qos-aware energy-efficient resource allocation in … and ee are not contradictory in hetnets...
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INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMS
Int. J. Commun. Syst. 2014; 00:1–26
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/dac
QoS-Aware Energy-Efficient Resource Allocation in
OFDM-Based Heterogenous Cellular Networks
Li Zhou1∗, Chunsheng Zhu2, Rukhsana Ruby2, Xiaofei Wang2, Xiaoting Ji3,
Shan Wang14, Jibo Wei1
1College of Electronic Science and Engineering, National University of Defense Technology, Changsha, China2Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada3College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, China4Science and Technology on Information Transmission and Dissemination in Communication Networks Laboratory,
Shijiazhuang, China
SUMMARY
Recently, in order to satisfy the heavy demands of network capacity brought about by the proliferation of
wireless devices, service providers are increasingly deploying heterogeneous cellular networks (HetNets) for
boosting the network coverage and capacity. In this paper, we present an iterative energy-efficient scheduling
scheme (IEESS) for downlink OFDM-based HetNets with quality-of-service (QoS) consideration. We
formulate the problem as a nonlinear fractional programming problem aiming to maximize the QoS-aware
energy efficiency (QEE) in HetNets. In order to solve this problem, we first transform it into a parametric
programming problem, which takes QEE as an evolved parameter in the iterative procedure of IEESS. In
each iteration, for the given value of QEE, subchannel and power assignment sub-problem is a nonlinear
NP-hard problem. And hence we adopt dual decomposition method for obtaining the optimal assignment
of subchannels and power of the sub-problem for the given value of QEE. Simulation results depict that
both outer QEE parameter search and inner subgradient search can converge in a few iterations and the
resultant solutions outperform the equal power allocation scheme (EPAS) [1] and capacity maximization
scheme (CMS) [2] in terms of QEE. Copyright c⃝ 2014 John Wiley & Sons, Ltd.
Received . . .
KEY WORDS: QoS-aware; Energy Efficiency; Resource Allocation; Heterogenous Networks
∗Correspondence to: College of Electronic Science and Engineering, National University of Defense Technology,
Changsha, China. E-mail: [email protected]
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2 L. ZHOU ET AL.
1. INTRODUCTION
With the rapid increasing of mobile users and multimedia services, current cellular networks
encounter a great challenge to satisfy the suddenly increasing mobile traffic driven by the widely
used smart devices, such as smartphones, tablets, etc. Meanwhile, the rapidly growing number of
smart devices and the demand for data rate lead to an ever-increasing power consumption in cellular
networks which causes huge cost for operators. Among all emerging concepts and technologies
towards this challenge, HetNets have been proposed as an important evolutionary path for LTE and
future 5G networks. The HetNet is defined as a mixture of macrocells and small cells, e.g., picocells,
femtocell and relays. The small cells can potentially enhance the spectrum reuse and coverage while
providing high data rate service and seamless connectivity in cellular networks. Although HetNets
have the potential of meeting the growing capacity requirement, it brings severe interference at
the same time due to the increasing density and unplanned deployment of the base stations (BSs).
Consequently, energy-efficient resource allocation has drawn enormous attention for the deployment
of HetNets.
Resource allocation in orthogonal frequency division multiplex (OFDM) based wireless networks
has extensively been studied because of its potentiality to become the core technology of the next
generation wireless networks. In OFDM-based networks, subchannel and power are two main
resources that need to be optimally assigned in order to enhance the system performance. For
the downlink operation, given the channel state information (CSI) and QoS requirements, the BSs
assign subchannels and power among the users. The traditional resource allocation schemes can be
classified into two categories [3]: 1) Rate adaptation (RA) scheme which focuses on maximizing
the network capacity [4, 5]. It is envisioned to optimize the spectral efficiency (SE) given certain
constraints. 2) Margin adaptation (MA) scheme which aims to minimize the total transmit power of
BSs. Despite the fact that it is designed to reduce energy consumption, it still has some limitations
in terms of energy efficiency (EE). For example, it considers neither the power consumption of
circuits nor the power amplifier (PA) efficiency of BSs which are considered as the major sources
of energy consumption in the system [6]. More importantly, employing the unique objective of
minimizing transmit power cannot achieve the real EE. In some cases, it would even hamper the
overall performance of networks because of the ignorance of SE. As a result, an effective EE
metric has received paramount importance from the perspective of EE. In existing literature, several
different EE metrics are introduced [7–9]. [8] examines the EE of HetNets with different metrics,
such as energy saving (ES) and energy consumption ratio (ECR). It points out the importance of the
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
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QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 3
EE metrics on the energy consumption of HetNets. The most common one among them is bits/Joule,
which is defined as the achieved system capacity per joule of energy expenditure [7].
Typically, there are two ways to achieve QoS in resource allocation model of OFDM-based
systems: 1) The first way is to set minimum data rate for each user in the constraint set. Some
prominent works following this method are [4, 5, 10–12]. The drawback of this method is, the
number of constraints in the model increases proportionally with the increasing number of users in
the system, which makes the problem much more difficult to deal with. Moreover, when the channel
condition is bad, satisfying the minimum data rate of all users might be impossible, which leaves a
fairness issue for the system. 2) The second way is to adopt utility-based objective functions. The
utility can be the function of users’ throughput or the quantification of their QoS, fairness, etc. In
prior work [2], a gradient-based utility method is proposed for assuring QoS in resource allocation
of OFDM-based systems. With this method, the problem appears to be a weighted sum capacity
maximization problem, where the weight is determined from the instantaneous system utility in the
previous time slot. In our work, in order to achieve energy-efficient resource allocation with QoS
assurance in a simplified manner, we have adopted the similar gradient-based utility method.
In this paper, we have proposed an energy-efficient QoS-aware resource allocation scheme
namely IEESS for OFDM-based downlink HetNets systems. First, we formulate the resource
allocation problem which maximizes the sum utility of the users per unit of power. For QoS
assurance, we use the gradient based framework which defines the utility of a user as its weighted
instantaneous rate. The resultant formulated problem is a ratio of one nonlinear and one linear
function. Hence, we transform it into parametric programming problem which is a function of
one parameter named QEE. Our solution method IEESS is iterative. In each iteration, for the
given value of QEE, intermediate resource allocation problem is solve and the parameter QEE is
updated. For the given value of QEE, resource allocation problem is NP-Hard. After transforming
the problem to convex one, we adopt dual decomposition method to determine the subchannel-
user mapping and their power assignment across the system. This procedure continues until the
parameter QEE converges to the optimal one. Extensive simulation results verify the reduced
computational complexity of IEESS and show that IEESS performs much better compared to EPAS
and CMS in terms of QEE.
The rest of the paper is organized as follows. We briefly summarize the existing work on EE
in Section 2. Section 3 gives the overview of the network model and formulates our defined
problem. Section 4 proposes the IEESS scheme. In order to show the efficacy and effectiveness
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
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4 L. ZHOU ET AL.
of our proposed scheme, we offer simulation-based evaluation results in Section 5. Finally, Section
6 concludes the paper.
2. RELATED WORKS
Resource allocation considering EE is an emerging topic for the future wireless networks. In [13],
EE is studied in two-tier HetNets by assigning subchannels among macro and pico cells in disjoint
manner. The outcome of this work is an optimal ratio of macro-pico density while optimizing the
EE of the network which is considered as an important result for the deployment of HetNets. In
[14], active/sleep modes of macro cell and deployment of small cells are considered to improve EE
of HetNets, which is also a good measure for the deployment of future HetNets. [15] compared
the influence of random sleeping and strategic sleeping of BSs on the power consumption and
EE. The results verify the effectiveness of sleeping strategy and indicate that the deployment of
small cells can improve EE but the gain saturates as the density increases. Without considering
transmit power in HetNets [16], it is verified analytically that increasing BS density can enhance
EE of the network if the power expenditure of the BS is smaller than certain threshold. In [17], a
framework combining radio planning and resource management is proposed for cellular networks.
It shows that energy-efficient operations can be carried out based on the radio planning decisions.
[18] proposed a partial spectrum reuse (FSR) scheme which aims to reduce inter-cell interference
and improve EE in two-tier HetNets. A FSR factor is defined as the proportion of the spectrum
reused by small cells and it is observed that the optimal FSR factor can be obtained when the ratio
of users’ data rate requirement and the total bandwidth of the system is close to zero. In [19], a
cross-layer design combining admission control and resource allocation is considered to improve
EE of femtocells while assuming that macrocell users can connect to femtocell BSs for mitigating
inter-cell interference. The authors studied multiband opportunistic mechanism in medium access
control (MAC) layer and admission control at the network layer to achieve better EE.
In a single cell OFDM-based system, there is a trade-off between EE and SE. This is because
SE is always enhanced with the increased transmit power in a network if there is no interference
from the neighboring networks. While in HetNets environment, capacity improvement is not always
possible by increasing transmit power. Because, increased transmit power may decrease SE due to
the interference experienced from the nearby BSs. Therefore, it is essential to consider the impact
of interference while studying the EE in HetNets although incorporating interference makes the
problem much more challenging. [20] analyzed the tradeoff between SE and EE while considering
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QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 5
BS power consumption and dynamic network setup. The authors in this work have proved that
SE and EE are not contradictory in HetNets compared to conventional cellular networks without
interference.
Due to the unplanned small cell deployment and frequency reuse in HetNets, the scenario can be
highly dynamic and much more challenging. The existing works on EE in HetNets mostly focus
on the optimum deployment of small cells, including active/sleep strategies and spectrum reuse
schemes [13–18] while assuming static/semi-static assignment of subchannels. To the best of our
knowledge, energy-efficient resource allocation scheme while achieving optimal subchannel and
power allocation in highly dynamic HetNets is still missing. To fill the gap in the literature, in this
work, we have proposed an energy-efficient subchannel and power allocation scheme in downlink
OFDM-based HetNets where subchannels in the system are shared by all macro and small cells.
3. NETWORK MODEL AND PROBLEM FORMULATION
In this paper, we consider a downlink OFDM-based system that consists of a macrocell and several
small cells. Fig. 1 shows a sample of such network. The set of active cells including macrocell and
small cells can be represented by S = {1, 2, · · · , S}, among which index 1 represents the macrocell.
We consider co-channel deployments [21] in the system, which means that the macrocell and small
cells operate on the same spectrum. To emulate a real scenario, small cells are arbitrarily distributed
within the coverage of the macrocell. There are Mi(i ∈ S) users in cell i, and total number of users
in the network is denoted by M =∑
i∈S Mi. We define Mi(i ∈ S) as the set to hold the users in
cell i. The macrocell BS (m-BS) and all small cell BSs (s-BSs) are connected to a local gateway,
which works as a centralized controller and manages the resource allocation and scheduling task
in the system [22]. The physical channels between the BSs and users are modeled by frequency
selective Rayleigh fading, which is mainly determined by distance attenuation. At the beginning
of each time slot, a training process is conducted for the centralized controller or m-BS to obtain
global CSI. Research on efficient channel training and estimation can be found in [23,24]. Training
and estimation is performed at the users in order to obtain the channel gains from them to the BSs.
Finally, all these acquired channel gains are notified to the centralized controller by the feedback
method. Total spectrum is divided into N subchannels and the set of the subchannels can be denoted
by N = {1, 2, · · · , N}. We assume that all BSs are equipped with omni-directional antennas and
the maximum transmit power of BS i is denoted by Pmaxi . The notations used in this paper are
summarized in Table I.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
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6 L. ZHOU ET AL.
Local Gateway
(Central Controller)
m-BS s-BS user
Figure 1. Network model.
Table I. Notation summary.
Notation DescriptionS; S Set of active cells in the network; number of cells
Mi; Mi Set of users in cell i; number of users in cell iN; N Set of subchannels; total number of subchannels
p; pi,k,n Power allocation matrix; allocated power on subchannel n for user k by BS ix; xi,k,n Subchannel allocation matrix; indicator on subchannel n for user k by BS ig; gi,k,n Channel gain matrix; channel gain of subchannel n between BS i and user kR; Ri,k,n Channel capacity matrix; channel capacity of subchannel n between BS i and user kR(g) Instantaneous feasible data rate regionB Total bandwidth
Pmaxi Maximum transmit power of BS i
SINRi,k,n SINR for the signal that user k receives from BS i on subchannel nΓ SINR gap
BER Bit error rateδ2 Thermal noise power
Wi,k Average throughput of user k in cell i up to the current time slotUi,k(Wi,k) Utility function of Wi,k
wi,k Weight for user k in cell iα Factor for fairnessβi,k QoS weight for user k in cell iγi Power-amplifier inefficiency factor of BS iPCi Static power consumption of BS i
ηEE QoS-aware energy efficiency
If BS i transmits data to user k through subchannel n, the user receives interference from the
nearby BSs in the network. As a result, the signal to interference and noise ratio (SINR) for the
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QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 7
signal that the user receives can be written as
SINRi,k,n =pi,k,ngi,k,n∑
j∈S,j =i
∑q∈Mj
pj,q,ngj,k,n + δ2, (1)
where pi,k,n is the allocated power on subchannel n for user k by BS i, and the matrix p holds all
power allocation decision for all users on each subchannel in the network. gi,k,n is defined as the
channel gain of subchannel n when BS i transmits to user k. δ2 is the power of thermal noise.
The Shannon capacity obtained by user k in cell i on subchannel n is represented by
Ri,k,n =B
Nlog2(1 + Γ·SINRi,k,n), (2)
where B is the total bandwidth and Γ indicates the SINR gap under a given bit error rate (BER),
which is defined as Γ = −1.5/ln(BER) [25].
First, we consider the traditional capacity maximization problem with QoS. In the downlink
OFDM system, time is divided into a sequence of time slots. In each time slot, the objective is
to determine a data rate matrix R from the instantaneous feasible data rate region R(g), where g
represents the corresponding matrix of CSI at the centralized controller of the network. Ri,k denotes
the data rate of user k in cell i, which is an element of matrix R.
We adopt a gradient-based scheduling framework [26] in order to ensure QoS across the system
and the resultant objective function is
maxR∈R(g)
∑i∈S
∑k∈Mi
∂Ui,k(Wi,k)
∂Wi,kRi,k, (3)
where Wi,k is the average throughput of user k in cell i up to the current time slot and Ui,k(Wi,k) is
the corresponding utility function. We adopt the utility function given in [26], which is
Ui,k(Wi,k) =
βi,kln(Wi,k), α = 0,
βi,k
α (Wαi,k), α ≤ 1, α = 0,
(4)
where βi,k is a QoS weight for user k in cell i and α is a factor for the fairness. From the
above definition, we find that different users can achieve different levels of data rate based on the
weights they are assigned. Besides, it is proved that the average throughput of the users converges
asymptotically to the weighted proportional fair capacity as time approaches to infinity [26].
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
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8 L. ZHOU ET AL.
Substituting (4) into (3), the problem appears
maxR∈R(g)
F (R) = maxR∈R(g)
∑i∈S
∑k∈Mi
wi,kRi,k, (5)
where F (R) is the weighted capacity of the system. wi,k is defined as the weight for user k in
cell i, which is computed by wi,k = βi,k(Wi,k)α−1 according to (4). The values of the parameters
indicate different requirements of the system. For instance, α = 0 means that the system focuses
on achieving proportional fairness, while α = 1 implies that the system aims to maximize the total
capacity. The utility can also be designed based on other parameters, such as delay or queue size.
Following the utility rules, the weights can be calculated from the gradient of the proposed utility
at the current time slot. In the objective function, the weight and channel gain are two time-varying
factors.
We consider F (R) as the function of p and x, say F (R(p,x)). x is a matrix holding the variables
for all users and subchannels. xi,k,n is a binary variable that indicates whether subchannel n has been
allocated to user k in cell i. xi,k,n = 1 means that subchannel n is assigned to user k in cell i, and
0 implies otherwise. Thus we have Ri,k =∑
n∈N xi,k,nRi,k,n. Inserting it into F (R), we get the
expression of F (R(p,x)), which is
F (R(p,x)) =∑i∈S
∑k∈Mi
wi,k
∑n∈N
xi,k,nRi,k,n. (6)
Now, we take the total power consumption into account and transform the above problem into a
QEE maximization problem. The overall power expenditure of the system is defined as
P (p,x) =∑i∈S
(γi∑k∈Mi
∑n∈N
xi,k,npi,k,n + PCi ), (7)
which is adopted in the previous work [27]. γi is defined as a power-amplifier inefficiency factor
of BS i. For instance, if γi = 5, it means that the power consumption for the PA is 5 times of the
total power transmitted from BS i. PCi is the static power consumption of BS i, including baseband
processing, cooling, battery backup, etc.
For the sake of simplicity, we define QEE as ηEE , which is the objective of the system. And, the
QEE maximization problem can be given as follows.
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QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 9
max ηEE =F (R(p,x))
P (p,x).
s.t. C1 : xi,k,n ∈ {0, 1},∀i ∈ S, k ∈ Mi, n ∈ N,
C2 : pi,k,n ≥ 0, ∀i ∈ S, k ∈ Mi, n ∈ N,
C3 :∑k∈Mi
xi,k,n ≤ 1, ∀i ∈ S, n ∈ N,
C4 :∑k∈Mi
∑n∈N
pi,k,n ≤ Pmaxi , ∀i ∈ S,
(8)
where C1 and C3 are boolean combinatorial constraints for subchannel allocation. C2 and
C4 represent the constraints for power allocation, including individual power and total power
constraints. The unit of ηEE is bits/Joule.
4. THE PROPOSED RESOURCE ALLOCATION SCHEME
The problem (8) is considered as a nonlinear fractional programming problem [28]. Meanwhile, it
is also a non-convex and NP-hard problem. Using exhaustive search to obtain the optimal solution
of this problem is computationally infeasible. Dinkelbach method is an efficient method to solve
large scale fractional programming problem for which its optimality and convergence properties
are established. This method is well-known to reduce the computational complexity while solving
large scale fractional programming problem compared to other nonlinear solvers. In [11, 29–31],
Dinkelbach method is used to develop an energy-efficient scheduling schme by solving a fractional
programming problem. The objective function of our problem is the ratio of one nonlinear and one
linear function and hence, Dinkelbach method is very appropriate to solve this problem. In this
section, we elaborate the Dinkelbach method which is named as IEESS. First, we transform the
original fractional programming problem into a parametric programming problem. And then, for
the given parameter, we solve the subchannel and power allocation problem using a dual based
method.
4.1. Parametric Programming Problem
First, in order to eliminate the fraction in the objective function, we transform the problem (8) into a
parametric programming problem. For notational simplicity, we define the feasible power allocation
set and subchannel allocation set by P and X respectively. Then, the optimal ηEE can be expressed
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
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10 L. ZHOU ET AL.
as
η∗EE =F (R(p∗, x∗))
P (p∗, x∗)= max
p∈P,x∈X
F (R(p,x))
P (p,x), (9)
where p∗ and x∗ are the optimal power allocation set and subchannel allocation set respectively.
Based on the former definitions, we give the necessary and sufficient conditions for achieving the
optimal solution in Theorem 4.1.
Theorem 4.1. The optimal solution (p∗,x∗) of (8) is obtained if and only if
maxp∈P,x∈X
[F (R(p,x))− η∗EEP (p,x)] = 0. (10)
Proof
See Appendix A.
Theorem 4.1 indicates that if we know the optimal ηEE in advance, we can search the optimal
solution in the solution space when F (R(p,x))− η∗EEP (p,x) = 0 is satisfied. For other non-
optimal feasible solutions of (p,x), F (R(p,x))− η∗EEP (p,x) = 0 holds. Otherwise, when we
only have a non-optimal ηEE at the beginning, we obtain F (R(p,x))− ηEEP (p,x) = 0 for all
feasible solutions of (p,x).
Next, we study the characteristics of the parametric function in terms of ηEE . We define
f(ηEE) = maxp∈P,x∈X
[F (R(p,x))− ηEEP (p,x)] and check the monotonicity property of f(ηEE).
Based on the definition, we give the monotonicity property in Theorem 4.2 and present the
corresponding proof.
Theorem 4.2. f(ηEE) is a rigorously monotonically decreasing function of ηEE .
Proof
See Appendix B.
From Theorem 4.1 and 4.2 , we have f(η∗EE) = 0 and f(ηEE) is a rigorously monotonically
decreasing. Thus, when ηEE < η∗EE , we have f(ηEE) > 0, otherwise when ηEE > η∗EE , we have
f(ηEE) < 0.
In the above two theorems, we get the optimal solution in terms of given ηEE . Then, we study
how f(ηEE) looks when we fix arbitrary feasible solution (p′, x′). We provide the resultant theorem
and its proof as follows.
Theorem 4.3. Let p′ ∈ P , x′ ∈ X and η′EE = F (R(p′, x′))/P (p′, x′), then f(η′EE) ≥ 0.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
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QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 11
Proof
See Appendix C.
Based on Theorem 4.1, the steps of IEESS are given in Algorithm 1. In the algorithm, we give the
convergence condition and the rule of updating the parameter ηEE . In each iteration, for the given
ηmEE , the optimization problem in (8) can be transformed into
max F (R(p,x))− ηmEEP (p,x)
s.t. C1− C4
(11)
Algorithm 1: Iterative Energy-Efficient Scheduling Scheme (IEESS)Set initial value of η0EE = 0.Set iteration number m = 0 and the tolerance value ϵ > 0 for the loop.Set the maximum number of iterations Imax and convergence flag Flag := false.while m < Imax do
Solve the optimization problem (11) with the given ηmEE and achieve the relative optimalsolution (pm,xm) using the proposed dual-based method.if |f(ηmEE)| = |F (R(p,x))− ηmEEP (pm, xm)| ≤ ϵ then
p∗ := pm,x∗ := xm.η∗EE := F (R(pm,xm))
P (pm,xm) .Flag := true.break.
elseηm+1EE := 1
2 (F (R(pm,xm))P (pm,xm) + ηmEE).
m := m+ 1.
Finally, based on the above three theorems, we give the convergence proof of the proposed IEESS
as follows.
Theorem 4.4. The proposed IEESS converges.
Proof
See Appendix D.
4.2. Dual Based Method for (11)
This subsection is to solve the problem (11), which is the crucial step in Algorithm 1. (11) is a mixed
integer programming problem. The objective function is concave while the constraint set is convex.
According to the conclusion of [32], the duality gap for multiuser spectrum optimization always
tends to zero as the number of subcarriers goes to infinity no matter the optimization problem is
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12 L. ZHOU ET AL.
convex or not. This concludes that we can obtain the optimal solution from the dual problem instead
of the primal one.
In order to formulate the dual problem, we give the Lagrangian function based on (11), which is
L(p,x,µ,λ) =∑i∈S
∑k∈Mi
wi,k
∑n∈N
xi,k,nRi,k,n − ηmEE
∑i∈S
(γi∑k∈Mi
∑n∈N
pi,k,n + PCi )
+∑i∈S
∑n∈N
λi,n(1−∑k∈Mi
xi,k,n) +∑i∈S
µi(Pmaxi −
∑k∈Mi
∑n∈N
pi,k,n),
(12)
where µ and λ represent Lagrange multipliers. The differentiation of the Lagrangian w.r.t. pi,k,n
can be expressed by
∂L
∂pi,k,n=
∑j∈S
wj,kxj,k,n∂Rj,k,n
∂pi,k,n− ηmEEγi − µi, ∀i ∈ S, k ∈ Mi, n ∈ N. (13)
First, we optimize over p given x and µ. Equating (13) to 0, we can get a set of equations as a
function of variable p, which is
−∑
j∈S,j =i
wj,kxj,k,npj,k,ngj,k,ngi,k,n(∑
c∈S,c=j
∑q∈Mc
pc,q,ngc,k,n + δ2)2 + Γpj,k,ngj,k,n(∑
c∈S,c =j
∑q∈Mc
pc,q,ngc,k,n + δ2)
+wi,kxi,k,ngi,k,n∑
j∈S,j =i
∑q∈Mj
pj,q,ngj,k,n + δ2− µiln2− ηmEEγiln2 = 0,∀i ∈ S, k ∈ Mi, n ∈ N.
(14)
The equations and the boundary constraints C4 are part of the Karush-Kuhn-Tucher (KKT)
conditions [33]. Due to the zero duality gap between the primal and dual problems, we obtain
the closed-form power allocation solution for all BSs by solving the KKT conditions. It is obvious
that the number of equations and variables in (14) are equal, so the equations can be solved and the
resulting power allocation is as follows.
p∗i,k,n = xi,k,nhi,k,n(x,µ), ∀i ∈ S, k ∈ Mi, n ∈ N, (15)
where hi,k,n is a function of x and µ for user k in cell i on subchannel n. The power allocation
policy is strongly related to the subchannel allocation policy. Given xi,k,n = 1, p∗i,k,n = hi,k,n(x,µ),
whereas given xi,k,n = 0, p∗i,k,n = 0.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
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QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 13
Inserting p∗ into the Lagrangian function L(p,x,µ,λ), we have
L(p∗,x,µ,λ) =∑i∈S
[ ∑k∈Mi
∑n∈N
xi,k,n(wi,kRi,k,n − ηmEEγip∗i,k,n − µip
∗i,k,n − λi,n)
+ µiPmaxi − ηmEEP
Ci +
∑n∈N
λi,n
].
(16)
Then we optimize L(p∗,x,µ,λ) over x. Consequently, the objective function of the dual problem
can be written as
L(µ,λ) = L(p∗,x∗,µ,λ) =∑i∈S
[ ∑k∈Mi
∑n∈N
(wi,kRi,k,n − ηmEEγip∗i,k,n − µip
∗i,k,n − λi,n)
+
+ µiPmaxi − ηmEEP
Ci +
∑n∈N
λi,n
].
(17)
Therefore, the dual problem can be expressed by
min L(µ,λ)
s.t. µ ≽ 0,λ ≽ 0.
(18)
As we stated before, there is no duality gap, thus we can obtain an optimal solution of (11) by
minimizing the dual objective function (17) over µ and λ.
From equation (17), we can obtain the following equation for µ ≽ 0
L(µ) = minλ≽0
L(µ,λ) =∑i∈S
µiPmaxi − ηmEE
∑i∈S
PCi +
∑i∈S
∑n∈N
λ∗i,n. (19)
For any channel n in cell i, the optimal λi,n can be obtained by solving
λ∗i,n(µ) = max
k∈Mi
wi,kRi,k,n − ηmEEγip∗i,k,n − µip
∗i,k,n. (20)
Furthermore, we can find that for cell i, subchannel n is assigned to user k∗(i, n) in the cell, which
is decided by
k∗(i, n) = arg maxk∈Mi
wi,kRi,k,n − ηmEEγip∗i,k,n − µip
∗i,k,n, ∀i ∈ S, n ∈ N. (21)
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
14 L. ZHOU ET AL.
The subchannel allocation matrix is obtained in the following way
xi,k,n =
1, if k = k∗(i, n),
0, else.
∀i ∈ S, n ∈ N. (22)
Finally, the problem appears to find the value of µ which minimizes L(µ). We use subgradient
search method to solve the problem, and give the parameter updating rule as follows.
µz+1i =
[µzi − τzi
(Pmaxi −
∑k∈Mi
∑n∈N
pzi,k,n)]+
, ∀i ∈ S, (23)
where z represents the iteration index and τzi is the step size in each iteration.
The rule for computing the step size is
τzi =
Lz−L∑
i∈S
(Pmax
i −∑
k∈Mi
∑n∈N pz
i,k,n
)2 , if µ is feasible,
ζ∑i∈S
(Pmax
i −∑
k∈Mi
∑n∈N pz
i,k,n
)2 , otherwise,
(24)
where ζ is a positive constant and L is current best value of the Lagrangian. We choose different
step sizes based on µ to speed up the convergence.
In each iteration of the inner loop, assignment of each subchannel requires one max operation
over M users. Hence, the complexity of each inner loop is O(MN). If I1 and I2 are the number of
iteration in outer and inner loops, total computational complexity of IEESS is O(I1I2MN). To a
large extent, the complexity of the solution is determined by the number of iterations in outer QEE
parameter search and inner subgradient search. Finally, the computational complexity of IEESS can
be expressed by O(I1I2SNM), where I1 is the iteration number of the outer loop and I2 is the
iteration number of the inner subgradient search. As a consequent, the complexity of the problem
is determined by the iteration numbers to a large extent, which means that we can obtain high
computational efficiency only if both the outer and inner algorithms can converge in a few iterations.
5. SIMULATION RESULT
To evaluate the proposed scheme, we carry out extensive simulations and discuss the results in this
section. The simulation setup follows the guidelines of 3GPP technical reports [34]. We consider
the spectrum bandwidth is 6 MHz, which corresponds to 32 subchannels. Each subchannel contains
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
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QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 15
12 subcarriers and occupies the total bandwidth of 180 KHz. The OFDM symbol duration is
66.67µs with a normal cyclic prefix (CP) of 4.7µs. Besides, we adopt the power consumption
model from [27] and set the value of circuit power consumption for m-BS and s-BS accordingly.
The parameters are listed in Table II. We consider Rayleigh fading to model the subchannels
between BSs and users. We assume that all users have identical QoS weights , with the parameter
βi,k = 1, ∀i, k. We take the results from monte carlo simulation of 5000 trials. In each trial, we
generate the locations of small cells and users randomly. We consider that the users move at a
low speed (< 3km/h) and the users are saturated with best effort traffic. We denote the maximum
transmit power of m-BS and s-BS by Pmaxm and Pmax
s respectively.
We compare our proposed scheme with EPAS and CMS in the simulation. For EPAS, we allocate
equal power to all subchannels and use the total transmit power of the BSs. For the sake of better
comparison, we take the subchannel allocation result obtained by IEESS for EPAS. For CMS, we
adopt equation (6) as the objective function while the constraints remain the same in (8). The
problem of capacity maximization can be written as follows.
max F (R(p,x)).
s.t. C1,C2,C3,C4.
(25)
We follow the same dual-based method as described in subsection (4.2) to obtain the final
subchannel allocation and power allocation for CMS.
Table II. Simulation parameters.
Parameter ValueNumber of Small Cells 4, 68, 10, 12, 14, 16
Number of Users per Cell 2, 3, 4, 5, 6Path Loss Model for Macrocell 128.1 + 37.6log10(R) dB, R in kmPath Loss Model for Small Cell 140.7 + 37.6log10(R) dB, R in km
Traffic Model for Users Best Effort TrafficMaximum Tx Power of m-BS(Pmax
m ) 46 dBmMaximum Tx Power of s-BS(Pmax
s ) 28, 32, 36, 40 dBmPA Inefficiency Factor of m-BS 2.63PA Inefficiency Factor of s-BS 5
Cell Radius of Macrocell 289 mTotal Bandwidth 6 MHz
Bandwidth per Subchannel 180 KHzNumber of Subchannels 32OFDM Symbol Duration 66.67µs
CP Duration 4.7µsBit Error Rate(BER) 10−3
Thermal Noise −174 dBm/Hz
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
16 L. ZHOU ET AL.
5.1. Convergence of IEESS and Subgradient Search
Figure 2 shows the evolution of IEESS under different maximum transmit power of s-BS. We
assume that the average number of users per cell is 4 so that there are total 20 users in the network.
For different values of maximum transmit power of s-BS, we can achieve different optimal ηEE .
Figure 3 illustrates the convergence of the proposed subgradient search when ηEE = 7.64× 105
bit/Joule. It is seen from the figure that F (R(p,x))− ηmEEP (p,x) converges to zero for different
values of maximum transmit power of s-BS. As shown in Figure 2 and Figure 3, both IEESS and
subgradient search converge within 10 iterations.
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9x 10
5
Number of iterations
QE
E(b
it/J
ou
le)
Ps
max=40dBm
Ps
max=36dBm
Ps
max=32dBm
Ps
max=28dBm
Figure 2. QEE versus the number of iterations in IEESS.
0 2 4 6 8 10 12−30
−25
−20
−15
−10
−5
0
Number of iterations
F(R
(p,x
))−
ηE
E
m P
(p,x
)
Ps
max=40dBm
Ps
max=36dBm
Ps
max=32dBm
Ps
max=28dBm
Figure 3. F (R(p,x))− ηmEEP (p,x) versus the number of iterationsfor ηEE = 7.64× 105 bit/Joule in subgradient search.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 17
5.2. Fairness Measure
In this subsection we measure the fairness of the proposed scheme. As we described in section 3, α
is the fairness factor in our system model. We use Jain’s fairness index [35] for the evaluation. The
fairness index can be expressed as
Fairness Index =(∑
i∈S
∑k∈Mi
∑n∈N xi,k,nRi,k,n)
2
M∑
i∈S
∑k∈Mi
∑n∈N(xi,k,nRi,k,n)2
. (26)
where the fairness index is constrained within the interval [0, 1]. The larger the index, the better the
fairness. We assume that there are 4 small cells in the network and each small cell has 4 users. From
Figure 4, we observe that the fairness index decreases with the increasing value of α, while the
average SE grows on the contrary. It proves that in IEESS, we can achieve maximum proportional
fairness at α = 0 and achieve maximum average network SE at α = 1. As a result, we can select
a certain value of α to obtain a tradeoff of fairness and average SE considering the requirements
practical systems.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α
Fa
irn
ess In
dex
(a) Fairness index versus α.
0 0.2 0.4 0.6 0.8 15
5.5
6
6.5
7
7.5
8
8.5
α
Ave
rag
e S
E(b
it/s
/Hz)
(b) Average SE versus α.
Figure 4. Fairness index and average SE versus α.
5.3. QEE and SE versus Maximum Transmit Power of s-BS
We compare the performance of IEESS with EPAS and CMS in terms of QEE and SE for different
values of maximum transmit power of s-BSs in Figure 5 and Figure 6. We assume that the number
of small cells is 4 and each small cell contains 4 users. Figure 5 shows that IEESS can obtain
much better QEE compared to EPAS and CMS. For IEESS, with the increasing Pmaxs , the QEE
increases at first, however later at a decreasing speed. When Pmaxs exceeds 36 dBm, the QEE
saturates to a certain level. The trends observed for EPAS and CMS are different. It is seen that
the QEE grows when Pmaxs is small for EPAS and CMS. However, after certain peaks, the QEE
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
18 L. ZHOU ET AL.
results of EPAS and CMS start to decrease because of the interference experienced by the users
and limitless transmit power consumption. However, the QEE of IEESS does not decrease due to
the energy-efficient power control. In Figure 6, we observe that the average SE of IEESS remains
constant when Pmaxs surpasses 42 dBm, while the average SE of EPAS decreases after the peak at 36
dBm. The performances of IEESS and CMS are close when Pmaxs is small while the gap increases
as the maximum transmit power increases. We also show the total transmit power consumption
for different Pmaxs s in Figure 7. We find that the total transmit power of IEESS is much lower
comparing with EPAS and CMS. When Pmaxs is greater than 36 dBm, the total transmit power of
IEESS remains constant while the total transmit power of CMS continues to increase. After a certain
point, more transmit power does not help to enhance the QEE. Besides, we find that the SE gain that
can be obtained by increasing Pmaxs is limited.
24 26 28 30 32 34 36 38 40 42 440
1
2
3
4
5
6
7
8
9x 10
5
Ps
max(dBm)
Ave
rag
e Q
EE
(bit/J
ou
le)
IEESS
EPAS
CMS
Figure 5. Average QEE versus Pmaxs , α = 0.2.
24 26 28 30 32 34 36 38 40 42 440
1
2
3
4
5
6
7
8
Ps
max(dBm)
Ave
rag
e S
E(b
it/s
/Hz)
IEESS
EPAS
CMS
Figure 6. Average SE versus Pmaxs , α = 0.2.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 19
24 26 28 30 32 34 36 38 40 42 4446
47
48
49
50
51
52
Ps
max(dBm)
Ave
rag
e t
ota
l tr
an
sm
it p
ow
er
co
nsu
mp
tio
n(d
Bm
)
IEESS
EPAS
CMS
Figure 7. Average total transmit power consumption versus Pmaxs , α = 0.2.
5.4. QEE and SE versus Number of Users with Fixed Number of Small Cells
Figure 8 and Figure 9 illustrate how the QEE and SE of the system change with the growing number
of users in the network when the number of small cells is fixed. We compare the performance
of different maximum transmit power of s-BS in the figures. As seen from the figures, the QEE
and SE of the system increase with the increasing number of users for all schemes. However, the
rate of increment decreases at the same time. To summarize, the performance improvement is not
significant with the growing number of users when the number of small cells is constant.
20 25 30 35 40 45 504.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5x 10
5
Number of users
Ave
rag
e Q
EE
(bit/J
ou
le)
IEESS,Ps
max=36dBm
EPAS,Ps
max=36dBm
CMS,Ps
max=36dBm
IEESS,Ps
max=32dBm
EPAS,Ps
max=32dBm
CMS,Ps
max=32dBm
Figure 8. Average QEE versus different number of users in the network,the number of small cells is 4, α = 0.2.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
20 L. ZHOU ET AL.
20 25 30 35 40 45 504
4.5
5
5.5
6
6.5
7
7.5
8
Number of users
Ave
rag
e S
E(b
it/s
/Hz)
IEESS,Ps
max=36dBm
EPAS,Ps
max=36dBm
CMS,Ps
max=36dBm
IEESS,Ps
max=32dBm
EPAS,Ps
max=32dBm
CMS,Ps
max=32dBm
Figure 9. Average SE versus different number of users in the network,the number of small cells is 4, α = 0.2.
5.5. QEE and SE versus Number of Small Cells with Fixed Number of Users per Cell
Figures 10 and 11 reveal the trends of QEE and SE with the increasing number of small cells. We
consider, the number of users in each small cell is 4. As a result, when the number of small cells
increases, the number of users increases consequently. As shown in Figure 10, the average QEE
of IEESS increases when the number of small cells is less than 10. Above 10, the average QEE
stays at a constant level with the increasing number of small cells. For EPAS, the QEE follows a
decreasing trend when the number of small cells is greater than 6. The trend of CMS follows IEESS
tightly when the number of small cells is less than 10, while after that the gap between IEESS and
CMS starts to increase. In Figure 11, we observe that the average SE increases with the growing
number of small cells for IEESS, while the rate of rise declines. The SE of CMS is the best and the
trend is similar to IEESS. For EPAS, the average SE slightly increases at first. But after a certain
point, it starts to decrease towards to zero. The trends are similar for different power levels of s-BS.
Figure 12 shows the total transmit power consumption of IEESS comparing with EPAS and CMS.
For EPAS and CMS, it illustrates that although the total power consumption increases with the
increasing number of small cells, the incrementing rate decreases, because more small cells bring
more interference and different objective functions lead to different transmit power consumption.
6. CONCLUSION
Having noticed the increasing demand of energy-efficient scheduling in OFDM-based networks,
we have proposed a scheme named IEESS for HetNets. The objective of our scheme is to allocate
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 21
4 6 8 10 12 14 160
2
4
6
8
10
12
14x 10
5
Number of small cells
Ave
rag
e Q
EE
(bit/J
ou
le)
IEESS,Ps
max=36dBm
EPAS,Ps
max=36dBm
CMS,Ps
max=36dBm
IEESS,Ps
max=32dBm
EPAS,Ps
max=32dBm
CMS,Ps
max=32dBm
Figure 10. Average QEE versus different number of small cells in the network,the number of users per cell is 4, α = 0.2.
4 6 8 10 12 14 160
5
10
15
20
25
Number of small cells
Ave
rag
e S
E(b
it/s
/Hz)
IEESS,Ps
max=36dBm
EPAS,Ps
max=36dBm
CMS,Ps
max=36dBm
IEESS,Ps
max=32dBm
EPAS,Ps
max=32dBm
CMS,Ps
max=32dBm
Figure 11. Average SE versus different number of small cells in the network,the number of users per cell is 4, α = 0.2.
4 6 8 10 12 14 1646
46.5
47
47.5
48
48.5
49
49.5
50
50.5
Number of small cells
Ave
rag
e t
ota
l tr
an
sm
it p
ow
er
co
nsu
mp
tio
n(d
Bm
)
IEESS,Ps
max=36dBm
EPAS,Ps
max=36dBm
CMS,Ps
max=36dBm
IEESS,Ps
max=32dBm
EPAS,Ps
max=32dBm
CMS,Ps
max=32dBm
Figure 12. Average total transmit power consumption versus different number of small cells in the network,the number of users per cell is 4, α = 0.2.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
22 L. ZHOU ET AL.
subchannels and power among the users such that it maximizes QEE of the system. Regarding the
QoS assurance, the weight of each user is determined in each scheduling slot following the gradient
based scheduling framework. In order to achieve QEE, first, we have formulated the problem as
a fractional programming problem. To solve this problem, we have converted it into a parametric
programming where the parameter QEE is updated in an iterative way. In each iteration, for the given
QEE, we have applied a dual based method to find the optimal solution of subchannel and power
allocation with QoS assurance. Our extensive simulations proved that parameter update process
and subgradient search achieve convergence in a few iterations. Furthermore, we showed that our
scheme has better performance in terms of QEE in comparison with the EPAS and CMS method.
In the future work, we will study the problem of uplink resource allocation in HetNets, which is a
challenging work, especially when we consider severe mutual interference in the dynamic system.
Further, the combination of downlink and uplink resource allocation might be a good point to further
enhance the system performance.
ACKNOWLEDGEMENT
This research was supported in part by the National Natural Science Foundation of China (Grant No.
61002032), 863 project (No. 2014AA01A701) and sponsored by the foundation of Science and Technology
on Information Transmission and Dissemination in Comm. Networks Lab, National Key Laboratory of Anti-
jamming Communication Technology.
A. APPENDIX: PROOF OF THEOREM 4.1
a) Assuming (p, x) is the optimal solution. If (31) holds, we have
F (R(p,x))− η∗EEP (p,x) ≤ F (R(p, x))− η∗EEP (p, x) = 0, ∀p ∈ P,x ∈ X . (27)
F (R(p,x))
P (p,x)≤ η∗EE ,∀p ∈ P,x ∈ X ,
F (R(p, x))
P (p, x)= η∗EE .
(28)
Therefore, (p, x) is the optimal solution of (8) as well. The sufficiency proof is completed.
b) If the optimal solution (p∗,x∗) is obtained, then we have
F (R(p,x))
P (p,x)≤ F (R(p∗, x∗))
P (p∗, x∗)= η∗EE , ∀p ∈ P,x ∈ X . (29)
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 23
Rearranging (29) we can get
F (R(p,x))− η∗EEP (p,x) ≤ 0,∀p ∈ P,x ∈ X
F (R(p∗, x∗))− η∗EEP (p∗, x∗) = 0(30)
Thus, we have
maxp∈P,x∈X
[F (R(p,x))− η∗EEP (p,x)] = F (R(p∗, x∗))− η∗EEP (p∗, x∗) = 0. (31)
The necessity proof is completed.
B. APPENDIX: PROOF OF THEOREM 4.2
Provided η′EE > η′′EE > 0, and the respective optimal solutions (p′,x′) and (p′′,x′′). We have
f(η′EE) = maxp∈P,x∈X
[F (R(p,x))− η′EEP (p,x)]
= F (R(p′, x′))− η′EEP (p′, x′)
< F (R(p′, x′))− η′′EEP (p′, x′)
≤ F (R(p′′, x′′))− η′′EEP (p′′, x′′)
= f(η′′EE)
(32)
C. APPENDIX: PROOF OF THEOREM 4.3
f(η′EE) = maxp∈P,x∈X
[F (R(p,x))− η′EEP (p,x)] ≥ F (R(p′, x′))− η′EEP (p′, x′) = 0. (33)
D. APPENDIX: PROOF OF THEOREM 4.4
Theorem 4.3 has proved that f(ηmEE) > 0. Using the update rule in Algorithm 1, we have
F (R(pm,xm)) = (2ηm+1EE − ηmEE)P (pm,xm). (34)
Therefore, we have
f(ηmEE) = F (R(p,x))− ηmEEP (pm, xm) = 2ηm+1EE P (pm,xm)− 2ηmEEP (pm,xm) > 0. (35)
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
24 L. ZHOU ET AL.
Since P (pm,xm) > 0, thus ηm+1EE > ηmEE . It means ηmEE is a increasing sequence. Then we need to prove
that limm→∞ηmEE = η∗EE . We prove by contradiction. If this is false, we would have limm→∞ηmEE =
ηEE < η∗EE . There exists a sequence ηmEE , and limm→∞f(ηmEE) = f(ηEE) = 0. According to Theorem
4.2, f(ηEE) is strictly monotonic decreasing , thus we have,
0 = f(ηEE) > f(η∗EE) = 0. (36)
The equation (36) is a contradiction. Consequently it results in limm→∞f(ηmEE) = f(η∗EE). Since the
f(ηEE) is continuous for ηEE , then we have limm→∞ηmEE = η∗EE .
REFERENCES
1. Lee HW and Chong S. Downlink resource allocation in multi-carrier systems: frequency-selective vs. equal power
allocation. IEEE Transactions on Wireless Communications 2008; 7(10):3738–3747.
2. Huang J, Subramanian VG, Agrawal R and Berry RA. Downlink scheduling and resource allocation for ofdm
systems. IEEE Transactions on Wireless Communications 2009; 8(1):288–296.
3. Sadr S, Anpalagan A and Raahemifar K. Radio resource allocation algorithms for the downlink of multiuser ofdm
communication systems. IEEE Communications Surveys & Tutorials 2009; 11(3):92–106.
4. Zhao W and Wang S. Joint subchannel and power allocation in multiuser ofdm systems with minimal rate
constraints. International Journal of Communication Systems 2014; 27(1):1–12.
5. Bai L, Chen C, Wu B and He J. Adaptive resource allocation for multicast orthogonal frequency division multiple
access systems with guaranteed ber and rate. International Journal of Communication Systems 2013; 26(7):912–
926.
6. Han C, Harrold T, Armour S, Krikidis I, Videv S, Grant PM, Haas H, Thompson JS, Ku I, Wang CX et al.
Green radio: radio techniques to enable energy-efficient wireless networks. IEEE Communications Magazine 2011;
49(6):46–54.
7. Rodoplu V and Meng TH. Bits-per-joule capacity of energy-limited wireless networks. IEEE Transactions on
Wireless Communications 2007; 6(3):857–865.
8. Nasimi M, Hashim F and Ng CK. Characterizing energy efficiency for heterogeneous cellular networks. In 2012
IEEE Student Conference on Research and Development (SCOReD). IEEE, 2012; 198–202.
9. De Domenico A, Calvanese Strinati E and Capone A. Enabling green cellular networks: A survey and outlook.
Computer Communications 2014; 37:5–24.
10. Oh E and Woo C. Performance analysis of dynamic channel allocation based on the greedy approach for orthogonal
frequency-division multiple access downlink systems. International Journal of Communication Systems 2012;
25(7):953–961.
11. Ng DWK, Lo ES and Schober R. Energy-efficient resource allocation in ofdma systems with large numbers of base
station antennas. IEEE Transactions on Wireless Communications 2012; 11(9):3292–3304.
12. Liao J, Qi Q, Li T, Cao Y, Zhu X and Wang J. An optimized qos scheme for ims-nemo in heterogeneous networks.
International Journal of Communication Systems 2012; 25(2):185–204. ISSN 1099-1131. doi:10.1002/dac.1263.
URL http://dx.doi.org/10.1002/dac.1263.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
QOS-AWARE ENERGY-EFFICIENT RESOURCE ALLOCATION IN HETNETS 25
13. Quek TQ, Cheung WC and Kountouris M. Energy efficiency analysis of two-tier heterogeneous networks. In 11th
European Wireless Conference 2011-Sustainable Wireless Technologies (European Wireless). VDE, 2011; 1–5.
14. Liu L, Cao X, Cheng Y, Du L, Song W and Wang Y. Energy-efficient capacity optimization in wireless networks.
In Proc. IEEE INFOCOM. 2014; .
15. Soh YS, Quek TQ, Kountouris M and Shin H. Energy efficient heterogeneous cellular networks. IEEE Journal on
Selected Areas in Communications 2013; 31(5):840–850.
16. Li C, Zhang J and Letaief KB. Energy efficiency analysis of small cell networks. In 2013 IEEE International
Conference on Communications (ICC). IEEE, 2013; 4404–4408.
17. Boiardi S, Capone A and Sanso B. Radio planning of energy-aware cellular networks. Computer Networks 2013;
57(13):2564–2577.
18. Cao D, Zhou S and Niu Z. Improving the energy efficiency of two-tier heterogeneous cellular networks through
partial spectrum reuse. IEEE Transactions on Wireless Communications 2013; 12(8):4129–4141.
19. Le LB, Niyato D, Hossain E, Kim DI and Hoang DT. Qos-aware and energy-efficient resource management in
ofdma femtocells. IEEE Transactions on Wireless Communications 2013; 12(1):180–194.
20. He G, Zhang S, Chen Y and Xu S. Spectrum efficiency and energy efficiency tradeoff for heterogeneous wireless
networks. In 2013 IEEE Wireless Communications and Networking Conference (WCNC). IEEE, 2013; 2570–2574.
21. Damnjanovic A, Montojo J, Wei Y, Ji T, Luo T, Vajapeyam M, Yoo T, Song O and Malladi D. A survey on 3gpp
heterogeneous networks. IEEE Wireless Communications 2011; 18(3):10–21.
22. Zhang X, Zhang Y, Yu R, Wang W and Guizani M. Enhancing spectral-energy efficiency forlte-advanced
heterogeneous networks: a users social pattern perspective. IEEE Wireless Communications 2014; 21(2):10–17.
23. Sun S and Jing Y. Channel training design in amplify-and-forward mimo relay networks. IEEE Transactions on
Wireless Communications 2011; 10(10):3380–3391.
24. Sun S and Jing Y. Training and decodings for cooperative network with multiple relays and receive antennas. IEEE
Transactions on Communications 2012; 60(6):1534–1544.
25. Fung CH, Yu W and Lim TJ. Precoding for the multiantenna downlink: Multiuser snr gap and optimal user ordering.
IEEE Transactions on Communications 2007; 55(1):188–197.
26. Agrawal R, Bedekar A, La RJ and Subramanian V. Class and channel condition based weighted proportional fair
scheduler. Proc ITC Teletraffic Engineering in the Internet Era 2001; 17:553–565.
27. Chen Y, Zhang S and Xu S. Impact of non-ideal efficiency on bits per joule performance of base station
transmissions. In 2011 IEEE 73rd Vehicular Technology Conference (VTC Spring). IEEE, 2011; 1–5.
28. Dinkelbach W. On nonlinear fractional programming. Management Science 1967; 13(7):492–498.
29. Ng DWK, Lo ES and Schober R. Energy-efficient resource allocation in multiuser ofdm systems with wireless
information and power transfer. In 2013 IEEE Wireless Communications and Networking Conference (WCNC).
IEEE, 2013; 3823–3828.
30. Isheden C, Chong Z, Jorswieck E and Fettweis G. Framework for link-level energy efficiency optimization with
informed transmitter. IEEE Transactions on Wireless Communications 2012; 11(8):2946–2957.
31. Cheung KTK, Yang S and Hanzo L. Achieving maximum energy-efficiency in multi-relay ofdma cellular networks:
A fractional programming approach. IEEE Transactions on Communications 2013; 61(7):2746–2757.
32. Yu W and Lui R. Dual methods for nonconvex spectrum optimization of multicarrier systems. IEEE Transactions
on Communications 2006; 54(7):1310–1322.
33. Boyd S and Vandenberghe L. Convex optimization. Cambridge university press, 2009.
34. 3GPP. Evolved universal terrestrial radio access (E-UTRA): Radio frequency(RF) system scenarios. TR 36942
V11 0 0 2012; URL http://www.3gpp.org/ftp/specs.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac
26 L. ZHOU ET AL.
35. Jain R, Chiu DM and Hawe WR. A quantitative measure of fairness and discrimination for resource allocation in
shared computer system. Eastern Research Laboratory, Digital Equipment Corporation Hudson, MA, 1984.
Copyright c⃝ 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. (2014)
Prepared using dacauth.cls DOI: 10.1002/dac