ for binary phase shift keying (BPSK)-modulated and noncoherent frequency shift keying (NC-FSK)-modulated automatic
repeat request transmission with 1000-bit length packets.
1
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 .
0.9
NC − FSK
BPSK
0.8
0.7
0.6
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, 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 ˛1 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 0:01%. Two different functions that correspond to
BPSK and noncoherent frequency shift keying (NC-FSK)
modulation schemes are used, which respectively are
as follows:
p 1000
fBPSK . / D 1 Q 1000
1
2
1000 1000
1
1
fncFSK . / D 1 e 2
2
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.
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
Figure 10. Normalised throughput at equilibrium versus ˛ for
binary phase shift keying (BPSK) and noncoherent frequency
shift keying (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, 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 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 Pmax 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 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.
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
O. Ozel and E. Uysal-Biyikoglu
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 noncooperative 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 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 network energy efficiency and network 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
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
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
is possible. This result is consistent with the related energy
efficiency-related results of, for example, Meshkati et al.
[18–20], although the work in this paper has addressed vector power control in a network with multiple choices of
receiver stations.
We believe that the results motivate further work 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 networks. 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 links to pricing and suggested 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 mechanism. 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
P
f b
hkb pk
k
b f .kb /
P
>
8Œpk1 pk2 pkM 2 Sk
pk
b pkb
(A1)
Let W D Œ0; Pmax . Without loss of generality, assume
k D 1 (i.e. consider the first user) and bk D 1 (i.e. for
the first user, the maximum b
hbk parameter is obtained with
base station 1). Hence,
b
h1b 8b 2 f1; 2; : : : ; M g
h11 > b
(A2)
There are two cases to consider:
,
b
h11
(2) p11 D Pmax
D
(1) p11
Assume the first case. Note that 1b D 1b .p11 ; p12 ; : : : ;
p1M / is a function of user 1’s power strategy vector
Œp11 p12 p1M given the other pij . As for b D 1, we
have 8p11 2 W
f. / f Œ11 .p11 ; p12 D 0; p13 D 0; : : : ; p1M D 0/
>
p11
p11
(A3)
Then, the inequality follows:
f . / f Œ11 .p11 ; p12 ; p13 ; : : : ; p1M /
>
p11
p11
8Œp11 p12 p1M 2 S1
(A4)
O. Ozel and E. Uysal-Biyikoglu
Considering for b D 2, maximisation is at either p12 D
Pmax or p12 D . For the latter case, the proceb
h12
dure is similar to the previous one. In the former case, at
p12 D Pmax , then
f .12 .p11 D 0; p12 D Pmax ; p13 D 0; : : : ; p1M D 0/
Pmax
6
f b
h1b p1b
D
p1b
f b
h1b Pmax
Pmax
8b 2 f1; 2; : : : ; M g:
(A5)
(A11)
Again, using the maximality of b
h11 , we reach
Equation (A8). Hence, the desired result follows for the
second case.
; then, it is obvi-
APPENDIX B: THE BEST RESPONSE
WITH PRIORITY EXPONENT ˛
f . / f . /
6
0
p12
p11
0 >P
0
for some p12
max such that p12 D
D P
In the second case, p11
max . Using the maximal DP
ity assumption of b
h11 , we observe that p1b
max 8b 2
f1; 2; : : : ; M g.
b
h21
0 >P
ous that p12
max > p11 . Hence, it follows 8p12 2 W
that
f. / f Œ12 .p11 D 0; p12 ; p13 D 0; : : : ; p1M D 0/
>
p11
p12
(A6)
We immediately see that
f . / f Œ12 .p11 ; p12 ; p13 ; : : : ; p1M /
>
p11
p12
Let W D Œ0; Pmax . Assuming bk D 1 and without loss of
generality, letting k D 1, we claim that
f b
h11 p1
.p1 /˛
>
P
b f .1b /
P
8Œp11 p12 p1M 2 Sk
˛
b p1b
(B1)
where
(A7)
b
h1b 8b 2 f1; 2; : : : ; M g
h11 > b
(B2)
Again, there are two cases to consider:
8Œp11 p12 p1M 2 S1
By proceeding similarly for other base stations, the
inequalities obtained in Equations (A4) and (A7) can be
generalised as follows:
f . / f Œ1b .p11 ; p12 ; p13 ; : : : ; p1M /
>
p11
p1b
(A8)
8Œp11 p12 p1M 2 W M and 8b 2 f1; 2; : : : ; M g
,
b
h11
DP
(2) p11
max
D
(1) p11
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 1, we have 8p11 2 W
f. .˛// f Œ11 .p11 ; p12 D 0; p13 D 0; : : : ; p1M D 0/
/˛ >
˛
.p11
p11
(B3)
Then, the inequality follows:
Converting the inequalities to
f Œ1b .p11 ; p12 ; p13 ; : : : ; p1M /
p1b
>
p11
f . /
(A9)
8Œp11 p12 p1M 2 S1
and summing over b, we obtain
P
b
p1b
p11
P
>
b
f Œ1b .p11 ; p12 ; p13 ; : : : ; p1M /
f . /
(A10)
f . .˛// f Œ11 .p11 ; p12 ; p13 ; : : : ; p1M /
/˛ >
˛
.p11
p11
(B4)
8Œp11 p12 p1M 2 S1
Considering for b D 2, maximisation is at either p12 D
Pmax or p12 D .˛/ . Similar to the proof in Appendix A,
b
h12
we can show the following result:
f . .˛// f Œ1b .p11 ; p12 ; p13 ; : : : ; p1M /
/˛ >
˛
.p11
p1b
(B5)
8Œp11 p12 p1M 2 S1
Converting once more, we obtain the desired result.
8Œp11 p12 p1M 2 W M and 8b 2 f1; 2; : : : ; M g
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
O. Ozel and E. Uysal-Biyikoglu
Converting the inequalities to
˛
p1b
/˛
.p11
>
f Œ1b .p11 ; p12 ; p13 ; : : : ; p1M /
f . .˛//
(B6)
8Œp11 p12 p1M 2 S1
and summing over b, we obtain
P
˛
b p1b
/˛
.p11
P
>
b
f Œ1b .p11 ; p12 ; p13 ; : : : ; p1M /
f . .˛//
(B7)
8Œp11 p12 p1M 2 S1
Converting once more, we obtain the desired result.
ACKNOWLEDGEMENT
This work was supported by TUBITAK under grant
106E119.
REFERENCES
1. Verdu S. On channel capacity per unit cost. IEEE Transactions on Information Theory 1990; 36(5): 1019–1030.
2. Uysal-Biyikoglu E. Bounds and observations on the
transmission of data packets with minimum energy.
IEEE Communications Letters 2006; 10: 707–709.
3. Foschini GJ, Miljanic Z. A simple distributed
autonomous power control algorithm and its convergence. IEEE Transactions on Vehicular Technology
1993; 42: 641–646.
4. Zander J. Distributed cochannel interference control for
cellular radio systems. IEEE Transactions on Vehicular
Technology 1992; 41: 305–311.
5. Yates RD, Huang CY. Integrated power control and base
station assignment. IEEE Transactions on Vehicular
Technology 1995; 44: 638–644.
6. Goodman DJ, Mandayam NB. Power control for wireless data. IEEE Personal Communications Magazine
2000; 7: 48–54.
7. Xiao M, Shroff NB, Chong EKP. A utility-based power
control scheme in cellular wireless systems. IEEE/ACM
Transactions on Networking 2003; 11: 210–221.
8. Huang J, Berry R, Honig M. Distributed interference
compensation for wireless networks. IEEE Journal
on Selected Areas in Communications 2006; 24:
1074–1084.
9. Fudenberg D, Tirole J. Game Theory. The MIT Press:
Cambridge, MA, 1991.
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett
10. Letaief KB, Mallik RK, Shi Y, Wang J. A gametheoretic approach for distributed power control in interference relay channels. IEEE Transactions on Wireless
Communications 2009; 8: 3151–3161.
11. Musku M, Chronopoulos A, Popescu D, Stefanescu A.
A game-theoretic approach to joint rate and
power control for uplink CDMA communications.
IEEE Transactions on Communications 2010; 58:
923–932.
12. St. Jean C, Jabbari B. On game-theoretic power control under successive interference cancellation. IEEE
Transactions on Wireless Communications 2009; 8:
1655–1657.
13. Ozel O, Uysal-Biyikoglu E. Distributed power control
using non-monotonic reaction curves, In Proc. 2009
International Conference on Game Theory for Networks,
Istanbul, May 2009; 60–67.
14. Tsuo F-Y, Lee W-L, Wang C-Y, Wei H-Y. Power
control game with sinr-pricing in variable-demand
wireless data networks, In Vehicular Technology
Conference (VTC 2010-Spring), 2010 IEEE 71st, Taipei,
May 2010; 1–5.
15. Teneketzis D, Sharma S. An externalities-based
decentralized optimal power allocation algorithm
for wireless networks. IEEE/ACM Transactions on
Networking 2009; 17(9): 1819–1831.
16. Saraydar CU, Mandayan NB, Goodman DJ. Efficient
power control via pricing in wireless data networks.
IEEE Transactions on Communications 2002; 50:
291–303.
17. Alpcan T, Basar T, Srikant R, Altman E. Power control
in CDMA as a non-cooperative game. Kluwer Academic
Publishers, Wireless Networks 2002; 8: 659–670.
18. Meshkati F, Poor HV, Schwartz SC, Mandayam NB. An
energy-efficient approach to power control and receiver
design in wireless data networks. IEEE Transactions on
Communications 2005; 53: 1185–1194.
19. Meshkati F, Goldsmith AJ, Poor HV, Schwartz SC. A
game-theoretic approach to energy-efficient modulation
in CDMA networks with delay QoS constraints. IEEE
Journal on Selected Areas in Communications 2007; 25:
1069–1078.
20. Meshkati F, Guo D, Poor HV, Schwartz SC. A unified
approach to power control in large energy-constrained
CDMA systems. IEEE Transactions on Wireless Communications 2008; 7: 1208–1216.
21. Meshkati F, Chiang M, Poor HV, Schwartz SC. A
game-theoretic approach to energy-efficient power
control in multicarrier CDMA systems. IEEE Journal
on Selected Areas in Communications 2006; 24:
1115–1129.
22. Rasti M. Pareto and energy efficient distributed power
control with feasibility check in wireless networks.
O. Ozel and E. Uysal-Biyikoglu
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
IEEE Transactions on Information Theory 2011; 75:
245–255.
Leung KK. An opportunistic power control algorithm
for cellular networks. IEEE/ACM Transactions on Networking 2006; 14(3): 470–478.
Chen CS, Shum KW, Sung CW. Round-robin power
control for the weighted sum rate maximisation of
wireless networks over multiple interfering links.
European Transactions on Telecommunications 2011:
1–13.
Hanly S, Inaltekin H. Optimality of binary powercontrol in a single cell via majorization, In Proc. 2011
IEEE International Symposium on Information Theory,
St Petersburg, Russia, 2011; 2891–2895.
Schmeink A. On fair rate adaption in interferencelimited systems. European Transactions on Telecommunications 2011; 22(5): 200–210.
Dawy Z, Yaacoub E. Uplink scheduling in LTE systems
using distributed base stations. European Transactions
on Telecommunications 2010; 21: 532–543.
Ertug O. Spectral and power efficiency trade-offs for
multiple-antenna reception techniques under co-channel
interference. European Transactions on Telecommunications 2011; 22(7): 415–427.
Tse D, Viswanath P. Fundamentals of Wireless Communications, (1st edn). Cambridge University Press:
New York, 2005.
Polyzos GC, Douros VG. Review of some fundamental approaches for power control in wireless
networks. Computer Communications 2011; 34(13):
1580–1592.
Sundaresan R, Singh BC, Kumar A. Uplink power control and base station association in multichannel cellular
networks, In Proc. of GameNets, International Symposium on Game Theory for Networks, Istanbul, Turkey,
May 2009.
Hanly S. Capacity and power control in spread spectrum macrodiversity radio networks. IEEE Transactions
on Communications 1996; 44: 247–256.
Pfletschinger S, Bader F. OFDMA with subcarrier
sharing. In Multi-Carrier Spread Spectrum, Fazel K,
Kaiser S (eds). Springer: Netherlands, 2006; 319–326.
10.1007/1-4020-4437-23 4.
34. Thomas D, Costa E. An allocation strategy to
avoid crossed-slot interference in time division
duplex. IEEE 16th International Symposium on
Personal, Indoor and Mobile Radio Communications, 2005. PIMRC 2005 2005; 2: 878–882. DOI:
10.1109/PIMRC.2005.1651568.
35. Buzzi S, Poor H. Noncooperative games for spreading
code optimization, power control and receiver design
in wireless data networks. (Available from: http://www.
arxiv.org).
36. Saraydar CU, Mandayam NB, Goodman DJ. Pricing and
power control in a multicell wireless data network. IEEE
Journal on Selected Areas in Communications 2001;
19(10): 1883–1892. DOI: 10.1109/49.957304.
37. Lal D, Manjeshwar A, Herrmann F, Uysal-Biyikoglu E,
Keshavarzian A. Measurement and characterization
of link quality metrics in energy constrained wireless sensor networks. In IEEE Globecom 03, Vol. 1:
San Francisco, CA, 2003; 446–452.
38. Keshavarzian A, Uysal-Biyikoglu E, Lal D,
Chintalapudi K. From experience with indoor wireless
networks: a link quality metric that captures channel
memory. IEEE Communications Letters 2007; 11:
729–731.
39. Rodriguez V. Robust modeling and analysis for wireless
data resource management, In Proc. 2003 IEEE Wireless
Communication and Networking Conference (WCNC),
New Orleans, LA, March 2003; 717–722.
40. Rodriguez V. An analytical foundation for resource
management in wireless communication, In Proceedings
of the IEEE Global Telecommunications Conference
(GLOBECOM), San Francisco, CA, USA, December
2003; 898–902.
41. Yates RD. A framework for uplink power control in
cellular radio systems. IEEE Journal on Selected Areas
in Communications 1995; 13: 1341–1347.
42. Senata E. Non-Negative Matrices and Markov Chain.
Springer-Verlag: New York, 1981.
43. Tse D. Optimal power allocation over parallel Gaussian
broadcast channels. (Available from: http://degas.eecs.
berkeley.edu/dtse/pub.html).
44. Proakis JG. Digital Communications, (4th edn).
McGraw-Hill: New York, NY, 2000.
Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.
DOI: 10.1002/ett