IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. …bbcr.uwaterloo.ca/~xshen/paper/2017/opciud.pdf · 2019. 9. 23. · IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 7, JULY 2017 4139

Optimal Power Control in Ultra-Dense Small CellNetworks: A Game-Theoretic Approach

Jianchao Zheng, Member, IEEE, Yuan Wu, Senior Member, IEEE, Ning Zhang, Member, IEEE,Haibo Zhou, Member, IEEE, Yueming Cai, Senior Member, IEEE, and Xuemin Shen, Fellow, IEEE

Abstract— In this paper, we study the power control problemfor interference management in the ultra-dense small cell net-works, which is formulated to maximize the sum-rate of all thesmall cells while keeping tolerable interference to the macrocellusers. We investigate the problem by proposing a novel gamewith dynamic pricing. Theoretically, we prove that the Nashequilibrium (NE) of the formulated game coincides with thestationary point of the original sum-rate maximization problem,which could be locally or globally optimal. Furthermore, we pro-pose a distributed iterative power control algorithm to convergeto the NE of the game with guaranteed convergence. To reduce theinformation exchange and computational complexity, we proposean approximation model for the original optimization problemby constructing the interfering domains, and accordingly designa local information-based iterative algorithm for updating eachsmall cell’s power strategy. Theoretic analysis shows that thelocal information-based power control algorithm can convergeto the NE of the game, which corresponds to the stationarypoint of the original sum-rate maximization problem. Finally,simulation results demonstrate that the proposed approach yieldsa significant transmission rate gain, compared with the existingbenchmark algorithms.

Index Terms— Interference management, power control,ultra-dense small cell networks, game theory, distributed algo-rithm, local information.

I. INTRODUCTION

FUELED by the rising popularity of smart phones andtablets, mobile wireless communication networks have

been experiencing an explosive growth of data traffic, whichis predicted to increase at least 1000 times over the nextdecade [1], [2]. Along with the explosion of mobile datatraffic, it is also expected that almost 50 billion wirelessdevices will be connected by 2020 [3]. Therefore, great

Manuscript received July 26, 2016; revised November 11, 2016; acceptedDecember 26, 2016. Date of publication December 29, 2016; date of currentversion July 10, 2017. This work was supported in part by the Project ofNatural Science Foundations of China under Grant 61301163 and Grant61301162 and in part by the Natural Science and Engineering ResearchCouncil, Canada. The associate editor coordinating the review of this paperand approving it for publication was P. Wang. (Corresponding author:Jianchao Zheng.)

J. Zheng and Y. Cai are with the College of Communications Engineering,PLA University of Science and Technology, Nanjing 210007, China (e-mail:[email protected]; [email protected]).

Y. Wu is with the College of Information Engineering, Zhejiang Universityof Technology, Hangzhou 310023, China (e-mail: [email protected]).

N. Zhang, H. Zhou, and X. Shen are with the Department ofElectrical and Computer Engineering, University of Waterloo, Waterloo,ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]; [email protected]).

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

Digital Object Identifier 10.1109/TWC.2016.2646346

effort has been made to explore new ways to improve thenetwork coverage and boost the network capacity. One of themost promising solutions to achieve such a goal is networkdensification by deploying more infrastructure nodes in theserving area [1], [2]. As a result, ultra-dense small cellnetworks have been proposed as a novel efficient networkingparadigm by densely deploying short-range, low-power, andlow-cost base stations (BSs) underlaying the legacy macro-cellular networks [4]–[6].

Ultra-dense networks offer unprecedented capacity bybringing the network close to mobile users. Theoretically,the overall capacity scales with the number of small cellsdeployed [1]. However, in a ultra-dense deployment, notonly desired signal strength but also interference from othercells increase due to the reuse of spectrum. Thus, inter-cellinterference has been considered as the key limiting factorfor the system capacity in ultra-dense small cell networks.Moreover, small cells are usually opportunistically and irreg-ularly deployed in the hotspots [7]. The ultra-dense andunplanned deployment of small cells has imposed a greatchallenge on interference management and radio resourceoptimization [8], [9].

It has been shown that efficient power control can greatlymitigate inter-cell interference and improve system capac-ity [10]. However, the power control problem is highlynon-convex due to the presence of inter-cell interference [11].Several optimization methods (e.g., geometric program-ming [12], fractional programming [13]–[15], and succes-sive convex approximation [16], [17]) have been applied totransform the non-convex power control problem into variousconvex problems, which can be solved by the standard convexoptimization techniques. Nevertheless, these solutions usuallynecessitate centralized controllers which may not work wellin ultra-dense small cell networks due to the huge signalingand computational overhead for collecting information from allcells. Therefore, distributed power control schemes outperformthe centralized counterparts due to less information exchangeand computations [18].

As is well-known, game theory is an important mathemati-cal tool to study and analyze the strategic interactions amongindividual decision makers in decentralized networks [19].Considering that BSs could be deployed by different ser-vice providers and individual users of different interests [2],we study a non-cooperative game-theoretic setting where eachBS distributively performs resource allocation to maximizeits own utility. Notice that, game theory-based power control

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4140 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 7, JULY 2017

has been widely studied in the literature. For the traditionalcode-division multiple access (CDMA) systems, [20], [21] usegame theory to design distributed power control schemes forindividual users that adjust transmit power strategies to maxi-mize their own utilities. For the orthogonal frequency divisionmultiple access (OFDMA)-based cellular networks, [22], [23]formulate power control games to mitigate the co-channelinterference and improve the system throughput.

Compared with the centralized power allocation algorithms,these game-theoretic distributed schemes require less or evenno information exchange. However, the aforementionedworks [20]–[23] lack a mechanism to control the amount ofaggregate interference imposed on macrocells, and thus cannotguarantee the quality of service (QoS) of macrocell users.Reference [24] formulates a novel game-theoretic frameworksubject to the interference temperature constraints. By utilizingthe equivalence between the variational inequality and con-strained game theory, [24], [25] devise several distributed iter-ative waterfilling algorithms to find the Nash equilibrium (NE).Meanwhile, Stackelberg game has also been applied to capturethe hierarchical resource competition in two-tier heterogeneouscellular networks. In [26] and [27], price-based single-leadermulti-follower Stackelberg games are formulated, where themacrocell as the game leader maximizes its monetary revenueby pricing the interference from the femtocells. However, allthese studies are based on the common assumption of globalinformation, where all the parameters involved in the optimiza-tion problem can be obtained. Due to the limited coordinationin large-scale dense networks, information incompletenessusually arises and the assumption of global information maynot always hold in practice [28], [29].

Moreover, a number of pricing schemes have been pro-posed in [27] and [30]–[35] to improve the efficiency ofthe NE. Reference [31] studies a non-cooperative powercontrol game in a single-cell system, and shows the NE ofthe game is unique and Pareto-efficient. By setting dynamicprices for users, various design goals can be achieved.However, in multicell settings where transmit powers of dif-ferent cells need to be jointly optimized, inter-cell interfer-ence cannot simply be treated as noise, making the solutionobtained by [31] inapplicable. Ngo et al. [32] propose twoPareto-optimal power control algorithms for maximizingthe total utility of both macrocell and femtocell networkswhile robustly protecting the prescribed minimum signal-to-interference-plus-noise ratios (SINRs) of macrocell users.However, it remains unclear about the gap between thesesolutions and the global optima for sum-rate maximization.

In this paper, the power control problem in ultra-denseheterogeneous small cell networks is formulated to maximizethe sum-rate of all the small cells while keeping tolerableinterference to the macrocell users. We investigate the problemby proposing a novel game with dynamic pricing, where eachsmall cell is modeled as a game player that independentlyadjusts its power strategy in a distributed manner. The maincontributions of this paper are summarized as follows:• In order to improve the efficiency of the game, we moti-

vate and encourage the cooperation among individualgame players (i.e., small cells) by properly designing the

Fig. 1. System model. For clarity and simplicity, only part of the interferinglinks are plotted.

utility for each player. Specifically, the proposed utilityfunction consists of two parts: its own transmission rateand the external effects to other players’ transmissionrates, which is adaptively adjusted by a dynamic price.

• Theoretically, we have proved that the NE of the for-mulated game coincides with the stationary point of theoriginal sum-rate maximization problem, which could belocally or globally optimal. Moreover, we have devised adistributed iterative power control algorithm to convergeto the NE of the game.

• To reduce the information exchange and computationalcomplexity, we propose an approximation model for theoriginal sum-rate maximization problem by constructingthe interfering domains, and accordingly design a localinformation-based iterative algorithm for updating eachsmall cell’s power strategy. Both theoretic analysis andsimulation results are provided to validate and evaluatethe performance of the proposed approach.

The remainder of this paper is organized as follows.In Section II, we present the system model and problemformulation. In Section III, we propose a game-theoreticapproach for the distributed power control in ultra-denseheterogeneous small cell networks. In Section IV, we designa local information-based iterative algorithm to update eachsmall cell’s power strategy. In Section V, simulation resultsand discussions are presented. Concluding remarks are givenin Section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

We consider an OFDMA-based heterogeneous cellular net-work, where multiple small cells (such as micro-, pico- andfemtocells) underlay the macrocells, as shown in Figure 1.The sets of macrocells and small cells are denoted byM = {m1, m2, . . . , mL} and N = {1, 2, . . . , N}, respectively.In each cell, there are multiple mobile users served by aBS located in the center. This paper focuses on the uplinkcommunications.1 Each user and each BS are equipped withone transmit antenna and one receive antenna, respectively.

1Our approach can also be applied to the downlink communications withminor revisions.

ZHENG et al.: OPTIMAL POWER CONTROL IN ULTRA-DENSE SMALL CELL NETWORKS: A GAME-THEORETIC APPROACH 4141

TABLE I

SUMMATION OF USED NOTATIONS

Each user communicates with its associated BS in a single-hopfashion. We study the universal frequency reuse deploymentin which all macrocells/small cells share the whole spectrumto serve the mobile users.

In the OFDMA-based system, intra-cell multi-user accessis orthogonal, whereas inter-cell multi-user access is simplysuperposed owing to full reuse of spectrum. It is the super-position of the spectral resource slots (i.e., subchannels, orfrequency bands) that leads to severe inter-cell interference,which largely limits the system capacity. Thus, it is intuitive todecouple the optimization of resource allocations in differentspectral slots [22], [36]. In this paper, we focus on the powercontrol in a particular spectral slot.

For the sake of easy presentation, we use the transmitpower of small cell n (macrocell m) to refer to the transmitpower of the target mobile user which is served by smallcell n (macrocell m) in the considered spectral slot. Specif-ically, the uplink transmit power of small cell n ∈ N andmacrocell m ∈ M are denoted by pn and pm , respectively.In addition, the link power gain from the transmitter of cell i(i.e., the mobile user) to the receiver of cell j (i.e., itsassociated BS) is denoted by gi j =

(di j

)−κβi j , where di j is

the distance between the transmitter of cell i and the receiverof cell j , κ is the path loss exponent, and βi j is the randomfading coefficient. For better reading, Table I summarizes themainly used notations in this paper.

B. Problem Formulation

In the heterogeneous small cell network, in order to guaran-tee the QoS of macrocell users, we should protect them fromthe excessive interference caused by other cells using the samechannel [7], [9], [27], [37]. Interference temperature is usedto limit such a suffered interference from other cells, which isgiven by:

∑

i∈M \{m}pi gi,m +

∑

j∈N

p j g j,m ≤ ηm , ∀m ∈ M . (1)

Here,∑

i∈M \{m}pi gi,m represents the aggregated interference

from other macrocells,∑

j∈Np j g j,m denotes the interference

from small cells, and ηm denotes the interference temperaturelimit imposed by macrocell m.

Each small cell n ∈ N experiences interference from boththe macrocells and the other small cells, and thus the signal-to-interference-plus-noise ratio (SINR) at small cell n can beexpressed as:

SINRn = pngn,n∑

m∈Mpm gm,n + ∑

j∈N \{n}p j g j,n + σn

, (2)

where σn denotes the noise power at the receiver of small celln. As a result, the corresponding transmission rate achieved bysmall cell n can be given by the following Shannon’s formula:

Rn = log2 (1+ SINRn). (3)

In this paper, we aim to jointly optimize the transmit powervector of the small cells under the constraint of not exceedingthe interference temperature limit of macrocell users. Theoptimization vector is defined as follows.

Definition 1: A transmit power vector characterizes theset of uplink transmit power used by each small cell: p =(p1, . . . , pn, . . . , pN ), where

[p]

n = pn is the uplink transmitpower of small cell n. The maximal transmit power of smallcell n is given by Pmax

n .Then, the optimization problem can be formulated as:

(P1) : maxp

Rsum (p) =∑

n∈N

Rn (p), (4a)

s.t . 0 < pn < Pmaxn , ∀n ∈ N , (4b)

∑

i∈M \{m}pi gi,m +

∑

j∈N

p j g j,m ≤ ηm , ∀m ∈ M . (4c)

Due to the complicated interference relationship, the aboveoptimization problem (P1) is non-convex [11]. Thus, standardoptimization techniques cannot be directly applied to computethe globally optimal solution. Moreover, even if computationalissues could be resolved, to execute the computation neces-sitates a central controller to collect instantaneous channelstate information, which results in enormous signaling over-head, especially in ultra-dense small cell networks. Therefore,designing a local information-based distributed scheme to findthe optimal solution is urgently needed.

III. POWER CONTROL GAME WITH

GLOBAL INFORMATION

In this section, we discuss about the distributed optimizationof the above problem (P1) by using game theory [19], which isa powerful tool to analyze the interactions among distributeddecision makers and improve the performance of decentralizednetworks. We will first study the case of global informationin this section, and then discuss the case of local informationin the next section.


A. Game Model

In consideration of the self-organizing nature of small cells,we model each small cell as a self-interested game player.Specifically, the distributed power control across differentsmall cells is formulated as a game, formally denoted byG =

[N , {Pn}n∈N ,

{Ut

n

}n∈N , {Fn}n∈N

]. Symbol N is the set

of game players (i.e., small cells), Pn = {pn | 0 < pn ≤ Pmaxn }

is the set of available power strategy for player n, Utn is the

utility function of player n at time t , and Fn represents acorrespondence function for satisfaction of the constraint [38].A strategy profile of all the players is a vector, denoted byp = (p1, p2, . . . , pN ) ∈ P , where P = P1 ⊗ P2 ⊗ · · · ⊗ PN

represents the joint strategy space for all the players. Besides,the strategy profile of all the players excluding n is denotedby p−n = (p1, . . . , pn−1, pn+1 . . . , pN ) ∈ P−n , where P−n =P1 ⊗ · · · ⊗ Pn−1 ⊗ Pn+1 ⊗ · · · ⊗ PN .

In our game model, Fn : P−n → Sn is a mapping whichdetermines the set of player n’s satisfied actions givenother players’ actions. In order to guarantee the QoS ofmacrocell users, we define the correspondence by Fn (p−n) ={

pn ∈ Pn : ∑

i∈M \{m}pi gi,m + ∑

j∈Np j g j,m ≤ ηm,∀m ∈ M

}

.

Obviously, Sn = Fn (P−n) is a subset of the set Pn ,i.e., Sn ⊆ Pn .

In order to improve the efficiency of the game, we designeach player’s utility function at time t as:

Utn(pn, pt−n) = Rn(pn, pt−n)+ λt

n pn, (5)

where λtn =

∑

i∈N \{n}∂

∂ptn

Ri(pt

)can be regarded as a dynamic

price for the transmit power, and the superscript t is addedto specify the time t . In particular, we notice that this useddynamic pricing scheme is similar to the pigouvian tax inmicroeconomic theory [39], which is an effective approach toquantify and correct the negative externality due to a player’sindividual decision in a social welfare environment. In ourscenario, the key purpose of the proposed dynamic pricing isto correct the negative externality of each small cell’s powerstrategy to other small cells. Thus, by incorporating sucha pricing scheme, each small cell carefully determines itstransmit power by taking into account its impact on other cells.Such a reciprocal utility design can encourage the cooperationamong different cells, which will yield a locally or globallyoptimal solution to the sum-rate maximization problem byplaying a non-cooperative game.

With the utility function designed in (5), the power controlgame is expressed as:

(G) : max

pn∈Fn(pt−n)Ut

n

(pn, pt−n

), ∀n ∈ N , ∀t . (6)

Obviously, Utn(pn, pt−n) is a convex function with respect

to the variable pn . Thus, by solving the standard convexoptimization, we can achieve the (unique) optimal strategy forsmall cell n as follows:

B Rn(pt−n

)� arg max

pn∈Fn(pt−n)Ut

n

(pn, pt−n

), (7)

which is known as the best response of small cell n.

B. Analysis of NE

Definition 2 (Nash Equilibrium (NE)): A power allocationprofile pt = (

pt1, pt

2, . . . , ptN

) ∈ P is a pure-strategy NE ofthe game G, if and only if every player’s strategy is the best-response correspondence to others’ strategies, i.e.,

ptn = B Rn

(pt−n

), ∀n ∈ N . (8)

The above definition guarantees that no single player has anincentive to deviate from NE unilaterally.

Let B R(pt

) = (B R1

(pt−1

), . . . , B RN

(pt−N

))denote the

best-response vector of all players. According to (7), we canderive

B R(pt) = arg max

p∈PRsum

(p; pt), (9)

where P denotes the convex feasible set given by (4b) and (4c),and

Rsum(p; pt) �

N∑

n=1

Utn(pn, pt−n), (10)

which can be seen as a convex approximation of the systemsum-rate Rsum at pt .

Definition 3 (Stationary Point [40], [43]): A point pt is astationary point of the constrained optimization problem (P1)if the following condition is satisfied:

∇Rsum(pt )T (

pt − p) ≥ 0, ∀p ∈ P, (11)

where ∇ denotes the gradient operation with respect to thevariable p.

Condition (11) is the necessary condition for pt to belocally (and also globally) optimal. Every locally or globallyoptimal solution must be stationary. For convex problems,the stationary point coincides with the globally optimal solu-tion, and condition (11) is also sufficient for global optimalityof pt . For non-convex problems like (P1), it is challengingto achieve the global optimality. Thus, the stationary points,which are computational efficient to achieve, are generallydesired.

Theorem 1: Each stationary point of the optimization prob-lem (P1) is an NE of the game G, and each NE of the game Gis also a stationary point of the optimization problem (P1).

Proof: i) Assume pt is a stationary point of (P1)that satisfies the first-order optimality condition:∇Rsum

(pt

)T (pt − p

) ≥ 0,∀p ∈ P . According to (5) and (10),it is easy to know

∇ Rsum(pt ; pt) = ∇Rsum

(pt),

thus, we have

∇ Rsum(pt ; pt)T (

pt − p) ≥ 0, ∀p ∈ P . (12)

Since ∇ Rsum(; pt

)is convex, (12) indicates

∇ Rsum(p; pt ) ≤ ∇ Rsum

(pt ; pt) , ∀p ∈ P .

Therefore, Rsum(pt ; pt

) = maxp∈P

Rsum(p; pt

), and pt =

B R(pt

).


ii) Suppose that pt = B R(pt

). We readily get


pt − p) ≥ 0, ∀p ∈ P .

Thus, ∇Rsum(pt

) = ∇ Rsum(pt ; pt

)leads to

∇Rsum(pt )T (

pt − p) ≥ 0, ∀p ∈ P ,

which proves pt is a stationary point of (P1).Therefore, Theorem 1 is proved.According to the above analysis, the NE points exhibit very

desirable and attractive properties, and we know the globallyoptimal solution to the non-convex optimization problem (P1)is an NE point. Next, we will propose a distributed algorithmto find the NE of the game, namely, the stationary point ofthe optimization problem (P1).

C. Algorithm Design

The detailed algorithm process is illustrated as follows.Lemma 1: If pt is not the NE of the game G,˜B R

(pt

)− pt

is an ascent direction of the system sum-rate Rsum (p), i.e.,

∇Rsum(pt )T

(˜B R

(pt)− pt

)> 0. (13)

Proof: Based on the definition of B R(pt

),

Rsum(pt ; pt) ≤ Rsum

(B R

(pt ) ; pt). (14)

If Rsum(pt ; pt

) = Rsum(B R

(pt

) ; pt), we have pt =

B R(pt

), and thus pt is a stationary point of (P1) and the NE of

the game, which contradicts with the proposition. Therefore,Rsum

(pt ; pt

) = Rsum(B R

(pt

) ; pt), which with (14) yields

Rsum(pt ; pt) < Rsum

(B R

(pt ) ; pt).

Due to the convexity of ∇ Rsum(p; pt

),


B R(pt)− pt ) > 0.

Thus, ∇Rsum(pt

)T (B R

(pt

)− pt)

> 0.The proof is completed.Based on Lemma 1, we can immediately derive the follow-

ing theorem by following the similar proof in [16].Theorem 2: The iterative process of Algorithm 1 converges

to an NE of the game, namely, a stationary point of the sum-rate maximization problem (P1).

In the proposed algorithm, each small cell needsglobal information to compute the optimal power strategyB Rn

(pt−n

), since each small cell’s utility function (5) depends

on global information, e.g., all the small cells’ current powerstrategies and the global channel state information. Consider-ing that collecting global information would cause enormoussignaling overhead in ultra-dense small cell networks, we nextdesign a local information-based distributed algorithm to com-pute the optimal power strategy B Rn

(pt−n

).

IV. ACHIEVING OPTIMAL POWER STRATEGY

WITH LOCAL INFORMATION

In this section, we first propose an approximation model forthe optimization problem, and then design a low-complexity,distributed iterative algorithm to compute the optimal powerstrategy for each small cell based on local information.

Algorithm 1 Iterative Power Control for Achieving NEInitialization: At the initial time t = 0, set the initial powerstrategy p0

n , for all n ∈ N .1) Given the power allocation vector pt−1 =(

pt−11 , . . . , pt−1

N

), each small cell n ∈ N

independently computes its optimal power strategyB Rn

(pt−n

)by (7). Thus, the optimal power allocation

vector B R(pt−1

)can be achieved;

2) Update the power vector by

pt = pt−1 + αt(

B R(

pt−1)− pt−1

). (15)

Here, αt ∈ (0, 1] is the updating step size generatedby the decreasing rule proposed in [16], specifically,

αt = αt−1(1− σαt−1), (16)

where σ ∈ (0, 1) is a constant;3) Stop the algorithm if the termination condition is

satisfied, i.e., |Rsum(pt+1) − Rsum(pt )| < δ, where δis a predefined sufficiently small constant. Otherwise,set t ← t + 1, and go back to step 1).

A. Approximation Model

Due to the path loss, the signal transmitted by a givencell causes a significant co-channel interference only to thesurrounding cells [36]. Thus, by using DI to represent theinterfering distance, we define two interfering domains2 foreach cell i ∈ N ∪M :

• Interfering domain in the small cell tier:

Bi = {n ∈ N : n = i, dni ≤ DI }. (17)

• Interfering domain in the macrocell tier:

Ci = {m ∈ M : m = i, dmi ≤ DI }. (18)

Based on the defined interfering domains, the transmissionrate of small cell n ∈ N can be approximated by:

Rn = log2

⎛

⎜⎝1+ pngn,n∑

i∈Bn

pi gi,n + ∑

m∈Cn

pmgm,n + σn

⎞

⎟⎠. (19)

Then, the optimization problem (P1) can be transformed into:

(P2) : maxp

∑

n∈N

Rn (p) , (20a)

s.t . 0 < pn < Pmaxn , ∀n ∈ N , (20b)

∑

j∈Bm

p j g j,m +∑

i∈Cm

pi gi,m ≤ ηm , ∀m ∈ M . (20c)

2It is worth noting that other metrics in [22] and [36] can also be used todecide the interfering domains (e.g., based on the received power, or furtherconsidering the traffic load, etc.). However, the optimal construction of theinterference domains is not the focus of this work, and it would not causelarge variations in the following game model and the main conclusions.


Accordingly, the pricing factor λtn in (5) is approximated

by:

λtn =

∑

i∈Hn

∂

∂ptn

Ri(pt ) = −

∑

i∈Hn

Qti gn,i

ln 2(Qt

i + W ti

)W t

i

, (21)

where Qti = pt

igi,i denotes the received power of small

cell i , and W ti =

∑

j∈Bi

ptj g j,i + ∑

m∈Ci

pmgm,i + σi represents

the interference-plus-noise experienced by small cell i . SetHn represents the set of interfered small cells by small cell n.Notably, there always exists n ∈ Bi ⇔ i ∈ Hn .

Thus, the utility function of each small cell n ∈ N can beapproximated by:

U tn (pn, pt−n) = Rn(pn, pt−n)+ λt

n pn. (22)

Since U tn(pn, p−n) is also a convex function with respect to the

variable pn , we can achieve its unique optimal (best-response)strategy, denoted by:

B Rn(pt−n

)� arg max

pn∈Fn(pt−n)U t

n

(pn, pt−n

), (23)

where the correspondence function Fn (p−n) is definedbased on the interfering domains, specifically Fn (p−n) ={

pn ∈ Pn : ∑

j∈Bm

p j g j,m + ∑

i∈Cm

pi gi,m ≤ ηm ,∀m ∈ M

}

.

Let B R(pt

) = (B R1(pt−1

), . . . , B RN

(pt−N

)) denote the

best-response vector of all the small cells. According to (23),we can derive

B R(pt) = arg max

p∈PRsum

(p; pt), (24)

where P denotes the convex feasible set given by (20b)and (20c), and

Rsum(p; pt) �

N∑

n=1

U tn(pn, pt−n). (25)

B. Algorithm Design

Since the optimization objective Rsum(p; pt

)and the con-

straint set P are convex, we can adopt the standard convexoptimization techniques to achieve the best-response powerallocation vector B R

(pt

).

The Lagrangian dual function associated with optimizationobjective Rsum

(p; pt

)and constraint P is given by:

L(μ; pt) � max

p∈P

{

Rsum(p; pt)−

∑

m∈M

μmϕm (p)

}

, (26)

where ϕm (p) = ∑

i∈Cm

pi gi,m + ∑

j∈Bm

p j g j,m − ηm , and μ =(μ1, . . . , μM ) denotes the dual vector.

Since the approximate sum-rate function Rsum is separablein the individual’s variable pn , the computation of (26) can bedecoupled across different small cells. The optimal solutionof (26) can be denoted by �

(μ; pt

)�

(�n

(μ; pt

))Nn=1, where

�n(μ; pt

)represents small cell n’s unique optimal transmit

power given by:

�n(μ; pt) � arg max

pn∈Pn

⎧⎨

⎩U t

n

(pn; pt)−

∑

m∈Dn

μm pngn,m

⎫⎬

⎭.

(27)

Set Dn represents the set of interfered macrocells by smallcell n. Notice that there always exists n ∈ Bm ⇔ m ∈ Dn .

Thus, based on the dual-decomposition, the optimal powerallocation vector can be computed in a distributed manner,as shown in Algorithm 2, which belongs to the class of well-known gradient algorithms [41]. Each small cell autonomouslycomputes its optimal transmit power in a distributed manner,which only involves local information exchange among neigh-boring cells. Specifically, small cell n only needs the localinformation including:

• Its own power gain gn,n and the experienced interference-plus-noise W t

n .• The received power Qt

i and the interference-plus-noiseW t

i experienced by n’s interfered neighboring smallcells i ∈ Hn , and the power gains gn,i from n toi ∈ Hn .

• The power gains gn,m from n to n’s interfered neighbor-ing macrocells m ∈ Dn , and the price factor μm set by n’sinterfered neighboring macrocells.

Algorithm 2 Distributed Computation of Optimal PowerAllocation Vector Based on Local Information

Input: pt , {εv} > 0, μ0m ≥ 0 for all m ∈ M .

Loop for v = 0, 1, 2, . . .1) Distributed power computation: Each small cell n ∈

N distributedly computes its optimal transmit power�n

(μv ; pt

)by (27), and reports its calculated transmit

power to neighboring cells.2) Updating price factor: Each macrocell m ∈ M

distributedly updates its price factor μvm according to

ϕvm =

∑

i∈Cm

pi gti,m +

∑

n∈Bm

�n(μv ; pt) gt

n,m − ηm,

(28)

μv+1m = [

μvm + εvϕv

m

]+, (29)

where εv is the step size for updating the price factor,[·]+ denotes the Euclidean projection onto R

+, i.e.,[x]+ � max {0, x}.

End loop until the maximum number of iterations isreached.Output: The optimal power allocation vector B R

(pt

)�

(B Rn

(pt

))Nn=1, where B Rn

(pt

) = �n(μv ; pt

).

Besides, it is easy to know each small cell n’s computationalcomplexity for distributed power computation in Step 1)is O(|Hn | + |Dn |), and each macrocell m’s computationalcomplexity for updating price factor in Step 2) is O(|Bm | +|Cm |). Thus, the total computational complexity for eachiteration is O

(∑n∈N (|Hn | + |Dn|)+∑

m∈M (|Bm | + |Cm |)),


which depends on the scale of the interfering domains. In con-trast, if we adopt the global interference model (i.e., each cell’stransmission signal causes interference to all the other cells),the interfering domains will be |Hn | = N − 1, |Dn| = M ,|Bm | = N , and |Cm | = M−1, leading to a larger computationalcomplexity as O

(N2 + N M + M2

).

Theorem 3: If the step size {εv } is chosen such that εv > 0,εv → 0,

∑v εv = ∞, and

∑v (εv)2 < ∞, the sequence

of price factor {μv} generated by Algorithm 2 converges toa solution to the dual problem minμ≥0 L

(μ; pt

), and the

sequence{�

((μv) , pt

)}converges to the unique optimal

power allocation solution B R(pt

), defined in (24).

The proof of the theorem follows the standard convergenceresults of gradient projection algorithms [41], which is omittedhere. To guarantee convergence, the step-size sequence {εt } isgenerated by the decreasing rule

εt = εt−1(1− σεt−1), (30)

where σ ∈ (0, 1) is a constant. For more proper selection rulesof the step size sequence {εv }, interested readers are referredto [16] and [17] for more details.

By constructing the interfering domain, we present anotherapproximate function for the network sum-rate, and then usethe Lagrangian dual-decomposition technique to design a localinformation-based distributed algorithm (i.e., Algorithm 2) tocompute the optimal power allocation vector B R

(pt

). As ana-

lyzed before, the computation of B R(pt

)relies only on local

information and smaller computational complexity. In contrast,the computation of B R

(pt

)in the previous Section requires

global information and larger complexity, but it could guar-antee the convergence to a stationary solution, as proved inTheorems 1 and 2 before. Thus, the most important problemhere is to prove whether the proposed local information-basedapproximation approach can guarantee the optimality of thesolution or not. In what follows, we will provide an answerfor this question.

C. Convergence Analysis

Notably, the construction of the interfering domainsin (17) and (18) ignores the very weak interference fromremote BSs, which inevitably brings computation inaccuracy.In contrast, the computation of B R

(pt

)is based on the

exact interference relationship (including the very weak inter-ference). For the convenience of analysis, we model thecomputation inaccuracy by

∥∥B R

(pt

)− B R(pt

)∥∥ ≤ ξ t .Theorem 4: Suppose that the following conditions are

satisfied: i)∑

t αtξ t <∞; and ii) αt → 0. Then,{

Rsum(pt

)}

converges to a stationary solution which is eitherlocally or globally optimal.

Proof: For clear presentation, the proof is divided intotwo steps.

1) By definition (7), for a given pt ∈ P , B R(pt

)satisfies

the maximum principle: ∀p � (pn)Nn=1 ∈ �,

(B R

(pt )− p

)T∇ Rsum(B R

(pt) ; pt) ≥ 0.

That is,

N∑

n=1

(B Rn

(pt )− pn

) [∇pn fn(B Rn

(pt) , pt−n

)

+∑

i =n

∇pn fi(pt )−

N∑

i=1

∇pn hi(pt)− τn

(B Rn

(pt)− pt

n

)]

≥ 0, (31)

where we used the equation B R(pt

)�

(B Rn

(pt

))Nn=1.

Furthermore, by letting pn = ptn , and adding and subtract-

ing ∇pn fn(pt

)for each term n of the sum (31), we derive

N∑

n=1

(B Rn

(pt)− pt

n

) [∇pn fn(B Rn

(pt) , pt−n

)+∇Rsum(pt)

−∇pn fn (x)− τn(B Rn

(pt)− pt

n

)] ≥ 0,

and thus(B R

(pt )− pt)T∇Rsum

(pt)

≥N∑

n=1

(pt

n− B Rn(pt ))·[∇pn fn

(B Rn

(pt ), pt−n

)−∇pn fn(pt)

− τn(B Rn

(pt)− pt

n

)].

Since fn is a concave function, we have the following result:

N∑

n=1

(pt

n − B Rn(pt)) [∇pn fn

(B Rn

(pt ) , pt−n

)−∇pn fn(pt )]

≥ 0,

then we have

(B R

(pt)− pt )T∇Rsum

(pt) ≥

N∑

n=1

τn(B Rn

(pt )− pt

n

)2.

By defining cτ = minn∈N{τn} ≥ 0, we can get

(B R

(pt )− pt )T∇Rsum

(pt ) ≥ cτ

∥∥B R

(pt)− pt

∥∥2

. (32)

2) According to the Descent Lemma in Appendix and letting� (x) = −Rsum (x) (obviously L∇� = L∇R ), we have

Rsum (x1)− Rsum (x2) ≤ −∇Rsum(x1)T (x2 − x1)

+ L∇R

2‖x2 − x1‖2.

Thus, we obtain

Rsum (x2)− Rsum (x1) ≥ ∇Rsum(x1)T (x2 − x1)

− L∇R

2‖x2 − x1‖2.

For any given t ≥ 0, there exists

Rsum

(pt+1

)− Rsum

(pt)

≥ ∇Rsum(pt )T

(pt+1 − pt

)− L∇R

2

∥∥∥pt+1 − pt

∥∥∥

2

= αt∇Rsum(pt )T (

B R(pt )− pt)

− L∇R

2

(αt )2∥∥B R

(pt )− pt

∥∥2

, (33)


where AT denotes the transpose of the matrix A. Notice that,in the last equation in (33), we have used the iterative updatingrule pt+1 = pt + αt

(B R

(pt

)− pt).

Notably, the following inequality holds:∥∥B R

(pt)− B R

(pt )∥∥2 + ∥

∥B R(pt)− pt

∥∥2

≥ 1

2

∥∥B R

(pt)− pt

∥∥2

, (34)

which consequently leads to∥∥B R

(pt )− pt

∥∥2 ≤ 2

∥∥B R

(pt)− pt

∥∥2 + 2

(ξ t )2

.

Moreover, we have

∇Rsum(pt )T (

B R(pt)− pt )

= ∇Rsum(pt)T (

B R(pt )− B R

(pt))

+∇Rsum(pt )T (

B R(pt)− pt ). (35)

Since

∇Rsum(pt)T (

B R(pt )− B R

(pt))

≤∣∣∣∇Rsum

(pt )T (

B R(pt)− B R

(pt ))

∣∣∣

≤∥∥∥∇Rsum

(pt )T

∥∥∥

∥∥B R

(pt )− B R

(pt)∥∥ ≤ ξ t

∥∥∇Rsum

(pt)∥∥,

we can derive the following result:

∇Rsum(pt )T (

B R(pt)− B R

(pt )) ≥ −ξ t

∥∥∇Rsum

(pt )∥∥.

(36)

Then, (32), (35), and (36) lead to

∇Rsum(pt )T (

B R(pt )− pt)

≥ cτ

∥∥B R(pt)− pt

∥∥− ξ t∥∥∇Rsum

(pt)∥∥. (37)

Therefore, based on (33), (34), and (37), we can get

Rsum

(pt+1

)− Rsum

(pt)

≥ αt∇Rsum(pt)T (

B R(pt )−pt)− L∇R

2

(αt )2∥∥B R

(pt )−pt

∥∥2

≥ αt cτ

∥∥B R

(pt)− pt

∥∥− αtξ t

∥∥∇Rsum

(pt)∥∥

− L∇R(αt )2

(∥∥B R

(pt )− pt

∥∥2 + (

ξ t )2)

=(αt cτ − L∇R

(αt )2

) ∥∥B R

(pt)− pt

∥∥− θ t , (38)

where θ t = αtξ t∥∥∇Rsum

(pt

)∥∥ + L∇R(αtξ t

)2. Under theassumptions of the theorem, there exists

∑∞t=0 θ t < ∞.

Moreover, since αt → 0, we have αt cτ − L∇R(αt

)2> 0 when

t is sufficiently large.Thus, on the basis of Lemma 3 in Appendix, we have either{

Rsum(pt

)}→∞ or{

Rsum(pt

)}converges to a finite value.

Due to the limitation of system capacity,{

Rsum(pt

)}is a

bounded function. Thus, we know{

Rsum(pt

)}converges to

a finite value. That implies the sequence{pt

}converges to

a fixed point (i.e., NE) of the game. According to Lemma 1,we know

{Rsum

(pt

)}converges to a stationary solution which

is locally or globally optimal.According to Theorem 4, convergence is guaranteed if∑t αtξ t <∞ and αt → 0. We generate the step size sequence{αt } by the decreasing rule (30), which satisfies

∑t αt < ∞

Fig. 2. Network topology with 4 macrocells and 50 small cells. The trianglesdenote the macrocell BSs, and their coverage is plotted by large green circles.Besides, the coverage of small cells is plotted by small blue circles, andthe edge connecting two small cells represents their interference relationship.We assume that the interference exists only when the distance of two smallcells are smaller than the interfering distance DI = 600 m.

and αt → 0. Thus, only if the computation error sequence {ξ t }is upper bounded, the convergence conditions can be satisfied.In the approximation model, the construction of the interfer-ing domains in (17) and (18) only ignores the very weakinterference from remote BSs, which would not cause largecomputation error. Therefore, the proposed local information-based distributed power control approach can converge to astationary point of the sum-rate maximization problem, whichcould be locally or globally optimal. It is worth noting thatdevising distributed algorithms that definitely converge to theglobally optimal solution of the system-wide optimizationproblem is up to date an open problem that is worth to beinvestigated [24].

V. SIMULATION RESULTS

In this section, we evaluate the performance of the proposeddistributed power control algorithm in ultra-dense heteroge-neous small cell networks.

A. Simulation Settings

In the simulation, we study a densely deployed heteroge-neous cellular network with 4 macrocells and 50 small cellsin a 2000m-by-2000m square area, as shown in Figure 2.Each BS is located at the center of its serving cell, andmobile users are randomly generated in each cell according toa uniform distribution. Rayleigh fading model is consideredin the simulation, and the random fading coefficient β isexponentially distributed with unit-mean. The interferencetemperature limit is set to be η = 10−6 W. Further simulationparameters are summarized in Table II.

The average transmission rate achieved by each smallcell (the network sum-rate divided by the number of smallcells) is adopted as the main performance metric. For theproposed distributed power control algorithm, the step sizesequences {αt } and {ξ t } are both generated by the decreasingrule, and the parameters are selected to be α0 = ξ0 = 1, andσ = 10−2 [16]. The proximal parameter τn is set to 10 for all


TABLE II

PARAMETERS SETTING

Fig. 3. The evolution of average transmission rate of each small cell fordifferent power control schemes.

small cells. As for performance comparison, we also conductsimulations for the Stackelberg power control scheme in [27].

B. Convergence Analysis

For convergence evaluation of the proposed power controlscheme, Figure 3 shows the evolution of system transmissionrate achieved by each small cell for different numbers of smallcells. The system transmission rates achieved by both theproposed power control scheme and the Stackelberg powercontrol scheme are updated in each iteration and signifi-cantly improved at the convergence time. It can be seen thatthe Stackelberg power control scheme has a faster conver-gence speed, but it converges to a relatively inferior stablesolution. Although the convergence time for the proposedscheme can even get a little longer when the number ofsmall cells increases, the convergence can all be achievedwithin 50 iterations. In addition, when the number of smallcells increases (from 40 to 50), the system transmission rateachieved by each small cell is reduced due to more severeinterference.

Figure 4 shows the interference received by the 4 macro-cells updating versus the number of iterations. Specifically,the received interference of each macrocell updates with theiteration step and remains unchanged after about 50 iterations,thus validating the convergence of the proposed algorithm.In order to improve the transmission rate, the small cellsincrease the transmit power, which leads to the increase in thereceived interference at macrocells. However, we can see thatafter convergence, the received interference at each macrocell

Fig. 4. The evolution of received interference by macrocells when the numberof small cells is 50.

Fig. 5. The performance evaluation of average transmission rate for differentsolutions versus the number of small cells.

is below the interference limit η = 10−6 W. The reason canbe explained as follows: if a macrocell m receives interferencelarger than the interference limit, it will increase its pricingfactor μm which leads to the decrease of the transmit powerof its neighboring small cells.

C. Performance Evaluation

Figure 5 shows performance comparison results for thedifferent solutions versus the number of small cells. Thepresented results are obtained by simulating 100 independenttrials and then taking the average value. As shown in Figure 5,the average transmission rate achieved by each small celldecreases with the increase of the number of small cells dueto more severe interference. Moreover, it can be seen that theproposed power control scheme outperforms the Stackelbergpower control scheme in [27].

Figure 6 shows the performance of the proposed localinformation-based distributed power control algorithm withrespect to the interfering distance DI . The iterative powercontrol algorithm (illustrated in Algorithm 1) is consideredfor comparison, which is irrespective of the defined interferingdistance, since it collects global information for computation.


Fig. 6. The performance evaluation of average transmission rate for differentsolutions versus the scale of interfering domain.

As shown in Figure 6, the achieved average transmissionrate increases with the interfering distance. That is becausea larger scale of interfering domain can approximate the realinterference relationship more accurately. Thus, the obtainedsolution is closer to the optimal solution with minor computingerror, as proved in Theorem 4. However, when the interferingdistance increases, more algorithm cost (including informationexchange overhead and computational complexity) would beincurred, which is consistent with our analysis in Section IV.Therefore, there is a tradeoff between the signaling andcomputational cost and the performance gain of proposedsolution. Besides, we can see that when the interfering distanceis set to be larger than 600 m, the performance of thelow-complexity distributed algorithm with local informationexchange is quite close to the optimal solution obtained by theglobal information-based power control algorithm. Therefore,in consideration of both system performance and algorithmcost, it is quite cost-effective to set the interfering distance tobe 600 m, which achieves a near-optimal solution but withlower computational complexity and only a small amount oflocal information exchange.

In order to better understand the impact of ultra-dense onthe problem and the advantage of our proposed optimizationscheme, we plot the signaling overhead3 and computationalcomplexity for different optimization schemes in Figure 7and Figure 8, respectively. We can see from Figure 7 thatas the number of small cells increases (i.e., the small cellsare more densely deployed), the signaling overhead of theglobal information-based optimization scheme increases sig-nificantly, while the signaling overhead of the proposed localinformation-based power control scheme increases slowly.Figure 8 also clearly shows the advantage of our proposedlocal information-based optimization scheme in terms of effec-tively reducing the computational complexity.

Figure 9 shows the average transmission rate of eachsmall cell versus the interference temperature limit ηm .Intuitively, the performance of both schemes is improved

3In the simulations, we assume that power control is executed once every5 ms (the WiMAX frame duration) and transmitting a binary number (and afloating number) needs 1 bit (and 16 bits).

Fig. 7. Signaling overhead analysis.

Fig. 8. Computational complexity analysis.

Fig. 9. The performance evaluation of average transmission rate for differentsolutions versus the interference temperature limit.

with the increased interference temperature limit. However,the increase of the interference temperature limit would leadto the increase in the interference received by macrocell users,and thus reduce the macrocell transmission rate. Therefore,there is a tradeoff between the small-cell transmission rate andmacrocell transmission rate, which can be adjusted by properlysetting the interference temperature limit for the macrocell


users. Moreover, Figure 9 also demonstrates that our proposeddistributed power control scheme outperforms the Stackelbergpower control scheme.

VI. CONCLUSION

In this paper, we have proposed a novel game to studythe optimal power control for interference management inultra-dense small cell networks. By properly designing theutility function with dynamic pricing, different small cellsare motivated to cooperate with each other, leading to desir-able game outcomes. We have proved that the NE of thegame coincides with the stationary point of the sum-ratemaximization problem, which could be locally or globallyoptimal. Furthermore, a low-complexity distributed powercontrol algorithm, which involves only local informationexchange, has been proposed with guaranteed convergenceperformance. Simulation results have demonstrated the effec-tiveness of our proposed algorithms. For the future work,we will study the optimal power control in dynamic environ-ments, e.g., dynamic BS switching on/off and time-varyingchannels.

APPENDIX

Lemma 2 (Descent Lemma [40]): For a continuously differ-entiable function � : RN → R, if ∇� is Lipschitz continuouswith constant L∇�, then ∀x1, x2 ∈ R

N ,

� (x2)−� (x1) ≤ ∇�(x1)T (x2 − x1)+ L∇�

2‖x2 − x1‖2.

(39)

Lemma 3 (Robbins-Siegmund Lemma [42]): For threesequences of numbers

{Xt

},

{Y t

}, and

{Zt

}, if Y t ≥ 0,

Xt+1 ≥ Xt + Y t − Zt , ∀t = 0, 1, . . ., and∑

t Z t < ∞, theneither Xt → ∞ or else

{Xt

}converges to a finite value and∑

t Y t <∞.

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Jianchao Zheng (M’16) received the B.S. degree incommunications engineering and the Ph.D. degreein communications and information systems fromthe College of Communications Engineering, PLAUniversity of Science and Technology, Nanjing,China, in 2010 and 2016, respectively. From 2015 to2016, he was a Visiting Scholar with the Broad-band Communications Research Group, Departmentof Electrical and Computer Engineering, Universityof Waterloo, Canada. He is currently an AssistantProfessor with the College of Communications Engi-

neering, PLA University of Science and Technology. He has authored orco-authored several papers in international conferences and reputed journalsin his research area. His research interests focus on interference mitigationtechniques, green communications and computing networks, game theory,learning theory, and optimization techniques.

Yuan Wu (M’10–SM’16) received the Ph.D. degreein electronic and computer engineering from theHong Kong University of Science and Technol-ogy, Hong Kong, in 2010. From 2010 to 2011, hewas the Post-Doctoral Research Associate with theHong Kong University of Science and Technology.From 2016 to 2017, he is the Visiting Scholar withthe Broadband Communications Research Group,Department of Electrical and Computer Engineering,University of Waterloo, Canada. He is currently anAssociate Professor with the College of Informa-

tion Engineering, Zhejiang University of Technology, Hangzhou, China. Hisresearch interests focus on radio resource allocations for wireless communi-cations and networks, and smart grid. He was a recipient of the Best PaperAward at the IEEE International Conference on Communications in 2016.

Ning Zhang (M’16) received the B.Sc. degree fromBeijing Jiaotong University, Beijing, China, in 2007,the M.Sc. degree from the Beijing University ofPosts and Telecommunications, Beijing, in 2010, andthe Ph.D. degree from the University of Waterlooin 2015. From 2015 to 2016, he was a Post-DoctoralResearch Fellow with the Broadband Communi-cations Research Group Laboratory, University ofWaterloo. His current research interests include nextgeneration wireless networks, software defined net-working, green communication, and physical layersecurity.

Haibo Zhou (M’14) received the Ph.D. degree ininformation and communication engineering fromShanghai Jiaotong University, Shanghai, China,in 2014. He is currently a Post-Doctoral Fellow withthe Broadband Communications Research Group,University of Waterloo. His current research interestsinclude resource management and protocol design incognitive radio networks and vehicular networks.

Yueming Cai (M’05–SM’12) received the B.S.degree in physics from Xiamen University, Xiamen,China, in 1982, and the M.S. degree in micro-electronics engineering and the Ph.D. degree incommunications and information systems fromSoutheast University, Nanjing, China, in 1988 and1996, respectively. His current research inter-est includes cooperative communications, signalprocessing in communications, wireless sensor net-works, and physical layer security.

Xuemin (Sherman) Shen (M’97–SM’02–F’09)received the B.Sc. degree from Dalian MaritimeUniversity, China, in 1982, and the M.Sc. and Ph.D.degrees from Rutgers University, New Brunswick,NJ, USA, in 1987 and 1990, respectively, all in elec-trical engineering. He is currently a Professor and theUniversity Research Chair with the Department ofElectrical and Computer Engineering, University ofWaterloo, Canada. He is also an Associate Chair forGraduate Studies. His research focuses on resourcemanagement in interconnected wireless/wired net-

works, wireless network security, social networks, smart grid, and vehicular adhoc and sensor networks. He is an elected member of the IEEE ComSoc Boardof Governors and the Chair of Distinguished Lecturers Selection Committee.He served as the Technical Program Committee Chair/Co-Chair of the IEEEGlobecom’16, the Infocom’14, the IEEE VTC’10 Fall, and the Globecom’07;the Symposia Chair at the IEEE ICC’10; the Tutorial Chair at the IEEEVTC’11 Spring and the IEEE ICC’08; the General Co-Chair at the ACMMobihoc’15, the Chinacom’07, and the QShine’06; and the Chair of the IEEECommunications Society Technical Committee on Wireless Communicationsand the P2P Communications and Networking. He also serves/served asthe Editor-in- Chief of the IEEE NETWORK, Peer-to-Peer Networking andApplication, and IET Communications; an Associate Editor-in-Chief of theIEEE INTERNET OF THINGS JOURNAL; a Founding Area Editor of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS; an Associate Editorof the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ComputerNetworks, and ACM/Wireless Networks; and Guest Editor of the IEEE JOUR-NAL ON SELECTED AREAS IN COMMUNICATIONS, the IEEE WIRELESS

COMMUNICATIONS, the IEEE Communications Magazine, and ACM MobileNetworks and Applications. He received the Excellent Graduate SupervisionAward in 2006 and the Outstanding Performance Award in 2004, 2007, 2010,and 2014, respectively, from the University of Waterloo, the Premiers ResearchExcellence Award in 2003 from the Province of Ontario, Canada, and theDistinguished Performance Award in 2002 and 2007, respectively, from theFaculty of Engineering, University of Waterloo. He is a registered ProfessionalEngineer of Ontario, Canada, an Engineering Institute of Canada Fellow,a Canadian Academy of Engineering Fellow, a Royal Society of CanadaFellow, and a Distinguished Lecturer of the IEEE Vehicular TechnologySociety and Communications Society.

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. …bbcr.uwaterloo.ca/~xshen/paper/2017/opciud.pdf · 2019. 9. 23. · IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO