Resource Allocation Scheme for Energy Saving in Heterogeneous … · 2019-06-22 · in Heterogeneous Networks Xun Sun and Shaowei Wang, Senior Member, IEEE Abstract—Energy efﬁciency

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 8, AUGUST 2015 4407

Resource Allocation Scheme for Energy Savingin Heterogeneous Networks

Xun Sun and Shaowei Wang, Senior Member, IEEE

Abstract—Energy efficiency in communication networks has re-ceived increasing attention in both industry and academia. In thispaper, we investigate the energy saving issue in a heterogeneousnetwork (HetNet), which is introduced to cellular radio networksto improve capacity and enhance coverage. A HetNet consists ofbase stations (BSs) with different transmission powers, resultingin systematic power control that is more complex than that in theconventional cellular networks. The main difficulty is addressingthe mutual interference between different kinds of BSs. In thispaper, we try to minimize the power consumption of an OFDM-based HetNet while satisfying all users’ rate requirements, aswell as considering the inter-cell interference. Our general prob-lem formulation leads to a nonconvex optimization task that isgenerally hard to tackle. We derive a concave lower bound ofuser’s achievable rate for a given power allocation, based onwhich an efficient iterative algorithm is developed to solve theformulated problem efficiently. Numerical results show that ourproposed resource allocation scheme works well in different sce-narios. The energy consumption of the cellular system is reduceddramatically compared to other schemes. Moreover, our proposedalgorithm converges quickly and stably, showing great potentialfor applications.

Index Terms—Energy efficiency, heterogeneous network, inter-ference management, nonconvex optimization.

I. INTRODUCTION

W IRELESS data traffic is dramatically growing and themonthly demand is forecasted to reach 6.3 EB on

2015, a 26-fold increase over 2010 [1]. However, as a scarcenatural resource, radio spectrum is very crowded in the bandfor mobile communications, which is becoming a bottleneck todevelop diverse wireless applications. Though some promisingspectrum utilization schemes have been proposed to addressspectrum crisis [2], [3], most of them are far from implemen-tation because of technical limitations. A practical solution isHeterogeneous Network (HetNet), which consists of various ra-dio access nodes, such as macro Base Stations (BSs), pico BSs,femto BSs and relays. These nodes have different capacitiesand operating functionalities, which can potentially improve

Manuscript received December 23, 2013; revised September 17, 2014,January 5, 2015, and March 19, 2015; accepted March 25, 2015. Date ofpublication April 6, 2015; date of current version August 10, 2015. Thisresearch was partially supported by the NSFC and JiangsuSF. Part of this workwas presented at the IEEE WCNC 2013, Shanghai, China, April 7–10, 2013.The associate editor coordinating the review of this paper and approving it forpublication was S. Cui. (Corresponding author: Shaowei Wang).

The authors are with the School of Electronic Science and Engineering,Nanjing University, Nanjing 210023, China (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2015.2420558

spectrum efficiency significantly by enhancing area spectrumreuse. On the other hand, HetNet also entails a new paradigmshifting from traditional centralized macro BS only frameworkto a more autonomous, uncoordinated, and intelligent one,resulting in more complex interference management and powercontrol issues that should be properly addressed.

Besides improving system throughput, HetNet also has apotential from the viewpoint of energy saving. Statistical datashow that Information and Communication Technology (ICT)industry is an increasingly key contributor with an 8% of theworldwide energy consumption in 2008, and is expected todouble by 2020. Particularly, mobile communication systemsoccupy 0.5% of global energy consumption [4]. The potentialenergy saving of the HetNet comes from the fact that low poweraccess nodes, such as pico BSs, femto BSs and relays, aregenerally much closer to end users than macro BSs, thus theradio links between these low power access nodes and the endusers suffer lower path losses as compared to the links betweenthe macro BSs and the end users. As a result, the low poweraccess nodes can not only offload the traffic of the macro BSs,but also reduce the total power consumption of the cellularsystem while satisfying the same users’ rate requirements.

Energy consumption issue has been investigated extensivelyin recent years. To increase the energy efficiency (EE) of thecellular systems, traffic-aware transmission strategies have beenproposed in [5]–[7], where underutilized BSs are recommendedto switch to sleep mode or be shut off during off-peak time of traf-fic loads. It is practical because the deployments of existingcellular networks are always designed for peak load traffics,leading to very inefficient usage of BSs during off-peak time. Thecell discontinuous transmission strategies proposed in theseworks are recognized as promising approaches to improve theEE of the cellular system. There are also energy saving schemesfrom the perspective of network deployment. In [8], bit/Jouleis introduced to evaluate the EE of a cognitive radio system,based on which an efficient resource allocation algorithm isdeveloped. In [9], the optimal BS density for both homogeneousand heterogeneous cellular networks is analyzed, where all BSstransmit data with fixed power and the objective is to minimizethe total energy cost. With variable BS transmission powers, theBS density should be re-optimized and the network energy con-sumption can be significantly reduced, as show in [10], whichfocuses on the energy efficient deployment for both homo-geneous and heterogeneous cellular networks under coverageperformance constraints by using stochastic geometry tools.

As a widely investigated topic in Orthogonal FrequencyDivision Multiplexing (OFDM)-based cellular networks, re-source allocation also plays an important role in interference

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4408 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 8, AUGUST 2015

management and energy saving. In [11], fundamental tradeoffbetween EE and spectral efficiency (SE) in the OFDM systemis studied. Energy-efficient spectrum sharing in heterogeneouscognitive radio network with femto BSs is investigated in [12],where Stackelberg game is employed to address the formu-lated problem. In [13], an energy-efficient power optimizationscheme is presented for interference-limited communication. In[14], non-cooperative game is introduced to perform subchan-nel assignment, followed by joint multi-cell power allocationto minimize the power consumption of each user. Generally,full channel state information (CSI) is necessary for the men-tioned resource allocation schemes. In a HetNet, femto BSsare typically consumer deployed (unplanned) access nodes forindoor application with a network backhaul facilitated by theconsumers home digital subscriber line (DSL) or cable modem[15]. However, due to the feedback latency and limited signaloverhead that can be exchanged among BSs, perfect CSI maynot usually be known by all BSs. In [16], the authors designeda game-theoretical resource allocation scheme considering bothEE and interference control in the HetNet with incompleteCSI. As for the femto BSs installed by users in an ad-hocmanner, the mutual interference between macro BSs and femtoBSs makes the resource allocation task challenging [17], [18].Interference management algorithms for femto BSs can befound in [19]–[21].

Most of the previous works focus on the improvement ofEE rather than directly reducing the energy consumption. Itis reasonable since maximizing the EE of the cellular networkcertainly reduces the energy consumption for a given systemthroughput target. As shown in [11], the optimal EE of anOFDM system can be reached in the case that the systemthroughput equals to a specific value. However, in most situa-tions, the total rate requirement of users is not equal to a specificone, which means that it is impossible to get the maximum EE.Moreover, even if the sum rate of all users happens to reachthe optimal point that yields the maximum EE, each user’s raterequirement may not be satisfied at this point. It is of greatconcern to satisfy the QoS requirements of users for serviceproviders. Therefore, it is more reasonable to minimize the totalenergy consumption of the cellular system while satisfying allusers’ QoS requirements, which differs from the objective ofmaximizing the system throughput and is also the motivationof this work. We try to save energy as much as possiblewhile keeping all users’ achievable rates above their requiredthresholds. By exploiting the properties of the lower bound ofthe user’s rate, the formulated nonconvex optimization task canbe relaxed to a convex optimization problem, which we developan iterative algorithm to address efficiently. At each iteration,we minimize the total power consumption to produce powerand subchannel allocation solutions with dual decomposition.It should be noted that in this work, perfect CSI exchangesbetween BSs are assumed, the study of imperfect CSI is out ofthe scope of this paper and the related research can be foundin [16]. Our proposed schemes not only reduces the energyconsumption of the system greatly but also converges quicklyand stably, which are verified by numerical results.

The rest of this paper is organized as follows. In Section II,we give system model and formulate our optimization task.

TABLE INOTATIONS

Fig. 1. An exemplary system setup, where the solid links and the dashed linksrepresent communication links and cochannel interfering links for users insystem.

In Section III, we propose resource allocation algorithms. Nu-merical results are given in Section IV, as well as discussions.Conclusion is given in Section V.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

The frequently used terminologies and notations are givenin Table I. Consider a heterogeneous cellular network shown inFig. 1. The sets of macro BSs and low power BSs (pico BSs inFig. 1) are denoted as Nmacro and Nlow, respectively. The totaltransmission power of BS n is limited to Pn,max. For practicalcellular systems, the maximum transmission power of a macroBS (e.g., 46 dBm) is much higher than that of a low powerBS (e.g., 30 dBm). As a result, the coverage area of the lowpower BS is much smaller than that of the macro BS. For thelow power BS, there are two kinds of access mode: open accessand closed subscriber groups (CSG) [22]. For the open accessmode, users are allowed to connect to either a macro BS or alow power BS. That is, if a user is within the coverage of a lowpower BS, it will be served by the low power BS (user 1, user 2in Fig. 1); otherwise, it will be served by the macro BS (users 3in Fig. 1). For the CSG mode, only subscribed users can connectto the low power BS. As shown in Fig. 1, user 4 is a subscribeduser which connects to the pico BS. However, user 5 can onlybe served by the macro BS even though it is in the coverage ofpico BS because it is not in the subscribed user group.

Denote K = {1, . . . ,K} and N = Nmacro ∪Nlow = {1,. . . , N} as the set of users and BSs, respectively. Each user can

SUN AND WANG: RESOURCE ALLOCATION SCHEME FOR ENERGY SAVING IN HETEROGENEOUS NETWORKS 4409

be served by only one BS. Denote Kn as the set of users asso-ciated with BS n, i.e., K = K1 ∪ · · · ∪ Kn and Kn ∩ Km = ∅,for n �= m. The total bandwidth is B and is available for bothmacro BSs and low power BSs. The bandwidth is divided into SOFDM subchannels. The set of subchannels is denoted as S ={1, . . . , S}. Each user k ∈ K requires a minimal transmissionrate Rk,min. We denote the channel power gain on subchannels from BS n to user k by gk,ns . The number of users servedby each BS is smaller than S, and each subchannel of anyBS can be allocated to only one user. We ignore the feedbacklatency in the message exchange and assume the BSs know{gk,ns , ∀n, ∀ s, ∀ k} precisely.

Each BS needs to determine: (i) which user should be sched-uled on subchannel n; and (ii) how much power should beallocated for the scheduled user on this subchannel. We definea binary variable An,s,k as subchannel assignment index,

An,k,s =

{1 subchannel s of BS n is allocated to user k,

0 otherwise.

We stack {An,k,s}Kk=1 into the vector An,s = [An,1,s, . . . ,An,K,s], and stack An,s’s of all users in BS n to a matrix An

column by column. Finally, we define a matrix A = [A1, . . . ,AN ], which indicates how subchannels are allocated to allusers. Since each subchannel for a BS can only be used by atmost one user, we have:∑

k∈Kn

An,k,s ≤ 1, ∀n ∈ N , s ∈ S. (1)

Define pns as the proportion of Pn,max allocated to subchan-nel s by BS n. We stack {pns }Nn=1 into an N × 1 vector ps =[p1s, . . . , p

Ns ]. Each BS has a power budget:

S∑s=1

pns ≤ 1, ∀n ∈ N . (2)

In this paper, we do not consider advanced multiuser de-tection or interference cancellation, and the interference fromother BSs is treated as noise. Obviously, if special interferencemanagement techniques were employed, the performance ofour proposed scheme could be improved further. For a givenpower vector ps, the rate of user k served by BS n can bewritten as

Rk =

S∑s=1

An,k,s ·Rk(ps), (3)

where Rk(ps) stands for the rate on subchannel s allocated touser k in BS n, and can be calculated as follows,

Rk(ps) =B

Sln

(1 +

γn,k,s(ps)

Γ

), (4)

where γn,k,s(ps) is the signal-to-interference-plus-noise ratio(SINR), and Γ represents the SINR gap to capacity which istypically a function of the desired bit error ratio (BER), the

coding gain and noise margin, e.g., Γ = − ln (5BER)1.5 in MQAM

[23]. We assume Γ=1 here for the study of modulation is out ofscope of this work. Particularly, γn,k,s(ps) can be calculated as

γn,k,s(ps) =gk,ns pnsPn,max∑

m �=n gk,ms pms Pm,max + σk

s

=pns

In,k,s(ps), (5)

where

In,k,s(ps) = xn,k,s +∑m �=n

pms yn,k,s,

xn,k,s =σks

gk,ns Pn,max

,

and

yn,k,s =gk,ms Pm,max

gk,ns Pn,max

.

where σks is the noise power spectrum density.

B. Problem Formulation

Stack {ps}Ss=1 into an N × S matrix P, and we refer to P asthe power allocation for the sake of convenience. Our target isto find the optimal A and P to minimize the total transmissionpower of BSs, while keeping the rate of each user abovea predefined threshold. Specifically, the objective function isexpressed as

f(A,P) =∑n∈N

∑s∈S

pnsPn,max. (6)

Mathematically, our optimization problem can be formulatedas follows,

minA,P

f(A,P)

s.t. C1 :∑S

s=1pns ≤ 1, ∀n ∈ N ,

C2 :∑k∈Kn

An,k,s ≤ 1, ∀n ∈ N , ∀ s ∈ S,

C3 : An,k,s ∈ {0, 1}, ∀n ∈ N , ∀ s ∈ S, ∀ k ∈ K,

C4 : Rk ≥ Rk,min, ∀ k. (7)

C1 is the power constraint of each BS. C4 specifies user’sminimum rate requirement Rk,min. It is notable that Rk is anonconvex function associated with power allocation variablespns ’s, which makes the formulated problem a nonconvex one.C2 and C3 are imposed to guarantee that each subchannel isused by only one user. Such optimization task of minimizingpower with rate constraints is closely related to the problemof maximizing rate with power constraints in [24], but the


TABLE IISUBCHANNEL ALLOCATION

interference introduced by other BSs makes this problem muchmore difficult.

III. OUR PROPOSED ALGORITHM

Equation (7) is a nonconvex problem since the transmitpower is coupled with integer variable An,k,s’s. The problemdefined by Eq. (7) is a mixed-integer nonlinear programming(MINLP), which is generally computationally intractable. Itis known that even if the simplified problem, in which sub-channel scheduling issue is eliminated by relaxing the integerconstraints to continuous ones, is hard to obtain the optimalsolution because the remaining optimization task is nonconvex.To find global optimal solution, we need to fully search thespace of the feasible power allocation for all BSs with a smallgranularity along with all possible combinations of subchannelallocations. Thus, it is infeasible for practical cellular sys-tems to solve Eq. (7) directly. In this paper, we deal with itwith a two-step procedure: subchannel assignment and poweroptimization.

A. Subchannel Assignment

For the case of open access, firstly we need to associateusers with BSs, where the total achievable rate that users canget from the BSs are adopted as the measure. This processis shown in line1–4 of Table II, where rk,n and rk,m denotethe maximum achievable rate that user k can get from macroBSs and low power BSs, respectively. Here the user rate fromone BS is calculated as the sum rate of all subchannels underthe condition of average power allocation. User k is served bythe BS that can supply the highest rate. In this way, the userset of each BS is determined. Such a BS association schemehas incorporated path-loss fading, shadowing and frequencyselective fading, which is more reasonable in realistic scenario.It is straightforward to extend our results to range extension,which increases the coverage of low power BS by adding apositive bias to their signal-strengths during BS association

[25]. For the case of CSG, user set of each BS is fixed, andthe subchannel assignment just starts from line 5 in Table II.

By initializing the power vector as [P0]n,s = (1/S), ∀n, ∀ s,we propose an efficient subchannel allocation method to figureout the binary variables An,k,s’s, specifying a subchannel as-signment for each BS n. Ωk is the set of subchannels occupiedby the kth user, which cannot be reused by other users servedby the same cell. The proposed subchannel allocation algorithmconsists of two steps: Step 1, we allocate subchannels to users tomeet their minimal rate requirements; Step 2, we allocate eachsubchannel in the remaining ones to the users with the highestSINR over it. The intuitiveness that lies in Step 1 is that theuser whose current rate is the farthest away from its target hasa priority to get a subchannel among all available ones. Andthe procedure continues until all users’ rate requirements aresatisfied. Preferably, the subchannel with the highest achievablerate associated with a user will be chosen at this step. At Step 2,each of the remaining subchannels is allocated to the user whohas the highest achievable rate over it to potentially maximizethe sum capacity of the system.

Notice that in [26], [27], subchannel assignment and powerallocation are implemented in an integral algorithm with a com-pact form. However, these algorithms have a prohibitively highcomputational complexity due to inner and outer loops. Also,it requires multiple information exchanges per slot betweenBSs to reflect the updated interference level followed by theupdated power. In this paper, we try to design a low complexityalgorithm without information exchanges during the running ofthe algorithm.

B. Power Allocation Optimization

For a given subchannel assignment A, the original problemdegenerates to the following power allocation one:

minP

∑n∈N

∑s∈S

pnsPn,max

s.t. C1 :∑S

s=1pns ≤ 1, ∀n ∈ N ,

C2 : Rk ≥ Rk,min, ∀ k. (8)

However, there is still no efficient algorithm to solve Eq. (8)because the user rate Rk is nonconvex, as a result of theexistence of interference. To tackle this problem, we establisha concave lower bound of user rate to approximate the originalproblem. If the lower bound is tight, we can obtain promisingsolutions to the considered optimization problem.

1) A Concave Lower Bound of User Rate: The concave lowerbound of user rate is associated with a given power allocation P∗

[27]. Denote p∗s as the sth column of P∗, and the corresponding

SINR and user rate are γn,k,s(p∗s) and Rn,k,s(p∗

s), respectively.Define

R∗n,k,s(e

q) = α∗n,k,s · ln (γn,k,s(eq)) + β∗

n,k,s, (9)

α∗n,k,s =

γn,k,s (p∗s)

Γ + γn,k,s (p∗s), (10)

β∗n,k,s = Rn,k,s (p∗

s)− α∗n,k,s · ln (γn,k,s(p∗

s)), (11)


where q is an N × 1 vector. Note that P∗ belongs to the setP+ = {P |∀n, ∀ s, pn,s ∈ (0, 1]}, to guarantee that ∀ k, ∀n, ∀s, γn,k,s(P∗

s) > 0, α∗n,k,s > 0.

To exploit the properties of R∗n,k,s(e

q), we first show thefollowing mathematical fact:

Fact 1: Given positive values z and z∗, we have

ln(1 + z) ≥ ln(1 + z∗) +z∗

1 + z∗(ln(z)− ln(z∗)) , (12)

where the equality holds when z = z∗ [28]. Then we can havethe following theorem:

Theorem 1: Rn,k,s(eq) ≥ R∗

n,k,s(eq) and the equality holds

when eq = p∗s; R

∗n,k,s(e

q) is a concave function of q.Proof: The first claim can be easily got by Fact.1.

Particularizing z and z∗ with γn,k,s(eq)/Γ and γn,k,s(p∗

s)/Γ,respectively. Rn,k,s(e

q) ≥ R∗n,k,s(e

q) follows and the equalityholds when eq = p∗

s.To prove the second claim, we expand R∗

n,k,s(eq) as

R∗n,k,s(e

q) = α∗n,k,s · (qn − ln (In,k,s(e

q))) + β∗n,k,s (13)

where qn is the nth entry of q. ln (In,k,s(eq)) = ln(xn,k,s +∑m �=n yn,k,se

qm) is a convex function of {qm}m �=n, details ofthis proof can be found in Appendix. Since α∗

n,k,s>0, it can beeasily shown that R∗

n,k,s(eq) is a concave function of q. �

Define

R∗k

(An,k,s, e

Q) = N∑n=1

S∑s=1

An,k,s ·R∗n,k,s(e

q), (14)

where Q is an N × S matrix whose the sth column is qs.Based on Theorem 1, we know that R∗

k(An,k,s, eQ) is a concave

function of Q, as well as a lower bound of Rk. Besides, thebound is tight when eQ = P∗. In this way, we can transform theoriginal nonconvex problem into a convex relaxation.

2) Power Allocation Algorithm: Define the sets P ε={P|∀n,∀ s, Pn,s ∈ [eε, 1]} and Qε = {∀n, ∀ s, qn,s ∈ [ε, 0]}, ε is aprescribed negative value. By replacing Rk with R∗

k(An,k,s,eQ), we can get a convex approximation to Eq. (8) as follows,

minQ

N∑n=1

S∑s=1

eqn,s · Pn,max

s.t. C1 : Q ∈ Qε,

C2 :∑S

s=1eqn,s ≤ 1, ∀n ∈ N ,

C3 : R∗k(An,k,s, e

Q) ≥ Rk,min, ∀ k. (15)

Once we get a feasible power allocation for the original prob-lem, define it as P∗ and solve Eq. (15) for optimal result Q0 ∈Qε. As Q0 is feasible for Eq. (15), and R∗

k(An,k,s, eQ0) ≤

Rk(An,k,s, eQ0), then P0 = eQ0 is guaranteed to be feasible for

Eq. (8). Since P∗ ∈ Pε, there must exist a unique element Q∗ ∈Qε which satisfies P∗ = eQ∗

. We can also find f(A, eQ0) ≤f(A, eQ∗) holds because Q0 and Q∗ are optimal and feasibleto Eq. (15), which means P0 is at least as good as P∗.

TABLE IIIPOWER ALLOCATION OPTIMIZATION ALGORITHM

Based on the analysis above, we proposed an iterativepower allocation algorithm for a given A, which is depicted inTable III, where i denotes the iteration number (i > 0), and Pi

denotes the tentative power allocation after the ith iteration. Ineach iteration, we solve Eq. (15) with P∗=Pi by a duality-basedalgorithm to get the optimal solution Q0, and set Pi+1 = eQ0 .

Then we develop a duality-based algorithm to solve Eq. (15).Denote the dual variables related to the power constraint of BSn and the rate constraint of user k by λn and νk, respectively.We stack all λn’s and νk’s into a vector ω = [λ1, . . . , λN , ν1,. . . , νK ].

Reformulate the dual function of Eq. (15) as

D(ω) = minQ∈Qε

N∑n=1

S∑s=1

eqn,sPn,max +

N∑n=1

λn

(S∑

s=1

eqn,s − 1

)

+

K∑k=1

νk(Rk,min −R∗

k(An,k,s, eQ))

= minQ∈Q

L(ω,Q), (16)

and the dual problem is

maxω

minQ

N∑n=1

S∑s=1

eqn,sPn,max +

N∑n=1

λn

(S∑

s=1

eqn,s − 1

)

+

K∑k=1

νk(Rk,min −R∗

k(An,k,s, eQ))

= maxω

D(ω). (17)

Define Qω = argminQ∈QεL(ω,Q). To solve Eq. (15), we

should first find out the optimal dual variable ω∗ = argmaxωD(ω), and then compute Qω∗ as the optimal solution toEq. (15). We can work out ω∗ and Qω∗ by using the methodproposed in [29]. To this end, we develop a duality-basedalgorithm as shown in Table IV. Here, i denotes the iterationnumber (i ≥ 0), ωi represents the dual variables produced afterthe ith iteration, Δ2 is a prescribed small positive values.

At the beginning, we initialize the dual variable ω0. Ineach iteration, Qωi can be computed by the gradient-projectionalgorithm proposed in the next section. When Qωi is workedout, we can update the dual variable ωi by solving Eq. (17).Equation (17) can be solved by ellipsoid method [30] or


TABLE IVTHE DUALITY-BASED ALGORITHM TO SOLVE EQ. (15)

TABLE VTHE GRADIENT-PROJECTION ALGORITHM

subgradient method [31]–[33]. In this paper, we use subgradientmethod to update the dual variables as follows,

λi+1n =

[λin + ϕi

(S∑

s=1

eqn,s − 1

)]+

, (18)

νi+1k =

[νik + ϕi

(Rk,min −R∗

k(An,k,s, eQ))]+

, (19)

where the superscript i indicates the associated dual variableis an entry in ωi, [x]+ = max{0, x}, and ϕi is a sufficientlysmall positive step size for the ith iteration. For example, ϕi

can be determined in a diminishing way, such as ϕi = 0.1√i. The

convergence of this duality-based algorithm is guaranteed if ϕi

is sufficiently small. The algorithm is terminated when ‖ωi −ωi−1‖ is smaller than a prescribed small positive value Δ2, thenQωi obtained in the last iteration can be taken as the optimalsolution to Eq. (15).

Other power updating methods can also be applied bysmoothing the objective function or optimizing step sizes, andsome of them may lead to even faster convergence. Theseadvanced updating techniques is beyond the scope of this paperand the readers interested in this aspect can refer to [34] andreferences therein.

3) Gradient-Projection Algorithm to Compute Qωi : To findQωi , we need to solve a constrained optimization problemover the convex set Qε, namely Qωi = argminQ∈Qε

L(ωi, Q).Because of the convexity of L(ωi, Q), we can adopt a gradient-projection based iterative algorithm, which features simplicityas well as guaranteed convergence to find Qωi [29].

The algorithm is shown in Table V, where t denotes theiteration times. The algorithm starts with initializing Qt by Q∗

if i = 0; otherwise by Qωi−1 . Then, Qt is iteratively updatedby Qt = [Qt−1 − τ ·B]Qε

, where B is a matrix containing thegradients of L(ωi, Q) with respect to every entry of Q, [·]Qε

isthe operator of projection into Qε, and τ is a prescribed small

TABLE VISIMULATION PARAMETERS

positive value that guarantees convergence of the gradient-projection based algorithm. By simple mathematical arrange-ment, every entry of Q is updated by

∀n, ∀ s, qn,s =

⎧⎪⎨⎪⎩ε if qn,s − τ · [B]n,s ≤ ε

0 if qn,s − τ · [B]n,s ≥ 0

qn,s − τ · [B]n,s otherwise

where [B]n,s = (∂L(ωi, Q))/∂qn,s is derived as

[B]n,s = eqn,sPn,max + λine

qn,s −K∑

k=1

νik

⎛⎜⎝An,k,sα

∗n,k,s

−N∑

m=1,m �=n

Am,k,sα∗m,k,s

eqn,sPn,maxgk,ns

N∑j=1,j �=m

eqj,sPj,maxgk,js + σk

s

⎞⎟⎠.

(20)

The iteration is terminated when ‖vec(Qt − Qt−1)‖ is smallerthan a prescribed value Δ3. Then Qt produced in the lastiteration is taken as Qωi .

IV. NUMERICAL EXPERIMENTS

Consider the downlink of an OFDM-based HetNet, wherepico BSs are placed with uniform intervals around a circle ofradius Rp = 400 m, and a macro BS is located at the centerof the circle. Simulation parameters are listed in Table VI andall results are obtained by averaging 1000 Monte Carlo experi-ments (unless otherwise specified).

We compare the energy consumption of the HetNet withdifferent modes (open access and CSG) and homogeneousnetwork that is only equipped with one macro BS. For theCSG mode, equal number of users are uniformly distributed inthe macrocell and picocells, which can simulate the scenarioof dense users distribution. Each user has the same rate re-quirement in the HetNet and the homogeneous network. FromFig. 2 we can see that for both modes in HetNet, the energyconsumption of the HetNet is less than the homogeneous one.Moreover, the gap between them becomes larger as the numberof users grows. The main reason is that more users will beserved by the pico BSs as the number of users increases.These users consume much less energy because of the shorter


Fig. 2. Energy consumption of homogeneous network and heterogeneousnetwork with different numbers of users.

Fig. 3. Energy consumption of heterogeneous network with different numbersof pico BSs.

distance between users and the pico BSs which serve them. Itis worthwhile to note that the homogeneous network cannotsupply services to all users when the number of users reachesa threshold because the energy consumption exceeds the maxi-mum transmit power (40 W for the macro BS); in contrast, forthe CSG mode, the HetNet has sufficient power margin to servemore users. We can conclude that deploying pico BSs in theareas with dense users can greatly save the energy.

Notice that the power consumption between the two accessmodes in the HetNet differs greatly as can be seen from Fig. 2.The reason is that we associate some specific users with thepico BSs for the CSG mode while maximum achievable rateassociation is adopted for the open access mode, which resultsthat the number of users served by the pico BSs in the openaccess mode is less than that of the CSG mode. In Fig. 3, we cansee that the energy consumptions of the HetNet with different

TABLE VIIENERGY CONSUMPTION OF OUR ALGORITHM,

UPPER BOUND AND REFIM

number of pico BSs is also less than that of the homogeneousnetwork. However, since the pico BSs for the CSG mode is notaccessible for all users, the power consumption gap between thetwo access modes becomes smaller when the number of picoBSs grows because more users can be served by the pico BSseven for the open access mode.

To show the effectiveness of our proposed algorithm, it ismore convincing to compare our results to the optimum oneor other existing schemes. However, the problem Eq. (7) is NPhard, which is difficult or impossible to get the optimal solutioneven for a middle scale case. On the other hand, as far as theauthors known, previous works in the literature generally focuson maximizing the system throughput or the energy efficiency,which are different from the formulated problem of this work.Instead, we compare our results with an upper bound that isobtained in the following way.

Obtain an upper bound: It is easy to know that an upperbound can be worked out if all mutual interferences are reducedto the least. In this case, the SINR of user k in subchannel s ofBS n can be calculated as follows:

γn,k,s(ps) =gk,ns pnsPn,max

xks + σk

s

. (21)

From Eq. (21) we can see that only the lowest interference inthe corresponding subchannel from other BSs is involved. It istreated as a constant and denoted by xk

s . Take Eq. (21) back intoEq. (3) and Eq. (4), then the users’ rates can be calculated easily.Moreover, the objective function in Eq. (7) is convex under thiscondition. Then the original power allocation problem Eq. (7)becomes a convex one and can be solved by using Water-Filling(WF) method [35]. Note that the result obtained in this way is arelaxed solution and can only serve as an upper bound.

Table VII shows the energy consumption of our proposedalgorithm with the CSG mode, the upper bound and the sub-channel allocation scheme proposed in [26] (REFIM). Wecan see that the energy consumption of our algorithm differsslightly from the upper bound for the macro BS while the gapbetween our proposal and the REFIM is large. The pico BSsemploying the REFIM need about twice energy as much asour proposed algorithm. Fig. 4 shows the energy consumptionsof pico BSs (index 1), the macro BS (index 2) and the cellu-lar system (index 3). We can find that the gap between ourproposal and the upper bound is less than 7% for the macroBS. Furthermore, our proposed algorithm can reduce more than12% energy consumption as compared to that proposed in [26].We can conservatively conclude that our proposed algorithmcan produce solutions close to the upper bound and reduce theenergy consumption of the cellular system efficiently.


Fig. 4. Energy consumption of pico BSs, macro BS and the cellular system.

Fig. 5. Percentage of energy consumption as a function of the number of usersin each BS.

Fig. 5 shows the percentage of energy consumption for themacro BS and the pico BSs as the number of users served byeach BS increases with the CSG mode. The macro BS needs31% of the maximum power budge for the case of 15 users,which corresponds to 17% for the pico BSs. As the number ofusers served by each BS increases, the percentage of energyconsumption for the macro BS increases more quickly than thatfor the pico BSs. Even if the the macro BS uses up its powerbudget, the pico BSs still have power margin to serve moreusers. It can be explained that the pico BSs usually consumemuch less power compared to the macro BS to serve a user dueto the proximity to the user.

Fig. 6 illustrates the convergence performance of our pro-posed algorithm. When the convergence precision is 10−1, thenumber of iterations varies in a narrow range with an averageof 3. It is still stable when the convergence precision decreasesto 10−2, for which the average number is about 10. Fig. 7 de-picts the energy consumption of each BS during iterationswith different convergence precisions. We can observe that theenergy consumption of the macro BS decreases as the number

Fig. 6. Number of iterations with different convergence precision. Upper:Δ1 = 10−1; Lower: Δ1 = 10−2.

Fig. 7. Power consumption with different convergence precision. Upper:Δ1 = 10−1; Lower: Δ1 = 10−2.

of iteration increases. Though the transmit power of the picoBSs may increase in some certain iterations (even higher than100%), the sum power consumption of all BSs keeps decreasingas the number of iterations increases, and the final energyconsumptions of the pico BSs are all lower than 100% whenthe iteration ends as shown in Fig. 7.

V. CONCLUSION

In this paper, we studied the energy saving problem in het-erogeneous networks, which extended our preliminary research[36]. Our optimization objective is to minimize the total energyconsumption while guaranteeing the demands of users’ ratesand considering the mutual interference between different kindsof BSs. We give the lower bound of the user’s rate and proveits convexity, based on which we develop efficient algorithmsfor subchannel assignment and power distribution. The ef-fectiveness and the efficiency of our algorithms are validated


by numerical experiments. Since the problem formulation isbased on practical scenarios of cellular networks, our proposedresource allocation scheme is competitive for applications.

APPENDIX

We prove the second claim of Theorem 1, where the keyis to show that g(q) = ln (In,k,s(e

q)) = ln(xn,k,s +∑

m �=n

yn,k,seqm) is a convex function of q = [q1, . . . , qN−1]. De-

fine Θ= [θ1, . . . , θN−1]T = [yn,k,se

q1 , . . . , yn,k,seqN−1 ]T , and

η = xn,k,s +∑

i θi. With intuitive mathematical arrangements,we can derive the Hessian of g(q) with respect to q as

�2q g(q) = η−2

(η · diag(Θ)−ΘΘT

), (22)

where diag(Θ) is an N − 1×N − 1 diagonal matrix with theith diagonal entry equal to θi. For any vector v = [υ1, . . . ,υN−1]

T , we can show that

vT�2q g(q)v = η−2

(η∑

iθiυ

2i −

(∑iθiυi

)2)

> η−2

(∑iθi ·

∑iθiυ

2i −

(∑iθiυi

)2)

= η−2

(∑i(√

θi)2 ·∑

i(√θiυi)−

(∑iθiυi

)2)

≥ 0, (23)

where the last term follows from the Cauchy-Schwarz inequal-ity. This means that �2

qg(q) > 0, therefore, g(q) is convex withrespect to q.

ACKNOWLEDGMENT

The authors want to thank the associate editor and anony-mous reviewers for helpful comments and suggestions.

REFERENCES

[1] “ Cisco visual networking index: Global mobile data traffic forecastupdate, 2010–2015,” Cisco, San Jose, CA, USA, Feb. 2011. [Online].Available: http://www.cisco.com/en/US/solutions/collateral/ns341/ns525/ns537/ns705/ns827/whitepaperc11-520862.html

[2] M. Dohler, R. Heath, A. Lozano, C. Papadias, and R. Valenzuela, “Is thePHY layer dead?” IEEE Commun. Mag., vol. 49, no. 4, pp. 159–165,Apr. 2011.

[3] Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadbandsystems,” IEEE Commun. Mag., vol. 49, no. 6, pp. 101–107, Jun. 2011.

[4] M. Pickavet et al., “Worldwide energy needs for ICT: The rise of power-aware networking,” in Proc. IEEE ANTS, 2008 pp. 1–3.

[5] Z. Niu, “Tango: Traffic-aware network planning and green operation,”IEEE Wireless Commun., vol. 18, no. 5, pp. 25–29, Oct. 2011.

[6] K. Son, H. Kim, Y. Yi, and B. Krishnamachari, “Base station oper-ation and user association mechanisms for energy-delay tradeoffs ingreen cellular networks,” IEEE J. Sel. Areas Commun., vol. 29, no. 8,pp. 1525–1536, Sep. 2011.

[7] L. Correia et al., “Challenges and enabling technologies for energy awaremobile radio networks,” IEEE Commun. Mag., vol. 48, no. 11, pp. 66–72,Nov. 2010.

[8] S. Wang, M. Ge, and W. Zhao, “Energy-efficient resource allocation forOFDM-based cognitive radio networks,” IEEE Trans. Commun., vol. 61,no. 8, pp. 3181–3191, Aug. 2013.

[9] D. Cao, S. Zhou, and Z. Niu, “Optimal combination of base station densi-ties for energy-efficient two-tier heterogeneous cellular networks,” IEEETrans. Wireless Commun., vol. 12, no. 9, pp. 4350–4362, Sep. 2013.

[10] J. Peng, P. Hong, and K. Xue, “Energy-aware cellular deployment strat-egy under coverage performance constraints,” IEEE Trans. WirelessCommun., vol. 14, no. 1, pp. 69–80, Jan. 2015.

[11] C. Xiong, G. Li, S. Zhang, Y. Chen, and S. Xu, “Energy—And spectral-efficiency tradeoff in downlink OFDMA networks,” IEEE Trans. WirelessCommun., vol. 10, no. 11, pp. 3874–3886, Nov. 2011.

[12] R. Xie, F. Yu, and H. Ji, “Energy-efficient spectrum sharing andpower allocation in cognitive radio femtocell networks,” in Proc. IEEEINFOCOM, Mar. 2012, pp. 1665–1673.

[13] G. Miao, N. Himayat, G. Li, and S. Talwar, “Distributed interference-aware energy-efficient power optimization,” IEEE Trans. WirelessCommun., vol. 10, no. 4, pp. 1323–1333, Apr. 2011.

[14] Z. Han, Z. Ji, and K. Liu, “Non-cooperative resource competition gameby virtual referee in multi-cell OFDMA networks,” IEEE J. Sel. AreasCommun., vol. 25, no. 6, pp. 1079–1090, Aug. 2007.

[15] A. Damnjanovic et al., “A survey on 3gpp heterogeneous networks,”IEEE Wireless Commun., vol. 18, no. 3, pp. 10–21, Jun. 2011.

[16] S. Bu, F. Yu, and H. Yanikomeroglu, “Interference-aware energy-efficientresource allocation for heterogeneous networks with incomplete chan-nel state information,” IEEE Trans. Veh. Technol., vol. 64, no. 3,pp. 1036–1050, Mar. 2014.

[17] “3G home nodeb study item technical report,” 3rd Generation PartnershipProject, Sophia Antipolis Cedex, TR25.820, v8.2.0, Aug. 2008.

[18] R. Giuliano, C. Monti, and P. Loreti, “WiMAX fractional frequencyreuse for rural environments,” IEEE Wireless Commun., vol. 15, no. 3,pp. 60–65, Jun. 2008.

[19] A. Gjendemsj, D. Gesbert, G. Oien, and S. Kiani, “Binary power controlfor sum rate maximization over multiple interfering links,” IEEE Trans.Wireless Commun., vol. 7, no. 8, pp. 3164–3173, Aug. 2008.

[20] L. Venturino, N. Prasad, and X. Wang, “Coordinated scheduling andpower allocation in downlink multicell OFDMA networks,” IEEE Trans.Veh. Technol., vol. 58, no. 6, pp. 2835–2848, Jul. 2009.

[21] K. Zheng, Y. Wang, C. Lin, X. Shen, and J. Wang, “Graph-based interfer-ence coordination scheme in orthogonal frequency-division multiplexingaccess femtocell networks,” IET Commun., vol. 5, no. 17, pp. 2533–2541,Nov. 2011.

[22] S. Parkvall, A. Furuskar, and E. Dahlman, “Evolution of LTE towardIMT-advanced,” IEEE Commun. Mag., vol. 49, no. 2, pp. 84–91,Feb. 2011.

[23] A. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM forfading channels,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1218–1230,Oct. 1997.

[24] J. Bazerque and G. Giannakis, “Distributed scheduling and resource allo-cation for cognitive OFDMA radios,” in Proc. IEEE CROWNCOM, 2007,pp. 343–350.

[25] I. Guvenc, “Capacity and fairness analysis of heterogeneous networkswith range expansion and interference coordination,” IEEE Commun.Lett., vol. 15, no. 10, pp. 1084–1087, Oct. 2011.

[26] K. Son, S. Lee, Y. Yi, and S. Chong, “REFIM: A practical interferencemanagement in heterogeneous wireless access networks,” IEEE J. Sel.Areas Commun., vol. 29, no. 6, pp. 1260–1272, Jun. 2011.

[27] T. Wang and L. Vandendorpe, “Iterative resource allocation for maximiz-ing weighted sum min-rate in downlink cellular OFDMA systems,” IEEETrans. Signal Process., vol. 59, no. 1, pp. 223–234, Jan. 2011.

[28] J. Papandriopoulos and J. Evans, “Low-complexity distributed algorithmsfor spectrum balancing in multi-user DSL networks,” in Proc. IEEE ICC,2006, pp. 3270–3275.

[29] D. P. Bertsekas, Nonlinear Programming, 2nd ed. Singapore: AthenaScientific, 2003.

[30] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimiza-tion of multicarrier systems,” IEEE Trans. Commun., vol. 54, no. 7,pp. 1310–1322, Jul. 2006.

[31] K. Seong, M. Mohseni, and J. Cioffi, “Optimal resource alloca-tion for OFDMA downlink systems,” in Proc. IEEE ISIT , 2006,pp. 1394–1398.

[32] N. Mokari, M. Javan, and K. Navaie, “Cross-layer resource allocation inOFDMA systems for heterogeneous traffic with imperfect CSI,” IEEETrans. Veh. Technol., vol. 59, no. 2, pp. 1011–1017, Feb. 2010.

[33] X. Zhou, G. Li, D. Li, D. Wang, and A. Soong, “Probabilistic resourceallocation for opportunistic spectrum access,” IEEE Trans. WirelessCommun., vol. 9, no. 9, pp. 2870–2879, Sep. 2010.

[34] P. Tsiaflakis, I. Necoara, J. Suykens, and M. Moonen, “Improved dualdecomposition based optimization for DSL dynamic spectrum manage-ment,” IEEE Trans. Signal Process., vol. 58, no. 4, pp. 2230–2245,Apr. 2010.

[35] D. Palomar and J. Fonollosa, “Practical algorithms for a family ofwaterfilling solutions,” IEEE Trans. Signal Process., vol. 53, no. 2,pp. 686–695, 2005.

[36] X. Sun and S. Wang, “Interference management for energy saving inheterogeneous networks,” in Proc. IEEE WCNC, 2013, pp. 1204–1208.


Xun Sun received the B.S. degree in communica-tion engineering from Nanjing University, China, in2012. He is currently pursuing the M.S. degree atthe School of Electronic Science and Engineering,Nanjing University, China. His research interestsinclude wireless communications and convex opti-mization. Currently, his research focuses on resourceallocation in wireless networks.

Shaowei Wang (S’06–M’07–SM’13) received theB.S., M.S. and Ph.D. degrees in electronic engineer-ing from Wuhan University, China, in 1997, 2003,and 2006, respectively. From 1997 to 2001, he waswith China Telecom as an R & D Scientist. Hehas been with the School of Electronic Science andEngineering at Nanjing University from 2006, China,where he is currently a full Professor. From 2012to 2013, he was also with Stanford University andThe University of British Columbia as a VisitingScholar/Professor. His research focuses on wireless

communications and networking. In these areas he has published more than60 papers in leading journals and conference proceedings. He organized theSpecial Issue on Enhancing Spectral Efficiency for LTE-Advanced and BeyondCellular Networks for IEEE Wireless Communications, and the Feature Topicon Energy-Efficient Cognitive Radio Networks for IEEE CommunicationsMagazine. He is on the editorial board of IEEE Communications Magazine, andserves/served on the technical or executive committee of reputable conferencesincluding IEEE INFOCOM, IEEE ICC, IEEE GLOBECOM, IEEE WCNC etc.

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Resource Allocation Scheme for Energy Saving in Heterogeneous … · 2019-06-22 · in Heterogeneous Networks Xun Sun and Shaowei Wang, Senior Member, IEEE Abstract—Energy efﬁciency