QoS-Aware Energy-Efﬁcient Resource Allocation in … and EE are not contradictory in HetNets compared to conventional cellular networks without interference. Due to the unplanned

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]

Copyright c⃝ 2014 John Wiley & Sons, Ltd.

Prepared using dacauth.cls [Version: 2010/03/27 v2.00]

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)

Prepared using dacauth.cls DOI: 10.1002/dac

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



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




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.



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




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].



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.




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



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.




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



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.




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)



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




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



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.




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



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.




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.



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




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


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


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.



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)




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)



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 .

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Documents

QoS-Aware Energy-Efﬁcient Resource Allocation in … and EE are not contradictory in HetNets compared to conventional cellular networks without interference. Due to the unplanned