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10.1109/TMC.2016.2622263, IEEE Transactions on Mobile Computing
EIRINI ELENI TSIROPOULOU ET AL.: SUPERMODULAR GAME-BASED DISTRIBUTED JOINT UPLINK POWER & RATE ALLOCATION IN TWO-TIER
FEMTOCELL NETWORKS 1
Supermodular Game-Based Distributed Joint Uplink Power
& Rate Allocation in Two-Tier Femtocell Networks
Eirini Eleni Tsiropoulou, Member IEEE, Panagiotis Vamvakas and Symeon Papavassiliou,
Senior Member IEEE
Abstract— This paper tackles the problem of joint users’
uplink transmission power and data rate allocation in
multi-service two-tier femtocell networks. Each user - either
macrocell (MUE) or femtocell user equipment (FUE) - is
associated with a two-variable utility function that
represents his perceived satisfaction with respect to his
allocated resources (i.e., power and rate). User’s utility
function is differentiated based both on the tier that the
user belongs to and the service he requests. The joint
resource allocation problem is directly confronted as a two-
variable optimization problem and formulated as a non-
cooperative game. The theory of supermodular games is
utilized towards treating the two-variable optimization
problem and the inherent multidimensional competition
that arises among the users. The existence of proposed
game’s Nash Equilibrium (NE) point is analytically shown,
while game’s convergence to its NE point is proven. A
distributed and iterative algorithm for computing the
desired NE is introduced, where the optimal values of each
user’s uplink transmission power and data rate are
simultaneously updated at the same step. The performance
of the proposed approach is evaluated via modeling and
simulation and its superiority compared to other state of
the art approaches is illustrated.
Index Terms—Two-tier femtocell networks; uplink
resource allocation; supermodular game; Nash equilibrium;
convergence.
1 INTRODUCTION
With the growing demand of mobile services for
high quality communications, small cell deployment
and optimal resource allocation have been recognized
among the key techniques to improve the capacity of
cellular wireless networks. Nowadays, more than
80% of data traffic and 65% of the phone calls take
place in indoor environments, thus femtocell technol-
ogy has emerged as a promising solution to guaran-
tee indoor coverage and users’ Quality of Service
(QoS) prerequisites satisfaction, while simultaneously
offloading macrocells [1].
Femtocells are usually deployed in indoor envi-
ronment (e.g., office, home etc.) by end users in a
plug and play fashion. They are characterized by
short range, i.e., 10-50m and consist of a Femtocell
Access Point (FAP) which is connected to service
provider’s internet network. They are cost-effective
low-power alternatives for wireless access, due to the
close transmitter-receiver proximity. Typically,
femtocells reside within the macrocell and share the
same frequency spectrum. The entire overlaid net-
work is referred to as two-tier femtocell network [2].
Within such an overall setting and in order to fully
exploit the benefits of the proposed two-tier architec-
ture, the problem of resource allocation (e.g., power,
rate, etc.) becomes of paramount importance to miti-
gate potential interference, while increasing system
capacity and satisfying users’ QoS requirements. In
this paper we treat exactly this problem in the uplink
of two-tier femtocell networks, focusing on the joint
power and rate allocation, in order to achieve signifi-
cant performance improvements and energy efficien-
cy.
1.1 UPLINK TRANSMISSION POWER & RATE ALLOCATION
IN WIRELESS NETWORKS
In recent literature, considerable research efforts
have been devoted to the problem of optimal re-
source allocation in the uplink of wireless networks
under different architectures and technologies (e.g.,
single-tier, two-tier, CDMA, OFDMA, SC-FDMA,
etc.), while special emphasis has been given on the
power control problems due to the inherent conven-
ience of single variable optimization problems. In [3],
the authors model the power control problem in a
single tier wireless network as a non-cooperative
game, where users act as individual players aiming at
maximizing selfishly their utilities and determining
their own transmission power in a distributed man-
ner. More energy-efficient power allocations are de-
termined in [4] and [5] via proposing a convex pric-
ing policy with respect to the user’s uplink transmis-
sion power and a linear pricing scheme as a function
of signal to interference ratio, respectively.
However, the joint problem of user’s uplink trans-
mission power and rate allocation in wireless net-
works has not yet been thoroughly examined in the
literature and especially under the paradigm of two-
tier femtocell networks. Some efforts have been de-
voted to this problem in multi-hop wireless networks.
A representative example is Heat Diffusion Protocol
————————————————
Eirini Eleni Tsiropoulou is with Erik Jonsson School of Engineering
and Computer Science, University of Texas at Dallas, Richardson,
75080, Texas, E-mail: eetsirop@utdallas.edu.
Panagiotis Vamvakas, Symeon Papavassiliou are with the School of
Electrical and Computer Engineering, National Technical Universi-
ty of Athens, Athens 15780, Greece. E-mail:
pvamvaka@central.ntua.gr, papavass@mail.ntua.gr.
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information.
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2 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID
[6], which provides a Pareto optimal resource alloca-
tion and packet latency in interference wireless net-
works. Moreover, in [7], a different derivative of Back
Pressure Routing Protocol has been proposed for joint
resource allocation and routing in time varying wire-
less networks. However, those approaches still re-
main to be investigated regarding their applicability
and extensibility in single-hop multi-tier wireless
networks. An initial approach to the joint power and
rate allocation problem in single tier wireless net-
works has been proposed in [8], where a specific form
of user’s utility function is adopted (i.e., the ratio of
the reliably transmitted bits to the base station divid-
ed by their corresponding transmission power) ena-
bling the authors to change the two variable optimi-
zation problem to a single variable problem. The lat-
ter goal is achieved via substituting the ratio of user’s
transmission rate to the corresponding transmission
power with a new variable and solve the correspond-
ing single variable optimization problem. Similar ap-
proach has been followed in [9] in two-tier femtocell
networks. However, though significant performance
improvements were obtained, the applicability of
such an approach is limited to special types of utility
functions, where the ratio of user’s transmission data
rate to power appears, in order to be able to amend
the two-variable optimization problem to a single-
variable one. In [10], the joint resource allocation has
been confronted as a two variable optimization prob-
lem in a single tier wireless network, where users’
optimal transmission power and rate are determined
simultaneously and independently. A thorough relat-
ed literature review is provided in Section 2.
In this paper, it is the first time that the joint users’
uplink transmission power and data rate allocation
problem in two tier femtocell networks is formulated
and solved as a two variable optimization problem.
The motivation and objective behind the proposed
approach is to determine users’ uplink transmission
power and rate in a distributed manner so as to max-
imize their perceived satisfaction. To better reflect the
latter, we design appropriate utility functions that
enable users to express their QoS demands via the
independent variables of power and rate, when con-
sidering not only multiple services but the tier of the
network that the user belongs to as well.
To tackle the inherent challenges stemming from
the joint two-variable consideration and the resulting
multidimensional competition among the users, the
theory of supermodular games is applied to the cor-
responding optimization problem and its Nash equi-
librium point is determined. The main novelty of our
work is that we provide two degrees of freedom to
the user, i.e., power and rate, enabling him to better
express his QoS prerequisites, achieve energy-
efficiency by obtaining high transmission rates with
low power consumption, while supporting both real
and non-real time services. Numerical results pre-
sented in the paper clearly indicate that the proposed
formulation, treatment and solution allows for more
energy efficient resource allocations that result in sig-
nificant performance improvements compared to the
current state of the art.
1.2 OUTLINE
The rest of this paper is organized as follows. In
Section 2, a brief overview of the related work in the
literature is provided, while in Section 3, we present
the system model and the proposed user’s two-
variable utility function. The joint Multi-service Up-
link transmission Power and data Rate Allocation
problem in two-tier Femtocell networks (MUPRAF
game) is formulated in Section 4.1, while its solution
based on the theory of supermodular games, i.e., NE
point, is provided in Section 4.2 and the convergence
of MUPRAF game to its NE point is shown in Section
4.3. In Section 5, a non-cooperative distributed low-
complexity and iterative algorithm is presented to
determine MUPRAF game’s solution, which is a vec-
tor of users’ uplink transmission power and data rate.
The performance of the proposed approach is evalu-
ated in detail through modeling and simulation in
Section 6, and its superiority compared to various
state of the art approaches is illustrated via detailed
comparative numerical results. Finally, Section 7 con-
cludes the paper.
2 RELATED WORK
The problem of optimal resource allocation in the
downlink of two-tier femtocell networks, where or-
thogonal frequency division multiple access (OFD-
MA) technology is adopted, has been thoroughly ex-
amined in the literature. The key problems that have
been studied target at interference mitigation [11],
[12], energy-efficient resource management [13], joint
subchannel and power allocation [14], and resource
allocation based on usage-pricing mechanisms [15],
and/or based on link-state propagation model [16].
However, all these approaches are centralized and
their goal is to guarantee system’s efficiency, e.g.,
maximization of system’s sum rate, system’s overall
interference mitigation etc.
On the other hand, some initial studies on the prob-
lem of resource allocation in the uplink of two-tier
femtocell networks have been proposed and mainly
focus on user’s uplink transmission power allocation
towards supporting user’s autonomicity and self-
optimization. The concept of utility function is adopt-
ed, which represents user’s satisfaction with respect
to the resource allocation. In [17], a common type of
utility function is adopted by all users, while in [18],
the users are organized in two classes, i.e., macro-
users (MUEs) and femto-users (FUEs), and based on
the class that the user belongs to, selects a utility
function which represents the tradeoff between the
reliably transmitted bits to the Base Station (BS) /
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/TMC.2016.2622263, IEEE Transactions on Mobile Computing
EIRINI ELENI TSIROPOULOU ET AL.: SUPERMODULAR GAME-BASED DISTRIBUTED JOINT UPLINK POWER & RATE ALLOCATION IN TWO-TIER
FEMTOCELL NETWORKS 3
FAP and the corresponding energy consumption.
Moreover, FUEs are penalized by a linear or convex
cost function [19], with respect to their uplink trans-
mission power, in order to give higher priority to the
satisfaction of MUEs’ QoS prerequisites. MUEs have
a priori worse channel conditions due to the fact that
they are more distant from the BS compared to the
FUEs from their corresponding FAP. The utility func-
tion is the objective function of a distributed power
allocation problem, where its solution is a stable us-
ers’ uplink transmission power vector achieving en-
ergy-efficiency and interference mitigation [17] – [19].
In [20], a distributed utility-based signal to interfer-
ence ratio (SIR) adaptation algorithm is proposed in
order to mitigate cross-tier interference at macrocell
from co-channel femtocells and concludes to users’
power allocation. In [21], the authors propose a
Stackelberg game-based spectrum allocation among
macro base stations and femto APs aiming at a dy-
namic optimal frequency allocation with minimum
interference to MUEs and FUEs.
Considering the joint resource allocation problem
in two tier femtocell networks, some initial approach-
es have been proposed that target mainly at system’s
efficiency and not at the maximization of each user’s
perceived satisfaction. In [22], the problem of sub-
channel, rate and power allocation in two-tier OFD-
MA femtocell networks using fractional frequency
reuse has been examined, towards maximizing only
FUEs throughput, while maintaining as little as pos-
sible reduction in the macrocell users’ performance.
In [23], the authors target at the maximization of the
sum users’ rate at each FAP, under the constraints of
cross-tier interference between macrocell and multi-
ple FAPs. In [24], a joint subchannel and power allo-
cation in two-tier OFDMA femtocell networks is pro-
posed. The formulated resource allocation problem
considers only the FUEs as candidate users in order
to determine the subchannels that are allocated to
them, as well as their transmission power towards
mitigating their co-channel and co-tier interference.
Furthermore, the subchannel and the power alloca-
tion problem is decomposed into two separate prob-
lems and solved. As a consequence, there is no guar-
antee that the obtained stable point is as efficient as
the one achieved if solving the actual joint subchan-
nel and power control problem where the two re-
sources are updated simultaneously at the same step.
Furthermore, in [25], the authors study the problem
of resource and power allocation in congestion cases
in which femtocell demands exceed the available re-
sources, while in [26], a distributed resource man-
agement framework that effectively manages inter-
ference across femtocells is proposed.
3 SYSTEM MODEL & UTILITY FUNCTION
3.1 SYSTEM MODEL
We consider the uplink of a two-tier femtocell wire-
less network, consisting of a macrocell base station
(BS) which serves a region
and covers an area of
radius
0
R
, providing multiple types of services to the
users. Furthermore, within the region
there are
deployed
F
co-channel femtocells. Each FAP sup-
ports a region
and consists of a disk of radi-
us
0c
RR
.
For simplicity in the following, cell
f
refers to the
cell served by one BS/FAP. Without loss of generali-
ty, the macrocell BS is indexed by
0
and FAPs
by
1,2,...,F
. Therefore, in the system a user i is associ-
ated (served) by base station Bi=0 if the user is MUE
or by FAP Bi=1,2,…,F, if the user is FUE. It is clarified
that at the beginning of each time slot, the number of
users per tier/cell is assumed to be known [27].
Though the cell selection problem itself is of high im-
portance it is not within the scope of this paper, while
several approaches exist in the literature to treat it
based on various criteria [27], e.g., maximum channel
gain/SINR/utility criteria. This assumption does not
restrict the applicability of the proposed approach
due to the fact that the cell selection component can
be easily included in our proposed framework prior
to the resource allocation component proposed in this
paper. Furthermore, it is noted that the FAPs consid-
ered in this paper in principle can be either of closed
access nature, i.e., the users should be registered to
the FAP in order to be served, or of open access na-
ture, where no user’s registration is needed. Never-
theless, this assumption can be more easily realized in
practice in the case of closed access femtocells.
We study a two-tier femtocell network, which sup-
ports users requesting either real-time (RT) services
or non-real time (NRT) services. Specifically, the
overall set of users is denoted by
S
and consists of
four subsets:
MRT
S
,
MNRT
S
,
FRT
S
and
FNRT
S
, represent-
ing macro (MUEs) and femto (FUEs) users, request-
ing real and non-real time services, respectively. The
corresponding numbers of macro and femto users,
requesting real and non-real time services are denot-
ed by
MRT
N
,
MNRT
N
,
FRT
N
and
FNRT
N
, respectively,
while the overall number of users residing in the two-
tier femtocell network is N, that is
N=
MRT
N
+
MNRT
N
+
FRT
N
+
FNRT
N
. Fig. 1 depicts a sam-
ple of users’ distribution within a two-tier femtocell
network, consisting of a macrocell base station and a
number of FAPs.
We assume an interference limited environment
where the available bandwidth (or a common band of
frequencies) is co-used and shared by MUEs and
FUEs for uplink communications. Let us denote by
Bi=l the BS/FAP of a user i under consideration.
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4 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID
Fig.1 Two-tier femtocell topology
The channel gain of the link between user
i
and
some BS/FAP
j
(j=0,1,2,..F) in the system is denoted
by
ij
h
. For simplicity these gains are modeled follow-
ing the simplified path loss model in the IMT-2000
specification [28], as follows:
i,0
i,
,0
2,
min ,1 ,
, 0,
min ,1 ,
min ,1 ,
min ,1 ,
a
b
c
a
ij j
a
i
a
ij
A d i is MUE, j=0
B R i is FUE, j j=l
h C D d i is MUE, j>0
A D d i is FUE, j=0
C D d i i
s FUE, j>0, j l
(1)
The exact values of the above described constants
have been adopted by [28] and are presented in Table
I. At each time slot, each user
,i i S
transmits with
an uplink transmission power
i
p
and an uplink
transmission data rate
i
r
, which are assumed to be
upper and lower bounded continuous variables, i.e.,
0Max
ii
pp
and
0Max
ii
rr
. The uplink transmis-
sion power and data rate vector of all users residing
in the two-tier femtocell network are denoted by
12
, ,..., N
p p pp
and
12
, ,..., N
r r rr
, respectively.
Therefore, the received signal-to-interference-plus-
noise ratio (SINR)
i
of user
i
at its serving BS/FAP
Bi=l (l=0,1,2,…F) is [3]:
2il i
ii kl k
ki
hp
W
r h p
(2)
where
W [Hz]
is the system’s available spread spec-
trum bandwidth and
2
denotes the Additive White
Gaussian Noise (AWGN) at the receiver, i.e., BS or
FAPs [3].
3.2 UTILITY FUNCTIONS & USERS’ QOS PREREQUISITES
To treat users’ diverse and multiple QoS prerequi-
sites under a common optimization framework, us-
ers’ utility functions are differentiated according to
the tier the user belongs to, as well as to the type of
requested service. It is noted that the overall pro-
posed approach and the utility function used here
allow for more general considerations compared to
other frameworks in the literature such as for exam-
ple the one in [9], where user’s perceived satisfaction
depended exclusively only on the ratio of the uplink
transmission data rate and power consumption.
TABLE I: PARAMETERS/NOTATIONS
PARAMETER
SYMBOL/VALUE
Macrocell BS
0
B
Femtocell AP
Bi=1,2,…,F
Macrocell’s region, radius
,
0
R
FAP’s region, radius
,
c
R
Number of FAPs
F
Set of MUEs and FUEs requesting RT and NRT services
MRT
S
,
MNRT
S
,
FRT
S
,
FNRT
S
Number of MUEs and FUEs requesting RT and NRT services
MRT
N
,
MNRT
N
,
FRT
N
,
FNRT
N
Channel gain between user
i
and BS/FAP
j
ij
h
Signal-to-interference-plus-noise ratio
i
User’s uplink transmission power
[W]
i
p
User’s uplink transmission rate
[ ]
i
r bps
System’s spread spectrum bandwidth
[Hz]W
Pricing factor
c
User’s strategy space
i i i
A P R
Macrocell and indoor femtocell path loss exponent
4a
Indoor to outdoor femtocell path loss exponent
3b
Distance of user
i
(if i is an MUE
0i
, otherwise
1i
) from BS/FAP
j
(if j is the BS:
0j
, otherwise if j is a FAP:
1j
)
,ij
d
Fixed decibel propagation loss during macrocell transmissions to the BS
28A dB
Fixed loss between FUE
i
to his corresponding FAP
ji
37B dB
Fixed path loss between FUE
i
to a different FAP
ji
.
37C dB
Partition loss during indoor to outdoor propagation
10D dB
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/TMC.2016.2622263, IEEE Transactions on Mobile Computing
EIRINI ELENI TSIROPOULOU ET AL.: SUPERMODULAR GAME-BASED DISTRIBUTED JOINT UPLINK POWER & RATE ALLOCATION IN TWO-TIER
FEMTOCELL NETWORKS 5
The latter was a strong assumption in the definition
of the utilities in [9] by design, in order to allow the
transformation of the two-variable optimization prob-
lem to a single-variable one.
In this work since we confront directly the two
variable (power and rate) optimization problem, we
remove this limitation and we assume a two-variable
utility function. Without loss of generality, in the fol-
lowing we consider the utility function presented in
(3), where it depends on one hand on the ratio of up-
link transmission rate to the uplink transmission
power in order to express energy efficiency, and on
the other hand depends directly on power
i
p
for the
case of FUEs. Other forms of the utility function can
be also treated under the general framework pro-
posed in this paper.
,
log 1 ,
( , )
1,
log 1 1,
pr
i
i
ii MRT
i
ii
MNRT
i
i
ii PFRT
i
ii
PFNRT
i
rf iS
p
rf
iS
p
Urf c e i S
p
rf
c e i S
p
(3)
where
i
f
denotes user’s efficiency function,
which represents the probability of successful packet
transmission. It is assumed increasing with respect to
user’s received SINR
i
and depends on the selected
transmission schemes (i.e., modulation and coding
schemes) [29]. Without loss of generality and for
demonstration purposes only, we adopt the following
form of efficiency function, which has been widely
utilized in the literature (e.g. [3], [29]).
1iM
A
i
fe
(4)
where
M
denotes the number of bits of transmitted
packets and
A
is a constant determining the slope of
the curve.
The physical meaning of the proposed formulation,
as detailed below, mainly stems from two fundamen-
tal observations: a) user’s (both MUE and FUE) per-
ceived satisfaction, i.e., utility, increases as his
achievable data rate increases - though in different
manner for real time and non-real time services - and
as his uplink transmission power decreases, the latter
due to the prolongation of his battery-life; and b)
MUEs should receive higher priority in being served
compared to FUEs. Specifically, in equation (3), with
reference to real time users, the achievable data rate
ii
rf
is assumed to be a sigmoidal-like function
with respect to
i
, where the inflection point is
mapped to real time user’s minimum QoS prerequi-
sites. Thus, if user’s SINR
i
results in achievable
data rate
ii
rf
lower than user’s desirable one,
then his utility decreases rapidly indicating user’s
starvation. Moreover, if user
,MRT FRT
i i S S
achieves
his desirable data rate then his utility is very slowly
increasing, thus indicating to the system that he has
already met his QoS prerequisites.
On the other hand, considering non-real time users’
(both MUEs and FUEs) QoS prerequisites, we express
the achievable data rate, i.e.,
log 1
ii
rf
via
adopting a log-based function with respect to user’s
efficiency function
i
f
. The proposed formulation
of achievable data rate properly represents non-real
time users’ greedy behavior, as they are characterized
by high-throughput (i.e., achievable data rate) expec-
tations. Their overall perceived satisfaction, i.e.,
( , )
i
Upr
, increases as they achieve higher SINR val-
ues, which in turn results in higher achievable data
rate via the strictly increasing function
log 1
ii
rf
with respect to
i
.
It should be noted that our framework does not lim-
it the applicability of the proposed approach to two
classes of services only (they are mainly used for
demonstration purposes). Each service class (i.e., RT
and NRT) could consist of other/many subclasses.
For example, considering a requested real-time ser-
vice, the proposed framework can support various
real-time services, i.e., voice, video, etc., which consist
the subclasses of real-time services, each one of them
with different QoS requirements. In this case different
applications can be considered and differentiated via
changing the slope of the sigmoidal function
ii
rf
(i.e., appropriately selecting A, M parameters) and
mapping the requested service minimum QoS re-
quirements to the appropriate inflection point.
Furthermore, FUEs are further penalized via a con-
vex pricing function with respect to their uplink
transmission power
i
p
, [9], [10], [19], [24], in order to
mitigate their caused interference within the overall
two-tier femtocell network. This way, higher priority
is given to the serving of MUEs, who a priori have
worst channel conditions due to the fact that in gen-
eral they are more distant from the base station com-
pared to the FUEs from the corresponding FAP. This
is of high importance as both FUEs and MUEs share
the same system’s available spectrum bandwidth in-
terfering with each other. The operator of the overall
network enforces the FAPs to announce the strict us-
age-based pricing policy to the FUEs being served by
them, in order the FAPs to be allowed to operate
within the overall network. Thus, each FAP announc-
es the proposed convex pricing policy to the served
FUEs.
Parameter c refers to the pricing factor as deter-
mined by the system and adopted by the users. This
parameter needs to be appropriately configured so
that, through user’s self-optimization, the best possi-
ble improvement in overall system’s performance is
achieved by forcing users to adopt a more social be-
havior. The determination of the appropriate value of
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pricing factor is discussed in Section 5. The convex
pricing policy is more realistic compared to other lin-
ear pricing schemes proposed in the literature [3],
due to the fact that the harm an FUE imposes on oth-
er users is not equivalent within the whole range of
transmission power. Previous attempts to consider
non-linear pricing are not directly applicable or ex-
tensible to our formulation, mainly due to the fact
that they were limited to treating cases where the ra-
tio of uplink transmission rate to uplink transmission
power appears in both users’ pure utility function
and their corresponding pricing function [5]. In our
work we aim at alleviating this limitation by consid-
ering a more general formulation with respect to both
utility and pricing functions, a fact that requires dif-
ferent treatment of the overall optimization problem.
4 A GAME THEORETIC APPROACH TO JOINT POWER &
RATE ALLOCATION
4.1 PROBLEM FORMULATION
As the system evolves, at the beginning of each time
slot, each user determines an appropriate uplink
transmission power and data rate level towards max-
imizing his overall perceived satisfaction, which is
appropriately represented via the corresponding val-
ues of the utility function. This is obtained as a result
of a user’s optimal joint utility-based Multi-service
Uplink transmission Power and data Rate Allocation
distributed algorithm in two-tier Femtocell wireless
networks (MUPRAF algorithm). In order to consider
users’ selfish and greedy behavior, game theory aris-
es as a powerful and appropriate tool [3] – [5] and [8]
– [10]. Specifically, the users act as players that com-
pete with each other and choose a strategy space of
uplink transmission power and data rate and subse-
quently achieve a payoff, which is represented by
their utility.
Let
,,
ii
i S i S
G S A U
denote the Multi-
service utility-based Uplink Power and data Rate Al-
location game in two-tier Femtocell wireless net-
works (MUPRAF game), where
MRT MNRT FRT FNRT
S S S S S
denotes the set of users
residing within the two-tier femtocell network,
i i i
A P R
is user’s
,i i S
strategy space, where
0, Max
ii
Pp
and
0, Max
ii
Rr
and
i
U
denotes user’s
payoff function. The strategy sets of each user, i.e.,
i
P
and
i
R
that consist his strategy space
i
A
are compact
and convex sets and as mentioned before are con-
strained with maximum and minimum values due to
user’s and system’s physical and technical limita-
tions.
The goal of each user is to maximize his utility via
selecting an appropriate strategy of uplink transmis-
sion power and data rate. Therefore, MUPRAF game,
is formulated as a distributed utility maximization
problem, as follows:
max , ,
. . 0 ,0
ii
ii
i
rR
pP
Max Max
i i i i
U i S
s t r r p p
p r
(5)
The solution of MUPRAF game should determine
the optimal equilibrium for the system, concluded by
the individual decisions of each user (either MUE or
FUE), given the decisions made by the rest of the us-
ers. A Nash equilibrium point of MUPRAF game is a
pair of vectors of users’ uplink transmission power
and data rate
,
**
pr
, i.e.,
*12
, ,..., T
N
p p p P p
12
... N
P P P
and
*12
, ,..., T
N
r r r R r
12
... N
R R R
, where the su-
perscript T denotes the transpose operation of a vec-
tor. More precisely, the Nash equilibrium point of the
presented two-variable non-cooperative MUPRAF
game can be defined as follows:
Definition 1: A pair of vectors
* * * * * * * *
1 2 1 2
, ,..., , , ,...,
NN
p p p r r rp ,r
in the strategy sets
*
ii
rR
and
*
ii
pP
, is a NE of the MUPRAF game if
for every user i the following condition holds true:
**
, , , , , ,
i i i i i i
U p r U p r
-i -i -i -i
p r p r
for all
ii
pP
and
ii
rR
.
The interpretation of NE is that no user has the in-
centive to change his strategy, due to the fact that he
cannot unilaterally improve his personal utility by
making any change to his own strategy, given the
strategies of the rest of the users. Moreover, based on
the above definition, it is concluded that the existence
of NE point guarantees a stable outcome of the
MUPRAF game, while on the contrary, the non-
existence of such an equilibrium point is translated to
an unstable and unsteady situation of the system.
4.2 PROBLEM SOLUTION VIA SUPERMODULAR GAMES
Towards proving the existence of at least one NE of
MUPRAF game, as mentioned before the theory of
supermodular games is adopted. Supermodular are
those games characterized by strategic complementa-
rities, i.e., when one player takes a more competitive
and aggressive action, then the rest of the users want
to follow the same behavior. Supermodular games
are of great interest as an optimization tool, due to
the fact that they encompass many applied models,
they tend to be analytically appealing and they have
the outstanding property that many solutions yield
the same predictions. More importantly, considering
multidimensional competition (as in MUPRAF game
due to the joint users’ uplink transmission power and
data rate allocation), supermodular games provide
another fertile ground for application, due to the fact
that they can readily and appropriately handle mul-
tidimensional strategy spaces. Considering a two di-
mensional game with strategy space
i i i
AX
, we
have the following definition [30].
Definition 2: The game
,,
i i i i iS
G S A X U
is smooth supermodular, if for all
iS
the following
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EIRINI ELENI TSIROPOULOU ET AL.: SUPERMODULAR GAME-BASED DISTRIBUTED JOINT UPLINK POWER & RATE ALLOCATION IN TWO-TIER
FEMTOCELL NETWORKS 7
conditions hold true:
A.
i i i
AX
is a compact cube in Euclidean
space.
B.
, , ,
i i i i i i i
U x y x X y Y
is twice continuously
differentiable:
1. supermodular in
,
ii
xy
for fixed
,
-i -i
xy
,
i.e.,
20
i
ii
U
xy
(6)
2. with increasing differences in
, , ,
ii
xy -i -i
xy
, i.e.,
20,
i
ij
Uji
xx
(7)
given that
20,
i
ij
Uji
xy
(8)
In a supermodular game, there always exist exter-
nal equilibria: a largest element
sup :a a A BR a a
and a smallest element
inf :a a A BR a a
of the equilibrium set,
where
BR
denotes player’s
,i i S
best response
strategy to other players’ strategies [29].
The MUPRAF game as formulated in Section 4.1 is
not a supermodular game in
,
i i i
A P R i S
, thus its
strategy space is appropriately modified, in order
conditions (6) - (8) to hold true. Therefore, in the fol-
lowing we examine separately and analytically each
condition, in order to determine the modified strate-
gy space
i
A
, where MUPRAF game is supermodular.
It should be noted that the original optimization
problem, as introduced in equation (5) is not modi-
fied, but we modify the strategy space where the po-
tential solution lies, so as to guarantee its existence.
Theorem 1: MUPRAF game’s
,,
ii
i S i S
G S A U
utility function
,
i
Upr
as
defined in (3) and (4) is supermodular in
,
ii
pr
for
fixed
,
-i -i
pr
, i.e.,
20
i
ii
U
pr
, if and only if
,
ln ,
i i k
M
A
(9)
where
, ,1 ,2
min ,
i k i i
,
,1 ,2
,
ii
are analytically de-
termined in the following proof.
Proof: See Appendix A.
Moreover, in the following theorem, we examine
the conditions (7) and (8), considering MUPRAF
game.
Theorem 2: MUPRAF game’s
,,
ii
i S i S
G S A U
utility function
,
i
Upr
as
defined in (3) and (4) has increasing differences in
, , ,
ii
pr -i -i
pr
, i.e., conditions:
20,
i
ij
Uji
pp
and
20,
i
ij
Uji
pr
hold true, if and only if
ln
iM
A
(10)
Proof: See Appendix B.
Combining Theorems 1 and 2, we easily conclude
the following.
Theorem 3: The MUPRAF game
,,
ii
i S i S
G S A U
is smooth supermodular in a
modified strategy space
i
A
, if for all
,i i S
,
,
ln ,
i i k
M
A
(11)
where
, ,1 ,2
min ,
i k i i
.
It should be noted that MUPRAF game is character-
ized by an exogenous parameter, i.e., the pricing fac-
tor, that cannot be controlled by the users. MUPRAF
game G with exogenous parameter
c
is said to
be supermodular, or it is a parameterized game with
complementarities, if
, , , ,
i i i
U p r
-i -i
pr
has increas-
ing differences in
,
i
p-i
p
and
,
i
p
, as well as in
,
i
r-i
r
and
,
i
r
for all
,i i S
.
Theorem 4: MUPRAF game with exogenous pa-
rameter
c
is a supermodular game.
Proof: See Appendix C.
Proving that MUPRAF game is supermodular in the
modified strategy space
ii
AA
, guarantees the ex-
istence of a non-empty set of Nash equilibria [30].
Therefore, the following holds:
Theorem 5: The modified MUPRAF game
,,
ii
iS
iS
G S A U
has at least one NE [31],
which is defined as:
**
,
, argmax ,
i i i
i i i
p r A
p r U
pr
(12)
It should be noted that Theorem 5 guarantees the ex-
istence of at least one NE point of the modified
MUPRAF game, while this point is not necessarily
unique. Practically, the best response equation in (12)
can be solved via multivariable calculus (i.e., second
partial derivative test) utilizing the Extreme Value
Theorem [33], and the most energy-efficient Nash
equilibrium (i.e., the NE characterized by less power
i
p
, while guaranteeing users’ QoS prerequisites’ sat-
isfaction, which is reflected by the value of uplink
transmission rate
i
r
) is adopted by each user. How-
ever, the NE point
,
**
pr
is a stable solution of opti-
mization problem (5) that maximizes each user’s utili-
ty function.
4.3 CONVERGENCE
In this section, the convergence of MUPRAF game
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to its NE point
,
ii
pr
is shown. Based on Theorem
5, user ’s best response strategy
,
i
BR -i -i
pr
to the
strategies
,
-i -i
pr
actually chosen by other players is
denoted as follows:
**
,
, , argmax , , ,
i i i
i i i i i i
p r A
BR p r U p r
-i -i -i -i
p r p r
(13)
Theorem 6: The modified MUPRAF game
,,
ii
iS
iS
G S A U
converges to its NE point
starting from any initial point.
Proof: By definition, the NE point of a non-
cooperative game has to satisfy:
**
,,
i i i
BR p r
-i -i
pr
.
Aiming at showing that the modified MUPRAF game
converges to its NE, we have to prove that user’s best
response strategy function is standard [32]. A func-
tion
f
is characterized as standard, if the following
conditions hold true:
i. Positivity:
( ) 0fx
;
ii. Monotonicity: if
'
xx
, then
( ) ( )ff'
xx
and
iii. Scalability: for all
1a
,
( ) ( )af f axx
,
for all
0x
, where
12
, ,..., N
x x xx
is a NE. Consid-
ering the modified MUPRAF game, we easily con-
clude that the above conditions hold true, as follows:
i.
()p,r 0
, thus
()BR p,r 0
, via relation (13),
ii. if
( ) ( )p,r p',r'
, then via (13) we conclude
that
( ) ( )BR p,r BR p',r'
,
iii. for all
1a
, since
()BR p,r
is a strictly in-
creasing function with respect to
()p,r
, then
we conclude that
( ) ( )a a aBR p,r BR p, r
. ■
5 RESOURCE ALLOCATION ALGORITHM
In this section we propose a distributed iterative
and low complexity algorithm, which determines the
NE point of MUPRAF game starting from any initial
feasible point. The algorithm is referred to as
MUPRAF algorithm and consists of two parts, i.e., the
Network Part and the User Part.
Considering the Network Part of MUPRAF algo-
rithm, it is implemented at each FAP and is responsi-
ble for obtaining the best pricing factor for FUEs. This
pricing factor is determined via an exhaustive search
following similar approaches proposed in the litera-
ture [3] – [5], [9]. Starting from an initial value of c,
e.g., c=0, an iterative approach is followed by increas-
ing c by Δc until an increase in c results in utility lev-
els worse than the previous equilibrium values for at
least one user. In other words, if the pure utility value
of at least one user (either macro or femto user) is
worse than the previous equilibrium utility, then the
Network Part of MUPRAF algorithm stops and sets
best
cc
. Following this process, it is noted that the
best pricing factor is determined via an exhaustive
search in the feasible interval of pricing factor’s val-
ues, i.e.,
0c
, however the processing time remains
low, due to the simplified calculations. Moreover, as
confirmed by the results presented later in Fig. 2, the
sum of equilibrium utilities for all users obtains its
highest value for
best
cc
.
On the other hand, the User Part is executed by each
user in the network (i.e., both macro and femto users)
in a distributed manner so that the NE point of
MUPRAF game is determined, i.e., each user deter-
mines its optimal uplink transmission power and rate
()p,r
.
MUPRAF Algorithm
A. Network Part
1
Each FAP announces the initial pricing factor
0c
to the FUEs residing in each cell.
2
for
1: ( )i N i S
3
Determine the NE point
**
,
ii
pr
(according to
User Part MUPRAF algorithm).
4
Compute the pure utility
**
,
i i i
U p r
(without the
penalty, i.e., cost function).
5
Increase the pricing factor
:c c c
, where
c
is a small positive constant.
6
Announce it to all FUEs.
7
If
,
ii
U c U c c i S
then go to step 2, else
stop and declare
best
cc
.
8
end.
B. User Part
1
Set ite=0, where ite denotes the iteration index.
Select a random feasible uplink transmission
power:
(ite 0)
i
p
and rate
(ite 0)
i
r
from the modified
strategy space
i
A
.
2
The BS/FAP broadcasts the overall users’ inter-
ference.
3
for
1: ( )i N i S
4
Compute the sensed interference by user i, as
( ) ( ) 2
()
ite ite
i i kl k
ki
I p h p
.
5
Refine
( 1)ite
i
p
and
( 1)ite
i
r
, in accordance to (12).
6
7
8
If
( 1) ( )ite ite
ii
pp
and
( 1) ( )ite ite
ii
rr
(
:
small positive constant), the powers and rates
have converged and stop.
end
otherwise, set ite:=ite+1 and return to step 2.
It should be noted that MUPRAF algorithm is execut-
ed in a distributed manner, due to the fact that the
decision making process lies on the user. Assuming
that the BS/FAP has perfect channel state infor-
mation of each user, the only information that is
broadcasted by the BS/FAP is the overall interference
(
( ) ( )
()
kk
Ip
), while each user calculates his sensed in-
terference (
( ) ( )
()
kk
ii
Ip
), excluding his contributing fac-
tor, i.e.,
ii i
hp
, from
( ) ( )
()
kk
Ip
. Moreover, the conver-
gence of MUPRAF algorithm, in terms of necessary
iterations, is thoroughly evaluated in the following
section, demonstrating the fast convergence property
of the algorithm.
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FEMTOCELL NETWORKS 9
6 NUMERICAL RESULTS
In this section we provide some numerical results
evaluating the operational features and performance
of the proposed MUPRAF algorithm. Initially, in Sec-
tion 6.1 we focus on the operation performance
achievements of MUPRAF algorithm, in terms of up-
link transmission power and achievable data rate
considering that all users, i.e., both MUEs and FUEs
request one type of service. Moreover, the fast con-
vergence of MUPRAF algorithm is shown and the
best pricing factor for FUEs is determined. Then, in
Section 6.2, we provide a comparative evaluation of
our proposed approach against other existing ap-
proaches in the literature with respect to several met-
rics and considering various scenarios. Specifically,
first in subsection 6.2.1 our proposed framework and
the framework presented in [9], which also aims at
jointly allocating users’ uplink transmission power
and data rate in two-tier femtocell networks, are
thoroughly compared in order to illustrate the ob-
tained gain in uplink transmission power and corre-
sponding energy-efficiency, when user’s service dif-
ferentiation is considered. Then, in Section 6.2.2, we
study cell’s capacity in terms of number of users that
meet their QoS prerequisites under increasing num-
ber of FAPs and users residing within the macrocell
for several approaches and scenarios. Specifically,
comparative results are presented that illustrate the
effectiveness of MUPRAF framework, in terms of de-
creased users’ uplink transmission power and in-
creased uplink transmission data rate compared to
other approaches proposed in the literature, i.e., [8] –
[10] (explained in brief below).
6.1 MUPRAF PROPERTIES AND OPERATION
In the following, we consider a two-tier femtocell
system supporting 20 backlogged MUEs. We assume
20 FAPs residing within the macrocell, where each
FAP serves 4 users. The considered topology sup-
ports a large set of users, i.e., 100 users in total. The
radius of the macrocell is set to
0290Rm
and of
each FAP
50
c
Rm
. FUE’s and MUE’s maximum up-
link transmission power is
0.2
Max
i
PW
, the adopted
efficiency function is
80
1.5
1i
i
fe
and the
spread spectrum bandwidth is
6
3.84 10W Hz
[3].
In this subsection, as mentioned before, we assume
that all users (FUEs and MUEs), request the same
type of service, i.e., target at the same uplink trans-
mission data rate, e.g., 128kbps (i.e., simple video up-
load), in order to better illustrate and comprehend the
operation characteristics and effectiveness of the
MUPRAF framework. A representative topology of
the scenario under consideration is the one presented
in Fig. 1.
Fig. 2 illustrates the sum of utilities at the equilib-
rium point
,
**
pr
as a function of the pricing factor
c. The diagram is constructed via MUPRAF algorithm
(Network Part) and estimates the “best” value of pric-
ing factor c, i.e.,
best
c
which is imposed by the system
to the users towards adopting a more social behavior
in terms of resources’ usage.
Fig. 3 illustrates users’ uplink transmission power
evolution as a function of the iterations required for
MUPRAF algorithm to converge at MUPRAF game’s
Nash equilibrium point
,
**
pr
at the time slot t un-
der consideration for the following indicative cases:
a) MUE closest to the BS, b) most distant MUE from
the BS, c) average n MUEs’ uplink transmission pow-
er and d) average FUEs’ uplink transmission power.
The results reveal that as MUEs channel conditions
become worse (i.e., distant users from the base sta-
tion) and in order to satisfy their QoS prerequisites
their uplink transmission power increases as ex-
pected.
Moreover, average FUEs’ uplink transmission
power is considerably lower compared to the corre-
sponding average value of MUEs, due to FUEs’ close
proximity to the femtocell AP and the convex pricing
policy that encourages FUEs to utilize more efficient-
ly their power and thus causing less interference to
the MUEs. Finally, the overall MUPRAF framework
is characterized as energy-efficient, due to the fact
that both FUEs and especially MUEs transmit with
low uplink transmission power, i.e., they do not ex-
haust their maximum uplink transmission power
0.2
Max
i
PW
, except for the edge MUEs due to their
increased distance from the BS.
Similarly, Fig. 4 presents the corresponding uplink
transmission data rate at the Nash equilibrium point
,
**
pr
of MUPRAF game, as a function of the itera-
tions that MUPRAF algorithm needs to converge to
the stable solution, for the same cases (i.e., a) MUE
closest to the BS, b) most distant MUE from the BS, c)
average MUEs’ transmission rate, and d) average
FUEs’ transmission rate). Based on the results, we
observe that the majority of MUEs even though they
suffer from worse channel conditions compared to
FUEs, achieve high uplink transmission data rate,
thus fulfilling their QoS prerequisites. Only the more
distant users from the base station (i.e., edge users)
achieve lower transmission data rate, due to their cor-
responding extremely bad channel conditions. Based
on Fig. 4, we observe that the average uplink trans-
mission data rate of FUEs is close to the target value,
i.e., 128 Kbps, while the corresponding average value
for MUEs is 120 Kbps. The percentage of satisfied
users, i.e., both MUEs and FUEs is 96%, while the
corresponding percentage value considering only the
MUEs is 90%.
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Fig. 2 Sum of equilibrium utilities as a function of the pric-
ing factor c.
Fig. 3 MUEs and FUEs uplink transmission power conver-
gence at Nash equilibrium (y-axis in logarithmic scale).
Fig. 4 MUEs and FUEs uplink transmission data rate
convergence at Nash equilibrium under MUPRAF algo-
rithm.
Furthermore, based on the results presented in Fig.
3 and Fig. 4, we observe that the convergence of the
proposed algorithm is very fast since less than thirty
iterations are required for reaching equilibrium for all
users, starting from randomly selected initial values,
while for all practical purposes we can observe that in
less than fifteen iterations the values of the powers, as
well as rates have very closely approximated their
corresponding equilibrium values. MUPRAF algo-
rithm – User Part as described in Section 5, is respon-
sible to determine users’ optimal uplink transmission
power and rate. Its distributed nature ensures the
scalability of the proposed framework, in the sense
that algorithm’s complexity does not directly depend
on the number of users in the overall network.
MUPRAF algorithm was tested and evaluated in an
Intel (R) Core (TM) 2 DUO CPU T7500 @ 2.20GHz
laptop with 2.00 GBytes available RAM and its
runtime was 0.4msec. The necessary time in order to
converge to the Nash equilibrium point is of similar
order of magnitude with the duration of a timeslot
(0.5msec), thus it can be easily adopted in a realistic
scenario. It should be also noted that the convergence
time of MUPRAF algorithm can further decrease, if
we adopt a more “educated” implementation of the
algorithm, i.e., utilize as initial values of the powers
and rates the corresponding values that were calcu-
lated in the previous time slot.
6.2 COMPARATIVE EVALUATION
6.2.1 SERVICE DIFFERENTIATION AND ENERGY-
EFFICIENCY
In this subsection, we provide some numerical re-
sults illustrating the performance of MUPRAF when
service differentiation is considered. For comparison
purposes, a simplified topology is considered consist-
ing of 10 MUEs and 8 FAPs, where each FAP serves 4
FUEs. Two types of services are considered, i.e., real-
time and non-real time service. The target rate for the
real-time users is
arg 128
t et
r Kbps
. Two different scenar-
ios are compared: (i) MUPRAF framework and (ii) the
proposed combined uplink transmission power and
data rate allocation in multi-service two-tier femtocell
networks, as it has been presented in [9]. It is noted
here that the two-variable problem in [9], as already
mentioned in Section 1 is amended to a single-
variable problem via substituting the ratio of user’s
uplink transmission data rate to transmission power
with a new single variable.
Fig. 5 illustrates MUEs uplink transmission power
as a function of their position within the macrocell,
considering the two comparative scenarios. Further-
more, comparing MUPRAF algorithm with the
framework of joint resource allocation in multi-
service two-tier femtocell networks, as presented in
[9], we observe that MUPRAF achieves an average of
56.70% reduction of users’ uplink transmission pow-
er. This benefit stems from (i) formulating better us-
ers’ QoS prerequisites in a two-variable utility func-
tion, (ii) solving the joint resource allocation problem
as a two-variable optimization problem, where user’s
uplink transmission power and data rate are deter-
mined independently and updated at the same step,
and (iii) imposing convex pricing policy with respect
to FUE’s uplink transmission power.
In Fig. 6 we present a comparative study of FUEs
mean uplink transmission power within each FAP
residing within the macrocell. The results reveal the
superiority of MUPRAF algorithm in terms of power
savings. Specifically, comparing the results of
MUPRAF algorithm with service differentiation to
the corresponding results of the proposed framework
in [9], we observe that MUPRAF framework achieves
on average 85.67% reduction with respect to power
consumption. It should be noted that these great im-
provements in power savings are obtained, while at
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/TMC.2016.2622263, IEEE Transactions on Mobile Computing
EIRINI ELENI TSIROPOULOU ET AL.: SUPERMODULAR GAME-BASED DISTRIBUTED JOINT UPLINK POWER & RATE ALLOCATION IN TWO-TIER
FEMTOCELL NETWORKS 11
Fig. 5 MUEs’ uplink transmission power as a function of
their distance from the base station (y axis in logarithmic
scale).
Fig. 6 FUEs’ mean uplink transmission power as a function
of the FAP’s ID.
Fig. 7 MUEs’ energy efficiency as a function of their dis-
tance from the base station (y-axis in logarithmic scale).
Fig. 8 FUEs’ mean energy efficiency as a function of FAP’s
ID (y-axis in logarithmic scale).
the same time similar performance with respect to the
transmission rate is achieved for the FUEs and even
better rates are obtained for the MUEs under the
MUPRAF framework.
The combined effect of power and rate improve-
ment is depicted in Fig. 7 and Fig. 8, where we pre-
sent MUEs’ energy efficiency [bits/Joule] as a function
of their distance from the base station and FUEs’
mean energy efficiency within each FAP residing in
the macrocell, respectively. Based on these results we
clearly confirm that MUPRAF algorithm outperforms
the approach in [9] in terms of energy efficiency as
well.
Fig. 9 Users’ mean uplink transmission power as a function
of the number of users.
Fig. 10 Users’ mean uplink transmission data rate as a
function of the number of users.
6.2.2 INCREASING NUMBER OF USERS AND FAPS
In the following, we present a comparative study il-
lustrating the benefits of the proposed MUPRAF al-
gorithm as the number of MUEs and FAPs residing
within the macrocell increases (thus respectively the
number of FUEs). Specifically, we compare four dif-
ferent approaches: (i) MUPRAF algorithm, (ii) the
algorithm presented in [9], (iii) the algorithm in [8],
which determines the power and transmission data
rate allocation in an one-tier macrocell wireless net-
work via solving the problem as a single variable op-
timization problem and (iv) the algorithm in [10],
which determines a joint multi-service power and
data rate allocation also in a single one-tier macrocell.
More specifically, in [8], the combined problem of
power and transmission data rate allocation is con-
sidered as a single-variable problem of the ratio of
uplink transmission data rate to the uplink transmis-
sion power in a single macrocell topology and both
power and transmission data rate are updated at the
same time. Moreover, in [10], the joint resource allo-
cation in a single macrocell topology has been con-
fronted as a two variable problem and users’ optimal
transmission power and transmission data rate are
determined simultaneously and independently.
Specifically, Fig. 9 and Fig. 10 present users’ mean
uplink transmission power and data rate as a func-
tion of the overall number of users in the cell, respec-
tively. The specific formation of users, i.e., number of
MUEs and FUEs, within the macrocell is presented in
Table II, along with the percentage reduction in mean
uplink transmission power and increase of
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information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
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12 IEEE TRANSACTIONS ON MOBILE COMPUTING, MANUSCRIPT ID
TABLE II: PERCENTAGE REDUCTION IN MEAN UPLINK TRANSMISSION POWER & INCREASE OF MEAN UPLINK TRANSMISSION DATA RATE UN-
DER FOUR COMPARATIVE SCENARIOS
Number of Users
Comparison to MUPRAF algorithm without service differentiation
% reduction in uplink transmission
power
% increase of mean uplink transmis-
sion data rate
Only
MUEs [8],
[10]
MUEs
FUEs
[8]
[10]
[9]
[8]
[10]
[9]
MUPRAF
Algorithm, [9]
16
8
8
96.69 %
95.72 %
91.80 %
410.02 %
117.44 %
9.62 %
32
8
24
96.84 %
96.54 %
91.98 %
929.02 %
569.62 %
9.56 %
48
12
36
82.36 %
80.79 %
60.68 %
1287.00 %
543.56 %
7.33 %
60
12
48
82.33 %
80.91 %
62.46 %
1660.90 %
547.39 %
7.71 %
72
12
60
82.02 %
80.89 %
66.60 %
2034.20 %
637.43 %
8.87 %
mean uplink transmission data rate achieved by the
proposed MUPRAF algorithm, when compared to [8]
– [10], respectively. The results reveal the superiority
of the MUPRAF algorithm in terms of both power
saving and increase of rate. Finally, it should be noted
that the consideration of two-tier femtocell architec-
ture is the key reason why both our framework and
[9] significantly outperform [8] and [10] that consider
single-tier macrocell architectures.
7 CONCLUDING REMARKS
In this paper the problem of joint power and rate
allocation in the uplink of two-tier femtocell networks
supporting multiple services is addressed. Each user
is associated with a two-variable utility function,
which represents his perceived satisfaction with re-
spect to the achieved uplink transmission rate and the
corresponding power consumption and is differenti-
ated based on the tier that the user belongs to (i.e.,
MUE or FUE) and the service he requests (i.e., real
and non-real time service). The joint resource alloca-
tion problem is confronted as a two-variable optimi-
zation problem and is formulated as a non-
cooperative game (MUPRAF game) that is solved via
adopting the theory of supermodular games. The ex-
istence of MUPRAF game’s Nash equilibrium has
been proven and the convergence of MUPRAF game
to its Nash equilibrium has been demonstrated. The
superiority of formulating and solving the joint re-
source allocation problem as a two-variable optimiza-
tion problem has been thoroughly evaluated via
analysis and simulation.
It is noted here that problem was treated as a two-
variable optimization one, under the assumption that
each user has already been associated with a specific
cell. This can be easily determined based on several
parameters such as the maximum achieved channel
gain or SINR or utility. Such an assumption does not
restrict the applicability of our proposed framework,
due to the fact that the cell selection component can
be easily included within our proposed framework,
in a collaborative and even iterative manner. Fur-
thermore, full consideration of the actual joint as-
signment problem (i.e., cell selection, power alloca-
tion and rate assignment) as a multi-variable optimi-
zation problem though would make the allocation
problem even more complex and pose several inter-
esting challenges, is of high research interest.
The introduced joint resource allocation approach
provides us with an enhanced flexible framework
which can be used to facilitate the uniform treatment
and analysis of different access technologies, i.e., de-
vice-to-device (D2D) or machine-to-machine (M2M)
communication, which both are part of the 5G wire-
less networks and are characterized by the nature of
two-tier architecture. Finally, it is noted that the pro-
posed approach facilitates the creation of a more flex-
ible and general framework, where the control intel-
ligence and the decision making process lie at the
mobile user, which enables the realization of mobile
node’s self-optimization functionalities.
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Eirini Eleni Tsiropoulou is a
postdoctoral researcher at Erik
Jonsson School of Engineering
and Computer Science, Uni-
versity of Texas at Dallas. She
obtained her Diploma, MBA in
Techno-economics and PhD
Degree in Electrical and Com-
puter Engineering from Na-
tional Technical University of Athens in 2008, 2010
and 2014 respectively. Two of her papers received the
Best Paper Award at IEEE Wireless Communications
and Networking Conference (WCNC 2012) in April
2012 and at the 7th International Conference on Ad
Hoc Networks (ADHOCNETS 2015) in September
2015. Her main research interests include wireless
and heterogeneous networking focusing on power
and resource allocation schemes, smart data pricing,
Internet of Things and dense wireless networks archi-
tectures.
Panagiotis Vamvakas received
his diploma Degree in Electrical
and Computer Engineering from
the National Technical Universi-
ty of Athens (NTUA), Greece
and an M.Sc. Degree in Man-
agement, Technology and Eco-
nomics from ETH Zurich, Swit-
zerland, in 2011 and 2015, re-
spectively. His main scientific
interests lie in the area of resource allocation in wire-
less networks, energy efficient and heterogeneous
networks, and smart data pricing. Currently he is a
research associate at the School of Electrical and
Computer Engineering, National Technical Universi-
ty of Athens.
Symeon Papavassiliou received
the diploma in Electrical Engi-
neering from the National
Technical University of Athens
(NTUA), Greece, in 1990 and
the MSc and PhD degrees in
electrical engineering from
Polytechnic University, Brooklyn, New York, in 1992
and 1995, respectively. Currently he is a professor in
the School of Electrical and Computer Engineering at
NTUA. From 1995 to 1999, he was a senior technical
staff member at AT&T Laboratories, New Jersey. In
August 1999 he joined the Electrical and Computer
Engineering Department at the New Jersey Institute
of Technology, USA, where he was an associate pro-
fessor until 2004. Dr. Papavassiliou was the Director
of the Broadband, Mobile and Wireless Networking
Laboratory (2000-2004) at the New Jersey Institute of
Technology, USA and a founding member and Asso-
ciate Director of the New Jersey Center for Wireless
Networking and Internet security (2002-2004, New
Jersey, USA). He has an established record of publi-
cations in his field of expertise, with more than 250
technical journal and conference published papers.
He received the Best Paper Award in IEEE INFO-
COM’94, the AT&T Division Recognition and
Achievement Award in 1997, the US National Science
Foundation Career Award in 2003, the Best Paper
Award in IEEE WCNC 2012, the Excellence in Re-
search Grant in Greece in 2012 and the Best Paper
Award in ADHOCNETS 2015. Dr. Papavassiliou also
served on the board of the Greek National Regulatory
Authority on Telecommunications and Posts (2006–
2009). His main research interests lie in the area of
communication networks, with emphasis on the
analysis, optimization and performance evaluation of
mobile and distributed systems, wireless networks
and complex systems.
