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Energy Efficient Resource Allocation in Two-Tier
OFDMA Networks with QoS Guarantees
Meysam Masoudi ·Hamidreza Zaefarani ·
Abbas Mohammadi ·Cicek Cavdar
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Abstract In this paper, we study joint power and subchannel allocation prob-
lem for OFDMA based femtocell networks with focus on uplink direction. We
minimize the aggregate power of all Femto user equipments and maximize the
total system energy efficiency while satisfying the minimum required rate of
all users. An interference limit constraint is considered to protect the QoS of
macrocells. The original problem is a mixed-integer non-convex optimization
problem which is converted to a convex problem using the time-sharing con-
cept. Three algorithms are proposed to provide a scheme to optimize the goal
function while meeting the constraints. The complexity order of all algorithms
was investigated and was compared to other alternative solutions. The ana-
lytic and simulation results have demonstrated that the proposed algorithms
could achieve significant power saving and better energy efficiency compared
to existing algorithms.
Keywords Femtocell ·Resource Allocation ·Power Minimizations ·Energy
Efficiency ·Convex Optimization
1 Introduction
Growing demand for high quality services in wireless cellular networks is one
of the most challenging issues to be met in near future. Furthermore, the study
in [1] shows that indoor usage comprises approximately 50% of all voice calls
and about 70% of data traffic. Consequently, operators are willing to pro-
vide high data rate services to the indoor users. This characteristic is vital
M. Masoudi, H.R. Zaefarani, A. Mohammadi
Microwave and Wireless Communication Research Laboratory,
Electrical Engineering Department, Amirkabir University of Technology (Tehran Polytech-
nic), Iran E-mail: meysammasoudi@aut.ac.ir; h-r-z@aut.ac.ir ; abm125@aut.ac.ir
C. Cavdar
Wireless@KTH, KTH Royal Institute of Technology, Sweden E-mail: Cavdar@kth.se
2 Meysam Masoudi et al.
in next generation networks such as LTE-A and probably 5G [2]. A promis-
ing solution to meet these requirements is to employ small cells overlaying a
macrocell [3]. In particular, femtocells as a heterogeneous network solution,
have received much more attention than other solutions from both academia
and industry [4]. Femtocell base stations (FBSs) are low power, short range
and low cost access points which are deployed by the end users. FBSs utilize
the infrastructures such as DSL, Cable Modem or separate RF as a back-
haul to connect to the operator primary network [5]. In addition, femtocells
can boost network performance in terms of high data rate services due to the
close proximity of femtocell user equipment (FUEs) to their indoor installed
FBS [6], [7]. Although femtocells are beneficial to both operators and users,
many technical challenges associated with femtocell deployment arise such as
high interference level in the network, radio resource utilization, fairness and
complexity [8]. Moreover, energy saving methods attract the attention of re-
searchers for next generation wireless networks such as 4G and 5G [9].
1.1 Literature Review
The resource allocation problem in orthogonal frequency-division multiple ac-
cess (OFDMA) cellular networks has been studied in the literature to deal with
the challenges and to enhance the network performance. Subchannel allocation
and power control are promising ways of dealing with an interference problem.
Especially co-channel spectrum allocation is a preferable spectrum utilization
from the industry standpoint due to its efficient spectrum utilization [10].
However, the subchannel assignment and power allocation are complicated
optimization problems [11]. Challenges of radio resource management for fem-
tocell networks are discussed in [4]. In [12] the authors proposed a scheme
to reduce the downlink interference while maintaining the QoS of users. The
authors in [13] proposed a distributed energy efficient resource management
algorithm using game theory to mitigate the excessive interference to achieve
a better throughput. Also in [14], a joint subchannel and power allocation al-
gorithm for the downlink of an OFDMA network deployment is proposed to
maximize the sum rate of all FUEs while guaranteeing macrocell user equip-
ment (MUEs) to reach their minimum QoS requirement. A novel spectrum
partitioning scheme using directional antennas is proposed in [15] to reduce
the interference.
To address the power allocation strategies, in [16] the authors proposed a
game theoretic approach for power allocation. The performance is evaluated
based on average femtocell and total throughput. Energy efficient power al-
location models are also proposed in [17] and performance of the femtocell
network is analyzed in terms of the blocking probabilities with partially open
channels. In [18], the authors proposed an algorithm to maximize the mini-
mum data rate of FUEs while MUEs must reach their required rate. Heuristic
approaches are adopted in [19] and [20]. In [19] a multiobjective approach us-
ing genetic algorithm is adopted. The authors in [20] proposed a generalized
Title Suppressed Due to Excessive Length 3
particle swarm optimization model for adaptive resource allocation for single
layer and cross layer optimizations. Power control strategies to manage min-
imum outage probability for femtocells are considered in [21]. In [22], based
on the Lagrangian dual decomposition method, the authors derived a subop-
timal resource allocation policy for femtocells in uplink direction. The authors
in [23] derived a closed-form relationship between two conflicting parameters,
spectral efficiency and energy efficiency.
Energy efficient resource allocation guarantees that maximum throughput
is achieved with minimum energy consumption. The studies in [9], [24–27]
analyzed their system from energy efficiency perspective. In [9], the authors
proposed two energy efficient algorithms but they do not consider neither
guaranteeing the QoS of macrocell users nor the channel allocation. Also they
consider only downlink power control. However, we focused on uplink and
channel allocation as well as more constraints is considered. In [24] instead of
energy efficiency, the total revenue of a system is maximized. They considered
cognitive network. Also their focus was only on power control and there is no
channel allocation. So it is not fair to compare our results with theirs. In [25] a
non-cooperative game is used to model the power control and numerical search
methods are adopted to solve the problem. In this study the user rate guarantee
was not taken into account as in our case. Moreover, the system model has only
one tier (say only macrocell). In [26] a multi-objective approach is considered
towards maximizing the individual energy efficiency rather than total energy
efficiency, focusing on single-cell OFDMA Systems and heterogeneous network
scenario with femtocells is not considered. In [27] a comprehensive survey on
energy efficiency metrics is provided. Therefore, the reader is referred to this
survey to access to the large number of papers related to energy efficiency.
The problem of power minimization in the OFDMA femtocell network has
been rarely discussed in the literature. Minimizing the transmit power of FBS
in downlink direction to reduce the interference level on the MUEs has been
considered in [28] under co-channel deployment while satisfying the minimum
requirements of FUEs. This work considers downlink Femtocell base station
transmit power. No subchannel assignment was considered in their model and
problems. Thus, this scenario is not applicable in practical cellular networks;
however, it has a valuable framework and insight on the power minimization
problems. Also, Lopez-Perez in [11] proposed a novel model for subchannel
and power allocation in downlink to minimize the cell transmit power while
taking realistic constraints into account. Although their proposed algorithms
offer significant performance improvements in terms of user outages and cell
capacity, these models are for downlink direction and uplink resource allocation
is not addressed there. However the analysis of energy efficiency have attracted
the attentions, in previous works, the problem of maximizing energy efficiency
and minimizing power are not jointly considered for femtocell networks in
uplink.
Different than the previous literature mentioned, in our study, we con-
sider joint power and channel allocation for uplink for two different objectives:
power minimization and energy efficiency maximization for a heterogeneous
4 Meysam Masoudi et al.
network scenario with overlay femtocell layer. For performance evaluation we
have chosen to compare our algorithm with an existing algorithm [29] and
equal power allocation algorithm as a benchmark.
1.2 Contribution
In this paper, to minimize the power consumption and maximize the energy
efficiency of the heterogeneous networks with femtocells we have formulated
two mixed-integer non-linear optimization problems. To satisfy the minimum
required rate of users, different constraints are imposed. In order to see the
effect of the high data rate users, two service classes are considered. The first
class includes users demanding a high data rate and the second class includes
the users with low data rate requirement. We also consider the co-channel
spectrum allocation for better spectral reuse and femtocells operating in closed
access mode. Furthermore, we consider a maximum acceptable interference
level on each subchannel to preserve the QoS of MUEs. The main contributions
of the paper are as follows:
–We have formulated the energy efficiency maximization and power mini-
mization problems for the uplink of heterogeneous networks. Since the for-
mulated problems for joint channel and power allocation are non-convex,
the problems are converted into a convex form by using the time sharing
concept. The optimal solutions for these problems are then obtained in
closed-form by KarushKuhnTucker (KKT) conditions on the Lagrangian
of the optimization problems.
–We have investigated the resource allocation problem in heterogeneous net-
works for given objectives. In order to perform the resource allocation in
the system, we have proposed two iterative power and channel allocation
algorithms. Moreover, a simplified algorithm is introduced to reduce the
complexity of proposed algorithms. In the uplink power model, we have also
considered the circuit power and its effects on the proposed algorithms. In
addition, the complexity of the proposed algorithms has been analyzed.
–Finally, we have analyzed and compared the proposed resource allocation
methods with energy efficiency criterion. It can be seen that significant per-
formance improvement is attained in comparison with existing algorithms.
The rest of the paper is organized as follows. In Section 2, the system
model and problem formulation is presented. Resource allocation strategies
are proposed in Section 3while three algorithms with their complexity anal-
ysis are presented in section 4. To verify the performance of the proposed
algorithms simulation results are presented in section 5. The concluding re-
marks are shown in section 6.
Title Suppressed Due to Excessive Length 5
Fig. 1: System model of Two-tier femtocell network.
2 System Model and Problem Formulation
2.1 System Model
According to Fig.1, a two tier OFDMA cellular network is considered in this
study. The system model parameters and descriptions are provided in Table
1. The FUEs and MUEs are distributed uniformly in the cell area. The total
bandwidth is shared by the FUEs and MUEs in uplink direction. In other
words, we consider the co-channel deployment network. Thus, it is imperative
to limit the interference caused by FUEs on MBS that might have higher pri-
ority to use the bands in networks [14]. In addition, FUEs that are connected
to the same FBS cannot share the same subchannel. We assumed closed ac-
cess mode for all femtocells which means only those who are defined by the
femtocell owner can connect to the femtocells [30].
Generally, all FUEs demand different QoS in terms of their data rate. We
consider two types of users corresponding to two different rate requirements.
|HDR|+|LDR|=Fand H DRk∩LDRK= Ø, where |H DR|,|LDR|denotes
the number of high data rate users and low data rate users. Our channel model
is composed of three parts; large scale fading that is chosen based on [31], small
scale fading and shadow fading. Let PM
w,n be the transmit power of MUE won
subchannel nand PF
k,u,n be the transmit power of FUE uconnected to FBS k
on subchannel n. Then, the received SINR of user uat the FBS kis calculated
as:
γk,u,n =pF
k,u,nhF
k,u,n
pM
w,nhF M
k,w,n +σ2.(1)
6 Meysam Masoudi et al.
Table 1: System model parameter definitions and notations
Notation Description
WBandwidth
NNumber of channels
KNumber of femtocells
MNumber of active MUEs
FNumber of users in each femtocell
PcCircuit power
N0Noise power spectral density
γk,u,n SINR of FUE uin femtocell kon subchannel n
Ith Maximum allowable interference on each channel
Rk,u Required rate by the uth user in kth femtocell
ak,u,n Binary channel allocation variable of femtocell k
in subchannel nfor user u
hF M
k,w,n
Channel gain from MUE wto femtocell kin sub-
channel n
hMF
k,u,n
Channel gain from FUE uof femtocell kin sub-
channel nto the MBS
hF
k,u,n
Channel gain from FUE uof femtocell kin sub-
channel nto the FBS
The first term in the denominator of γk,u,n is interference caused by MUEs on
the subchannel non the FBS kand the second term is thermal noise, which is
denoted by σ2. The interference caused by nearby FUEs of neighboring femto-
cells as well as other cells interferences on subchannel nis absorbed in thermal
noise. This assumption is realistic particularly in sparse deployed femtocells
with a high penetration loss of residential buildings, where the interference
level is negligible [29]. The spectral efficiency of each FUE in femtocell kon
subchannel n, which is measured in bps/Hz, follows the normalized Shannon
rule which is defined as:
CF
k,u,n = log2(1 + SINRF
k,u,n).(2)
The total energy efficiency of the system is defined as the ratio of the sum rate
and the total power consumption of all users:
EET=PK
k=1 PF
u=1 PN
n=1 ak,u,nCF
k,u,n
PK
k=1 PF
u=1(PN
n=1 ak,u,npk ,u,n +Pc)(3)
where Pcis a constant circuit power consumed at the UE representing the
power consumption of device electronics while the transmit power is used for
reliable data transmission [32]. As in [33], UE power model includes both the
circuit power and the transmit power.
Title Suppressed Due to Excessive Length 7
2.2 Problem Formulation
In this subsection, we aim to optimize our system from two standpoints, ef-
ficiency and power. To achieve our goal, two main optimization problems are
formulated. The first problem aims to minimize the aggregate power of all
FUEs while the second problem focuses on maximizing the energy efficiency
and they are referred to as problem Iand problem II respectively.
Minimum Power Problem Formulation:
The optimization problem Ican be formulated as:
min
ak,u,n,pk,u,n
K
X
k=1
F
X
u=1
N
X
n=1
ak,u,npk ,u,n (4)
s.t.
F
X
u=1
ak,u,n ≤1∀k, n (5)
ak,u,n ≥0∀k, u, n (6)
N
X
n=1
ak,u,npk ,u,n ≤Pmax.∀k, u (7)
N
X
n=1
ak,u,nCk ,u,n ≥Rk,u ∀u, k (8)
K
X
k=1
F
X
u=1
ak,u,npk ,u,nhM F
k,u,n ≤Ith
n.∀n(9)
The objective function to be minimized in (4) is the total aggregate power
consumed by all FUEs and ak,u,n ∈ {0,1}is a variable that indicates whether
a subchannel nis assigned to a user. In (4) we omit the circuit power because
we have assumed that it is constant during the transmission and therefore
it has no effect on the final solution of problem. Also, each subchannel must
be dedicated to at most one FUE in each femtocell. Thus, constraints (5)
and (6) are applied to the problem. The constraint shown in (7) is imposed
to ensure that the total transmit power of each FUE does not exceed the
maximum allowable value which is denoted by Pmax. The constraint shown
in (7) does not affect the solution significantly due to the power minimization
nature of the problem. However, the interference plays a key role in finding the
final solution; (9) guarantees that the total interference of all FUEs on each
subchannel is lower than the predefined threshold denoted by Ith
n. Therefore,
by satisfying (9), the transmitting power of FUEs does not disturb the MUEs
QoS significantly. In addition, each user requires a data rate which needs to
be satisfied by the constraint that is imposed in (8), where Ck,u,n is defined
in (2).
Energy Efficiency Problem Formulation:
The goal of this optimization problem is to maximize the total energy effi-
ciency, EET, defined in (3) and the constraints are the same as the previous
8 Meysam Masoudi et al.
optimization problem, as in the following:
max
ak,u,n,pk,u,n
EET(10)
s.t. (5)to (9)
Energy efficiency is a metric which quantifies how efficiently resources are
used. It is measured by bit/Hz/joule, which is the spectral efficiency over
power consumption. In some sense, this is a multi objective problem that
maximizes the throughput and minimizes power at the same time. Therefore,
this problem makes effort to transmit data with maximum possible data rate
with minimum power consumption, while satisfying the constraints.
3 Subchannel Assignment And Power Allocation Strategy
The optimization problems Iand II are mixed-integer non-convex problems,
therefore it is hard to find their optimal solution [34]. In this section, the time
sharing concept is used to transform the mixed-integer non-convex optimiza-
tion problem to a convex one. Then, the resulting convex optimization problem
is solved using the Lagrangian dual decomposition method [35]. First, the op-
timization problem Iis solved then to deal with optimization problem II, we
transform it to an equivalent problem which is similar to the first problem in
form. Hence, the solution to the second problem will be straightforward and
only the final results are given as we will explain later.
3.1 Minimizing the Aggregate Power
In the problem formulation, constraints (5) and (6) cause model to become
mixed integer non-convex because ak,u,n is a binary variable. To find the opti-
mal solution of such a problem an exhaustive search algorithm like brute-force
must be applied. These approaches have a high computational complexity [29].
By means of heuristic search methods like the genetic algorithm, near optimal
solutions can be obtained as well; however, global optimality of the solutions
cannot be guaranteed [36].
The time sharing concept has been widely used to transform these kind of
problems to convex optimization problems in multi channel multi user systems
[29], [37] and is introduced by Wong in [35]. Then, in [36] the zero duality gap of
multicarrier systems was proved once the time sharing condition was fulfilled.
To apply this concept to the proposed problem, instead of dealing with ak,u,n
as a binary variable, it is treated as a continuous variable in the range of
[0,1]. ak,u,n can be seen as a time-share factor which shows the fraction of
time that a subchannel is assigned to a user. After relaxing the ak,u,n to be
a continuous variable, and changing the variables ak,u,npk,u,n into a variable
sk,u,n the resulting optimization problem will be convex.
The goal function in (4) is concave due to negative semi-definite elements
of the Hessian matrix with respect to ak,u,n and sk,u,n . The feasible set of
Title Suppressed Due to Excessive Length 9
goal function is convex because the inequality constraints in (6) to (9) are
convex [34]. Therefore, the optimization problem becomes a convex optimiza-
tion problem that has a unique optimal solution. In other words, the local
solutions are optimal, and can be obtained in polynomial time [34]. In this
section we use the Lagrangian dual decomposition method to solve the convex
optimization problem. First we formed the Lagrangian function and then we
have decomposed the problem. The Lagrangian function can be formulated as:
L({ak,u,n},{pk ,u,n}, λ, ν, µ, η )
=
K
X
k=1
F
X
u=1
N
X
n=1
ak,u,npk ,u,n
+
K
X
k=1
F
X
u=1
λk,u(
N
X
n=1
ak,u,npk ,u,n −Pmax)
+
N
X
n=1
µn(Ith
n−
K
X
k=1
F
X
u=1
ak,u,npk ,u,nhM F
k,u,n)
+
K
X
k=1
N
X
n=1
ηk,n(1 −
F
X
u=1
ak,u,n)
+
K
X
k=1
F
X
u=1
νk,u(
N
X
n=1
ak,u,nCk ,u,n −Rk,u),(11)
where η, λ, ν, µ are dual variable vectors for (5), (7), (8) and (9), respectively.
The condition in 6will be considered in KKT conditions later. Now the dual
function is:
g(λ, ν, µ, η)) =
min
{ak,u,n},{pk,u,n }
L({ak,u,n},{pk ,u,n}, λ, ν, µ, η )
(12)
Thus, the dual problem becomes:
min
λ,ν,µ,η g(λ, ν, µ, η)
s.t. λ, ν, µ, η ≥0 (13)
To solve the problem, the idea of decomposition is used in which the main
problem is broken into several smaller problems. The sub-problems cannot be
solved separately, since they are coupled to each other with the constraints.
Thus, the overall problem must be solved iteratively. In our problem, each
femtocell solves the sub-problem for each subchannel of its users. Consequently,
the Lagrangian dual function in (12) is decomposed into a master problem and
K×Nsubproblems. We define Lk,n as:
Lk,n({ak ,u,n},{pk,u,n }, λ, ν, µ, η) =
10 Meysam Masoudi et al.
F
X
u=1
[ak,u,npk ,u,n +λk,uak ,u,npk,u,n +ηk ,nak,u,n
−νk,uak ,u,nCF
k,u,n +µnak,u,n pk,u,nhM F
k,u,n].(14)
Hence, the Lagrangian function introduced in (11) is as follows:
L({ak,u,n},{pk ,u,n}, λ, ν, µ, η )
=
K
X
k=1
N
X
n=1
Lk,n({ak ,u,n},{pk,u,n }, λ, ν, µ, η)
+
N
X
n=1
µnIth
n−
K
X
k=1
F
X
u=1
νk,uRk ,u
+
K
X
k=1
F
X
u=1
λk,uPmax +
K
X
k=1
N
X
n=1
ηk,n.(15)
Owing to KKT conditions to find the optimal solution of the subproblems,
we set the derivative of (14) with respect to {ak,u,n}and {pk ,u,n}to zero as
follows:
∂Lk,n
∂pk,u,n
= 0 (16)
∂Lk,n
∂ak,u,n
= 0.(17)
By considering (16), it can be shown that:
pk,u,n =νk,u
ln(2)(1 + λk,u +µnhM F
k,u,n)−Ik ,u,n
hk,u,n
(18)
Therefore, according to (18) the power assigned to the user becomes:
˜pk,u,n = (pk,u,n )+(19)
where (z)+is max(z, 0). It is noteworthy to see that the dual variable νk,u
is related to the required rate of users, so if some users do not demand any
rates, their dual variable νk,u will be zero. This forces the first term in (18) to
become zero and due to (19) the power allocated to the user is forced to be
zero. According to this interesting result, all the users are assumed to require
different data rates. Furthermore, this result is in line with power minimization
objective since if a user does not need any rate, in order to reduce the extra
power, the system forces the transmit power to be zero. Taking (17) into
account, the decision criterion to find the proper channel is derived as follows:
∂Lk,n
∂ak,u,n
=
pF
k,u,n(1 + λk ,u +µnhMF
k,u,n)−νk ,uCF
k,u,n +ηk,n .
(20)
Title Suppressed Due to Excessive Length 11
The last term in (20), which is calculated in MBS, is not related to the users
so it will be sent to each femtocell by MBS. This term does not play any
role in subchannel assignment, even though it has an effect on updating the
variables. Finally, to assign the subchannels, the channel criterion (CCk ,u,n) is
introduced to make a decision based on that. CCk,u,n is similar to (20) except
for the last term:
CCk ,u,n =pF
k,u,n(1 + λk ,u +µnhMF
k,u,n)
−νk,uCF
k,u,n (21)
Based on a channel criterion, CCk,u,n , we introduce the following rule to assign
a subchannel to a user. Each subchannel is allocated to a user with lower
CCk ,u,n in each femtocell. In addition, by using the time sharing concept and
a decision criterion, CCk ,u,n, we relax a condition; therefore, we seek for a
near optimal solution with semi distributed resource allocation strategy.
˜ak ,˜u,n = 1
˜u=minC Ck,u,n
∀k, n (22)
By allocating a subchannel to a user, FUE must transmit power on that
subchannel. Therefore, it is in contrast with the power minimization objec-
tive of this problem. As a result, the lowest increase in total power allocated
to FUEs is more preferable. In other words, a channel with lower CCk,u,n
regarding to (20) and (21) is superior to other channels. Another notewor-
thy result from considering (21) and (22) is that time sharing is ignored. In
other words, while this problem is solved using time-sharing, this system can
get better performance in terms of interference management. This result is
in accordance with the results shown by Zhang in [29]. They suggested that
improvement in the performance without time-sharing is due to preventing a
co-tier interference in femtocells. An iterative approach is used to update the
dual variables when applying the sub-gradient method as follows:
λ(t+1)
k,u = (23)
λ(t)
k,u −α(t)
1(Pmax −
N
X
n=1
ak,u,npk ,u,n)+
,∀k, u
ν(t+1)
k,u = (24)
ν(t)
k,u −α(t)
2(
N
X
n=1
ak,u,nCF
k,u,n −Rk,u )+
,∀k, u
µ(t+1)
n= (25)
µ(t)
n−α(t)
3(Ith
n−
K
X
k=1
F
X
u=1
ak,u,npk ,u,nhM F
k,u,n)+
,∀n
where α(t)
1, α(t)
2, α(t)
3are step sizes and must be chosen carefully because they
control the convergence of the algorithms. In addition, they have to follow
12 Meysam Masoudi et al.
some rules as:
lim
T→∞
T
X
t=1
α(t)
i→ ∞,(26)
lim
t→∞ α(t)
i= 0,(27)
where i∈ {1,2,3}and t∈ {1, ..., Tmax}which Tmax is the maximum number
of iterations. In each femtocell, (24) and (25) are updated iteratively and
all they need are provided by local information. In (21) the effect of ηis
neglected, thus FBSs do not need extra information except their local data
to perform channel assignment and power allocation. Also, (26) is updated in
MBS so it needs information about the interference of the users who share the
same subchannel. Information about interference can be obtained by channel
gain estimation as well as users power information, which is provided through
backhaul. Finally, ηwill be sent to each FBS over the backhaul to update
its parameters such as allocated power. Hence, by updating dual variables by
(24), (25) and (26) power and subchannels are assigned dynamically and once
they are converged, the ultimate solution is reported by the algorithm.
3.2 Maximizing Energy Efficiency
The optimization problem II is convex-concave fractional programming and
the Dinkelbach algorithm [38] can be used to find its optimal solution [26]. In
order to solve problem II, it should be converted to a convex form. To simplify
our notation, the nominator and denominator of (3) is redefined as CTand
PT, respectively. Then, the objective function is as follows:
ξ=CT
PT⇒CT−ξP T= 0.(28)
Therefore, the optimization problem II can be formulated as follows:
min
ak,u,n,pk,u,n
−(CT−ξP T) (29)
s.t. (5)to (9)
The fractional objective function is linearized in (29) and the problem for-
mation is similar to the problem I. Consequently, the solution procedure is
straightforward and is similar to that of the optimization problem Iwith some
adjustments. Performing the same procedure as in optimization problem I, the
final results are provided. It is worth mentioning that ξmust be calculated
and updated in MBS, solving (28) for ξand the value must be sent to each
FBS in each iteration. The channel criterion is:
CCk ,u,n = (30)
pF
k,u,n(ξ+λk ,u +µnhMF
k,u,n)−(1 −νk ,u)CF
k,u,n.
Title Suppressed Due to Excessive Length 13
The power assigned to a user on each subchannel is:
pk,u,n =(1 + νk,u )
ln(2)(ξ+λk,u +µnhM F
k,u,n)−Ik ,u,n
hk,u,n
(31)
4 Resource Allocation Algorithms and Complexity Analysis
Three algorithms are proposed to implement the procedure of power and sub-
channel allocation to the users while optimizing the objective function, mean-
while handling the constraints. In (18), (19), and (22) the equations to find
optimal subchannel and allocated power for each user are derived but the pro-
cedure of assigning resources is still a question. In order to answer this, three
algorithms are proposed. These methods are suitable to be used in FBS, be-
cause they both converge fast enough to be used in practical systems and need
very low information to be exchanged with MBS. Furthermore, a low complex
and distributed interference management scheme, called REFIM is proposed
in [39] to deal with feedback overhead over backhaul.
4.1 Near Optimal Algorithms
Algorithm 1and 2are presented to demonstrate the steps of the proposed
resource allocation methods as will be discussed later.
Algorithm 1 MinPower joint subchannel and power allocation
1: Initialize Tmax , λ, ν, µ and t= 0
2: Initialize pk,u,n with equal power distribution on each subchannel
3: Initialize ak,u,n based on (22)∀k , u, n
4: while t≤Tmax do
5: for k=1 to K do
6: for u=1 to F do
7: for n=1 to N do
a) Update pk,u,n according to (18) and (19)
b) Update CCk,u,n based on (21)
c) Update ak,u,n according to (22)
d) Update λ, ν, according to (24) and (25)
8: end for
9: end for
10: end for
11: MBS updates the µnbased on (26) and sends it to the FBS via backhaul.
12: end while
In Algorithm 1and 2, for each femtocell all the parameters are available
based on local data except µnand the channel gain hMF
k,u,n. These parameters
are required to compute the allocated power and channel criterion defined in
(18), (31), and (21). The channel gain can be measured by FBS locally or
MBS could approximate and send it to the FBS via backhaul. The complexity
analysis of both algorithms are discussed later.
14 Meysam Masoudi et al.
Algorithm 2 MaxEE joint subchannel and power allocation
1: Initialize Tmax , λ, ν, µ and t= 0, ξ= 0
2: Initialize pk,u,n with equal power distribution on each subchannel
3: Initialize ak,u,n based on (22) and (31)∀k , u, n
4: while t≤Tmax do
5: for k=1 to K do
6: for u=1 to F do
7: for n=1 to N do
a) Update pk,u,n according to (31) and (19)
b) Update CCk,u,n based on (31)
c) Update ak,u,n according to (22)
d) Update λ, ν, according to (24) and (25)
8: end for
9: end for
10: end for
11: MBS updates the µnand ξbased on (26) and (28)and sends them to each FBS via
backhaul.
12: end while
4.2 Sub Optimal Algorithm
The third proposed method is the simplified version of the first and second
algorithms so that it becomes more practical to be used by femtocells. The
simplified algorithm sacrifices the optimal solution to obtain a lower order
of complexity. The complexity analysis of the proposed algorithms will be
considered later. The simplified version of algorithm 1and algorithm 2contains
two main steps: Subchannel allocation and power allocation, as introduced in
algorithm 3. Algorithm 3is more preferable due to its simple implementation;
however, it is not able to find the near optimal solutions. The third or simplified
algorithm consists of two major parts. The first part is devoted to channel
allocation that allocates a suitable subchannel to the FUEs and the second
part is about power updating of each user on each subchannel, that is, FBS
adjusts FUE’s transmit power to minimize the aggregate power consumed by
FUEs. In other words, in the former part we make sure that by uniform power
allocation, all the users will reach their required rate meanwhile, in the latter
no channel allocation will be performed and only the power will be updated,
hence the complexity will be reduced which will be addressed in the next
section.
4.3 Complexity Analysis
Complexity analysis plays an important role in comparing the time efficiency
of different algorithms. In this subsection, the complexity order of the proposed
algorithms is discussed. Generally, Algorithm 1and 2offer similar complex-
ity because they are almost similar in procedure. Therefore, we focus on the
complexity order of 1. In Algorithm 1, during each iteration, calculating (21)
and updating (22) need K F N operations, independently. Furthermore, KF
calculations are required for updating the dual variables λand νaccording
Title Suppressed Due to Excessive Length 15
Algorithm 3 Simplified joint subchannel and power allocation
1: Subchannel Assignment Section
2: initialize same power to all subchannels ∀k, u
3: for k=1 to K do
4: set FemtoUser = F, FemtoSet = {1, ..., F }
5: set ChannelNumber = N, ChannelSet = {1, ..., N }
6: while FemtoUser 6= 0 do
7: find ˜u=argmax(hk,u,n
Ik,n ),
8: n= ChannelNumber
9: ak,˜u,n = 1
10: ChannelSet = ChannelSet -{n}
11: ChannelNumber = ChannelNumber - 1
12: if PN
n=1 ak,u,nCk,u,n ≥Rk ,u then
13: FemtoUser = FemtoUser -1
14: end if
15: end while
16: while ChannelNumber ≥1do
17: find ˜n=argmax(hk,u,n
Ik,n )
18: ak,u,˜n= 1
19: ChannelSet = ChannelSet -{˜n}
20: ChannelNumber = ChannelNumber - 1
21: end while
22: end for
23: Power Allocation Section
24: Initialize Tmax , λ, ν, µ and t= 0 ∀k, u, n
25: while t≤Tmax do
26: for k=1 to K do
27: for n=1 to N do
28: Update pk,u,n according to (18) and (19)
29: Update λ, ν, according to (24) and (25)
30: end for
31: end for
32: MBS updates the µnbased on (26) and sends it to the FBS via backhaul.
33: end while
to (24) and (25), respectively. The parameter µis also calculated in MBS
for each subchannel and its complexity order is O(N) per iteration. Let algo-
rithm 1converge after Tconv iterations, therefore, its total complexity becomes
O(Tconv(K F N )2). Although all K, F and Nare fixed network parameters,
Tconv is not a constant predefined value. The results of our preliminary tests
show that smart initialization of parameters such as dual variables and step
sizes can provide a smaller Tconv. It should be noted that updating the dual
variables after the execution of the iteration loop on the subchannels can re-
duce the complexity of algorithm 1. Furthermore, it is possible to update ak,u,n
for all the users of the same femtocell at the same time.These modifications in
the procedures not only reduce the complexity order, but also result in reduc-
ing the total execution time of the algorithm, which is critically important,
even though it requires more memory.
Algorithm 3suggests a simplified procedure to solve the proposed opti-
mization problem. Finding a proper user in each subchannel for every fem-
tocell requires total KN F operations while updating µfor each subchannel
16 Meysam Masoudi et al.
calls for total KN 2calculations. Therefore, the total complexity order of the
subchannel allocation section is O(KN (F+N)).The complexity analysis of
the second part of algorithm 3is almost similar to what is done in algorithm 1.
The loop on the number of femtocells is reduced and subsequently, the overall
required calculation is O(K2N2F Tmax). To further reduce the complexity, it
is also possible to update the dual variables for each femtocell instead of all
of them, therefore the complexity becomes O(KN 2F Tmax).To sum up the
analysis here, the total complexity order of algorithm 3is O(KN 2F Tmax ).
Femtocells can use both algorithms but the proposed simplified scheme pro-
vides lower complexity than algorithm 1and algorithm 2; however, in return
it sacrifices the optimality of the final solution.
To compare the complexity order of the proposed algorithms with exist-
ing schemes, we consider the exhaustive search and genetic algorithm as two
alternative approaches towards solving the optimization problem. O(KF N) is
a complexity order of exhaustive search whereas, NSGA II, as an example of
the genetic algorithm, has a complexity order of O((NpG)(F KN +ObNp))
where Gis the number of generations, Npis the population number and Obis
the number of objectives which is 1 in our case [19]. The proposed algorithms
have much lower complexity than NSGA II and exhaustive search knowing
the fact that generally the number of generations and populations is high in
NSGA II in comparison with F, K, N [19].
5 Simulation Results
In the previous sections, three resource allocation methods are proposed aim-
ing at total user power minimization and energy efficiency maximization. In
this section the performance of the proposed algorithms is investigated. The
scenario as depicted in Fig.1, is co-channel deployed femtocells that are over-
laid on a macrocell. The simulation parameters and their corresponding values
are summarized in Table 2. To make the system more realistic, we assume that
each femtocell can serve up to Fusers and they require two types of services
with high and low QoS. The carrier frequency is set to 2GHz and thermal
noise is considered as a zero mean Gaussian random variable with variance
of σ2and power spectral density of N0=−174dbm/Hz, so σ2= (W/N)N0.
Channel fading is composed of path loss, shadow fading and Rayleigh fading.
Pathloss models are chosen based on [31], shadow fading is modeled as zero
mean log normal distributions with variance of 10db and 8db for MUEs and
FUEs, respectively. Besides, Rayleigh fading channel gains are modeled as a
unit-mean exponential distribution. Femtocells have a coverage radius of 20m,
furthermore, they are distributed uniformly in a macrocell coverage with ra-
dius of 1000m. In our simulations, femtocell centers have a distance, at least
three times of their radius to that of the others; therefore femtocells coverage
cannot overlap each other. This assumption is in line with the ignorance of
the co-tier interference.
Title Suppressed Due to Excessive Length 17
Table 2: Simulation parameters values
Definition Notation Value
Bandwidth W20 MHz
Number of channels N50
Number of active MUEs M50
Number of femtocells K30
Number of users in each femtocells F4
Circuit power Pc0,100 mW
Maximum allowable interference Ith −101 dbm
Maximum transmit power of users Pmax 23 dbm
Noise power spectral density N0−174 dbm/Hz
High quality service requirements RHDR 60 bps/H z
Low quality service requirements RLDR 20 bps/Hz
0 50 100 150 200
Number Of Iterations
0
1
2
3
4
5
Total Consumed Power (mW)
P roposed A lgorithm 1
P roposed A lgorithm 3
P roposed A lgorithm 2
Fig. 2: Convergence performance of aggregate power consumption for proposed
algorithms.
In Fig.2and Fig.3, the convergence performance of the goal function for
all proposed methods are depicted in which the number of femtocells and
users in each femtocell are set to 30 and 6, respectively. Step sizes in updating
the dual variables in (24),(25) and (26) have effects on the convergence rate
of algorithms. By using the step size coefficient updating function t−t.p in
our simulations, we let the early steps have greater effects on updating the
variables. It can be seen that all algorithms converge rapidly proportional to
the total number of users; therefore they are all suitable to be utilized in
practical resource management in small cell scenarios. Also it can be seen
from Fig.2and Fig.3that the first algorithm outperforms the others in terms
of power consumption and the second one outperforms the others in terms of
energy efficiency. When it comes to the convergence performance, the third
algorithm performs better in total power consumption convergence and the
second algorithm performs worse because its objective function is not power.
18 Meysam Masoudi et al.
0 50 100 150 200 250 300
Number Of Iterations
0
2000
4000
6000
8000
10000
12000
14000
Energy Efficiency [bit/Hz]/joule
P roposed A lgorithm 1
P roposed A lgorithm 3
P roposed A lgorithm 2
Fig. 3: Convergence performance of total system efficiency for proposed algo-
rithms.
10 15 20 25 30 35 40 45 50
Number Of Femtocells
10-1
100
101
102
103
104
Consumed Power (mW)
F= 4 P ropose d Algor ithm 1
F= 4 P ropose d Algor ithm 3
F= 4 Eq ual P ow er All ocation
F= 4 M aximum T hroughp ut
F= 4 P ropose d Algor ithm 2
Fig. 4: Aggregate transmit power consumption Vs. number of femtocells
The total power consumption of the proposed algorithms for F= 4 is
demonstrated in Fig.4for different femtocell numbers. As a reference and
benchmark, we compare the power consumption of our algorithms with an
existing algorithm [29], referred to as maximum throughput algorithm and
equal power allocation. Equal power allocation is of industry interest because
it is simple and easy to implement. In equal power allocation, we set the value
of power to which the worst user reaches its required rate. As depicted in Fig.4,
significant energy saving is attained in comparison with the existing solution
and equal power allocation. All of our algorithms perform better than existing
ones and also it can be seen that algorithm 1and algorithm 2outperform
the simplified algorithm. Furthermore, in the proposed algorithms, when F
is set to 4 the aggregate power consumption does not increase dramatically
Title Suppressed Due to Excessive Length 19
10 15 20 25 30 35 40 45 50
Number Of Femtocells
10-2
10-1
100
101
102
Consumed Power (mW)
RHDR = 40 (bps/Hz) P ropo sed Algo rithm 1
RHDR = 60 (bps/Hz) P ropo sed Algo rithm 1
RHDR = 40 (bps/Hz) P ropo sed Algo rithm 3
RHDR = 60 (bps/Hz) P ropo sed Algo rithm 3
RHDR = 40 (bps/Hz) P ropo sed Algo rithm 2
RHDR = 60 (bps/Hz) P ropo sed Algo rithm 2
Fig. 5: Total transmit power consumption Vs. number of femtocells
Number Of Femtocell User
2345678
Efficiency [bit/Hz]/joule
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
P roposed A lgorithm 1
P roposed A lgorithm 3
Equa l P ower Allocation
M aximum T hr oughput Algort ihm
P roposed A lgorithm 2
Fig. 6: Load impact effect on energy efficiency Pc= 100mW
since there is enough channel to be allocated to the users in each femtocell.
Consequently, users do not need to use more power to reach their required
rate. Generally, the simplified algorithm uses more power than algorithm 1
and 2especially when the number of users in a cell is high e.g. F= 8. This
higher power consumption is due to the existence of more users and also the
lower number of allocated subchannels to each user, which makes them use
more power to reach their required rate. Although consumption power in the
simplified algorithm is higher than algorithm 1and algorithm 2, it is yet
acceptable to be utilized in practical systems.
Total power consumption in terms of number of femtocells for different
required rate of HDR users is depicted in Fig.5. In this simulation, the required
rate of LDR users, HDR users and number of users in each femtocell is set
to RLDR = 10 bps/H z,RH DR = 40 and 60 bps/Hz and F= 4, respectively.
20 Meysam Masoudi et al.
Number Of Femtocells
10 15 20 25 30 35 40 45 50
Efficiency [bit/Hz]/joule
1
1.5
2
2.5
3
3.5
P roposed A lgorithm 1
P roposed A lgorithm 3
Equa l P ower Allocation
M aximum T hr oughput Algort ihm
P roposed A lgorithm 2
Fig. 7: Total Energy efficiency of proposed algorithms Vs. Number of femtocells
Pc= 100mW
Interestingly, when the number of femtocells and the rate of high data rate
users are set to F= 10 and RH DR = 40 bps/H z the consumed power is
relatively high in algorithm 2. The reason is that when the number of femtocells
and required rate of users are low the interference level is not a critical issue
and the system has the freedom to allocate as much as power and channel to
the users to maximize the energy efficiency. Furthermore, it is expected that
demanding a greater data rate results in more power consumption, verified by
Fig.5. Algorithm 1outperforms the others with a gentle slope compared to
other methods where the power consumption slope is relatively steep.
In Fig.6the load impact effect on energy efficiency is investigated. It can
be seen that the more user in femtocell is active the lower energy efficiency
is achieved. This is because when there are more users, number of channels
are limited accordingly, therefore each user must transmit more power on each
channel and consequently the energy efficiency decreases.
The total energy efficiency of the system for different number of femtocells
is demonstrated in Fig.7and Fig.8. It can be seen that in Fig.7the best perfor-
mance is for algorithm 2while maximum throughput algorithm is the second
best algorithm. When considering the (3), Pcplays a critical role. However Pc
is circuit power and is not considered in transmission power, it has an effect on
the energy efficiency from user’s side. The circuit power dominates the trans-
mit power, when minimizing the power, in the denominator of (3) because
the transmission power is much lower than circuit power when minimizing the
power. That is why the maximum throughput algorithm outperforms the algo-
rithm 1and 3. From the network perspective, it is valuable to ignore the Pcand
investigate the algorithms performance by considering only the transmission
power. In Fig.7this issue is addressed. It can be seen that algorithm 2rep-
resents the best performance while algorithm 1and algorithm 3still perform
better than maximum throughput and the equal power algorithm. Due to the
Title Suppressed Due to Excessive Length 21
10 15 20 25 30 35 40 45 50
Number Of Femtocells
100
102
104
106
108
1010
1012
Efficiency [bit/Hz]/joule
F= 4 P ropose d Algor ithm 1
F= 4 P ropose d Algor ithm 3
F= 4 Eq ual P ow er All ocation
F= 4 M aximum T hroughp ut Alg ortihm
F= 4 P ropose d Algor ithm 2
Fig. 8: Total Energy efficiency of proposed algorithms Vs. Number of femtocells
Pc= 0
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Probability
100
102
104
106
108
Efficiency ((bps/Hz)/mW)
F= 2 P ropose d Algor ithm 1
F= 2 P ropose d Algor ithm 3
F= 2 Eq ual P ow er All ocation
F= 2 M aximum T hroughp ut Alg ortihm
F= 2 P ropose d Algor tihm 2
Fig. 9: Total Energy efficiency of proposed algorithms Vs. probability of being
high data rate FUE Pc= 0
high consumed power of the maximum throughput algorithm, it also fails to
provide high energy efficiency. Thus, it is not an interesting algorithm in terms
of efficiency, however it maximizes the throughput. Furthermore, the efficiency
of all algorithms except algorithm 2does not change dramatically with varia-
tion of femtocell numbers. There are two reasons for this phenomenon. Firstly,
we ignore the co-tier interference; therefore the lower interference level is ex-
perienced in our simulations while in the dense scenario it might be crucial.
In other words, if we consider the co-tier interference in our model, a steep
downward trend is expected here. Secondly, all algorithms except algorithm 2
are far lower than their optimum efficiency level and their level of interference
are relatively low and as a result by increasing the number of femtocells they
22 Meysam Masoudi et al.
10 15 20 25 30 35 40 45 50
Number Of Femtocells
600
700
800
900
1000
1100
1200
1300
Achieved Throughput (bps/Hz)
P roposed A lgorithm 1
P roposed A lgorithm 3
Equa l P ower Allocation
M aximum T hr oughput Algori thm
P roposed A lgorithm 2
Fig. 10: The effect of proposed algorithms on MUE’s rate
can still use their proper channel so their efficiency does not drop sharply. In
contrast to other algorithms, algorithm 2experiences a sharp decline. This is
because of constraints on power and interference level on each channel. By
increasing the number of femtocells there exist more users who are using the
same channel. Consequently, the cross-tier interference, which is more severe
with the higher number of femtocells, make users use their non optimal channel
with more power in order to satisfy the interference constraints on all chan-
nels. Hence, their efficiency level becomes lower. Interestingly, with the higher
number of femtocells, e.g. K= 40 the difference between the efficiency level of
algorithm 2and others become lower and lower but it is still an upper bound.
In Fig.9, the effect of HDR user probability in our system is depicted. The
more the number of HDR users are, the less efficiency we experience because
high data rate users need more resources and considering these users costs the
system. These users use more resources and the amount of available resources
decrease so this results in lower efficiency.
To preserve the MUEs QoS, the maximum allowable threshold level is im-
posed on each channel but it is noteworthy to see the effect of algorithms on the
MUEs rate. In Fig.10 this issue is explored. It can be seen that our proposed
algorithms degrade the MUEs rate by 10 per cent while max throughput algo-
rithm caused about 21 percent degradation in MUEs rate which means that
our algorithms performed two times better than existing algorithms. Moreover,
the total throughput of MUEs in our algorithms are twice to that of proposed
algorithms. Interestingly, algorithm 2outperform the others in terms of lower
impact on MUEs QoS degradation although the degradation slope of the pro-
posed algorithms are almost similar to each other.
In algorithm 3channel assignment is done just once and then only power
allocation is performed, while in algorithm 1and algorithm 2power allocation
and channel assignment is performed simultaneously. Normalized deviation
ratio (DR) of each user with regards to its required rate is defined as DRk,u =
Title Suppressed Due to Excessive Length 23
-1 -0.5 0 0.5 1
Deviation Ratio
0
0.2
0.4
0.6
0.8
1
Probability
K= 30, F= 4, P roposed Algorithm 3
Fig. 11: Deviation ratio of algorithm 3.
Ra
k,u
Rk,u −1, where the achieved rate by user uin femtocell kis denoted by Ra
k,u.
Fig.11 presents the histogram of DR for the algorithm 3. The Number of users
in each femtocell and number of femtocells are set to F= 4 and K= 30 in this
simulation, respectively. According to Fig.11, in algorithm 3, 95% of the users
reach 95% of their required rate. Therefore, algorithm 3behaves perfectly and
almost all users reach their predefined rate. By increasing F, the system may
become infeasible and thus the total number of users that do not reach their
rate increases.
6 Conclusion
In this paper, two critical aspects of a cellular networks namely power and
energy efficiency are considered. To utilize the available resources efficiently
and to optimize the system with the given goal functions, two optimization
problems were solved. First problem was dedicated to minimize the aggre-
gate power of all FUEs and the second problem was devoted to maximize
the energy efficiency of the system. In order to protect the QoS of macrocell
users, a maximum allowable interference level was imposed on each subchan-
nel and to guarantee the QoS of FUEs, the minimum required rate of them
are satisfied. Knowing the inherent non-convexity of our primary problems,
we applied the time-sharing concept to transform the non-convex problem to
a convex one. Accordingly, the problems were decomposed to sub problems
and were solved by the sub-gradient method. In addition, three resource allo-
cation schemes were proposed to examine the problems. The first and second
algorithm were proposed with a near optimal solution while they needed only
little local information to perform the resource allocation distributively and
limited data to exchange with MBS. Furthermore, to decrease the complexity
order, we proposed a simplified algorithm, which satisfies the minimum QoS
24 Meysam Masoudi et al.
of the users.Finally, simulation results demonstrated significant enhancement
in terms of energy efficiency and the proposed algorithms could achieve sig-
nificant power savings, while the first and second algorithm outperformed the
simplified algorithm. To extend this work joint resource allocation of MUEs
and FUEs will be considered while some subchannels are exclusively dedicated
to MUEs.
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