Energy-Efficient Matching for Resource Allocation in D2D ...source allocation problems for D2D communications are sum-marized and compared as follows. In [40], matching theory was

Energy-Efficient Matching for ResourceAllocation in D2D Enabled Cellular Networks

著者 ZHOU Zhenyu, OTA Kaoru, DONG Mianxiong, XUChen

journal orpublication title

IEEE Transactions on Vehicular Technology

volume 66number 6page range 5256-5268year 2017-06URL http://hdl.handle.net/10258/00009561

doi: info:doi:10.1109/TVT.2016.2615718

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. XX, NO. XX, XXX 2015 1

Energy-Efficient Matching for Resource Allocationin D2D Enabled Cellular Networks

Zhenyu Zhou, Member, IEEE, Kaoru Ota, Member, IEEE, Mianxiong Dong, Member, IEEE,and Chen Xu, Member, IEEE

Abstract—Energy-efficiency (EE) is critical for device-to-device(D2D) enabled cellular networks due to limited battery capacityand severe co-channel interference. In this paper, we addressthe EE optimization problem by adopting a stable matchingapproach. The NP-hard joint resource allocation problem isformulated as a one-to-one matching problem under two-sidedpreferences, which vary dynamically with channel states andinterference levels. A game-theoretic approach is employed toanalyze the interactions and correlations among user equipments(UEs), and an iterative power allocation algorithm is developedto establish mutual preferences based on nonlinear fractionalprogramming. We then employ the Gale-Shapley (GS) algorithmto match D2D pairs with cellular UEs (CUs), which is proved tobe stable and weak Pareto optimal. We provide a theoretical anal-ysis and description for implementation details and algorithmiccomplexity. We also extend the algorithm to address scalabilityissues in large-scale networks by developing tie-breaking andpreference deletion based matching rules. Simulation resultsvalidate the theoretical analysis and demonstrate that significantperformance gains of average EE and matching satisfactions canbe achieved by the proposed algorithm.

Index Terms—energy-efficient matching, resource allocation,nonconvex optimization, D2D communications, game-theoreticalapproach.

I. INTRODUCTION

DUE to the explosive growth of Internet of things (IoT)and cellular technology, it is predicted that billions of

devices will be interconnected and the corresponding datatraffic will grow more than 1000 times by 2020 [1]–[4].Device-to-device (D2D) communication that enables ubiqui-tous information acquisition and exchange among devices overa direct link [5], is a key enabler to facilitate future 5Gmobile systems [6]. D2D communications can be operated asan underlay to cellular networks through spectrum reusing [5],which brings numerous benefits and significant performance

Copyright (c) 2016 IEEE Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

Manuscript received January 2, 2016; revised May 16, 2016; acceptedJuly 16, 2016. This work was partially supported by the FundamentalResearch Funds for the Central Universities under Grant Number 2014MS08,2015MS04, and 2016MS17. This work was also partially supported by JSPSKAKENHI Grant Number 26730056, 15K15976, 16K00117, and JSPS A3Foresight Program.

Z. Zhou and C. Xu are with the State Key Laboratory of AlternateElectrical Power System with Renewable Energy Sources, School of Electricaland Electronic Engineering, North China Electric Power University, Beijing,China, 102206. (E-mal: zhenyu [email protected], [email protected])

K. Ota and M. Dong are with the Department of Information and ElectricEngineering, Muroran Institute of Technology, Muroran, Hokkaido, Japan. M.Dong is the corresponding aurthor. (E-mail: [email protected],[email protected])

improvements to network capacity and user experience [7],[8].

The operation of distributed D2D communications withincentralized cellular networks does give rise to new challengesin resource allocation optimization due to limited spectrum andbattery capacity. One major line of works aims at maximizingthe spectrum efficiency (SE) through resource allocation. Athree-stage joint optimization of admission control, powerallocation, and link selection was studied in [9]. An evaluationof SE performance under various resource sharing modes wasperformed in [10]. Resource allocation problems with queuingmodels and delay constraints were considered in [11]. In [12],the authors employed reverse iterative combinatorial auction(ICA) to solve the system sum-rate optimization problem.Spectrum-efficient resource allocation algorithms have beenproposed to address problems under different scenarios suchas relay-aided transmission [13], [14], constrained networkcapacity [15], wireless video networks [16], software-definedmulti-tier cellular networks [17], [18], and energy harvesting[19], etc. Comprehensive surveys and overviews of spectrum-efficient resource management for D2D communications wereprovided in [13], [20].

Although the above works are able to achieve significant SEperformance gains, energy consumption of user equipments(UEs) is completely ignored during the optimization process.Without energy-efficient resource allocation design, UEs haveto continually increase transmission power to satisfy stringentquality of service (QoS) requirements in an interference-limited environment, which in turn causes more interferenceto other UEs and leads to rapid depletion of battery energy.By exploiting the centralized architecture of cellular networksand distributed locations of UEs, energy-efficient resourceallocation problems have been addressed in both centralizedand distributed ways. Centralized schemes for device-to-multi-device (D2MD) and device-cluster scenarios were studied in[21] and [22], respectively. In [23]–[25], the authors pro-posed auction based power allocation, channel selection, andcooperative relay selection algorithms for energy efficiency(EE) optimization, respectively. A fractional frequency reuse(FFR) based two-stage channel selection and power allocationalgorithm was proposed in [26]. A joint optimization algorithmfor mode selection and resource scheduling was proposed byemploying a coalition game modeling approach [27]. Theoret-ical analysis of the tradeoff between EE and SE was developedin [28], [29], and was extended to multi-hop scenarios in [30].

However, most of the previous studies have ignored mutualpreferences and satisfactions of both D2D pairs and cellular


UEs (CUs). A common assumption is that every UE is willingto accept and obey a resource allocation decision even thoughit can achieve a better performance by disrupting this decision.Detailed modeling of preference from the perspective of EE ismissing. Several significant challenges arise when UEs’ prefer-ences and satisfactions are taken into consideration. First of all,it is difficult to model preferences of UEs because they varydynamically with channel states and aggregate interferencelevels. Second, it is impossible to make every UE always feelsatisfied with the same resource allocation decision becauseUEs may have different or even conflicting preferences. Lastbut not least, communication overhead and transmission delaycaused by obtaining the complete knowledge of every UE’spreference may lead to infeasibility and scalability issues withtwo-sided preferences.

To address these challenges, we propose an energy-efficientstable matching algorithm to solve the NP-hard resourceallocation problem by combining advantages of game theory,matching theory, and nonlinear fractional programming. Incomputational complexity theory, NP-hard problem representsthat the problem is at least as hard as any NP-completeproblem, which cannot be solved by using polynomial-timealgorithms [31]. Game theory that enables in-depth analysisof UEs with conflicting objectives has been widely applied foroptimizing resource allocation in D2D communications [13],[20], [32]. Despite its popularity and potential benefits, mostgame-theory based approaches such as the Nash equilibrium,only validate unilateral stability notions per player by showingthat any strategy deviating from the equilibrium cannot achievebetter performance [33]. Such one-sided stability may beimpractical when performing resource allocation between twodisjoint sets of players with individualized mutual preferences.

In comparison, matching theory provides a distributed self-organizing and self-optimizing approach for solving combi-natorial problems of matching resources in two disjoint sets[34]–[36]. In particular, it is suitable for wireless resourcemanagement due to its ability to handle complex objectivefunctions related to heterogeneous UEs through generallydefined preferences. Matching theory based resource allocationalgorithms have been developed for cellular networks [33],[37], cognitive radios [38], D2D communications [33], andmobile energy-harvesting networks [39], etc.

Related works that employ matching theory to solve re-source allocation problems for D2D communications are sum-marized and compared as follows. In [40], matching theorywas exploited to solve the resource allocation problem ina multi-tier heterogeneous network which consists of macroUEs, small cell UEs, and D2D UEs. The work was thenextended to the scenario of relay-aided D2D communicationsunder channel uncertainties, which were modeled as ellipsoidaluncertainty sets by exploiting robust optimization theory [41].In [42], matching theory was used for solving the relayselection problem in full duplex D2D communications. Theinteractions and interconnections between D2D UEs and CUswere not taken into consideration. In [43], cheating wasintroduced into the matching process to further improve UEs’satisfactions by falsifying certain UEs’ preference profiles.However, we believe that the energy-efficient joint partner

selection and power allocation problem with UEs’ preferencesand satisfactions considered in this paper has not been welladdressed in the above mentioned works.

The main contributions of this paper are summarized asfollows:

• A problem formulation for optimizing energy efficiencyof any D2D pair or CU under transmission power, channelreusing, and QoS constraints is derived. The formulationobtained is an mixed integer nonlinear programming(MINLP) problem, which uses a binary variable to in-dicate partner selection (which UE should be selectedto form a channel-reusing partnership), and a continuousvariable for power allocation (how much transmissionpower should be allocated for the potential D2D-CUpair). To solve the MINLP problem, we introduce aone-to-one matching model which matches D2D pairswith CUs according to mutual preferences. In this way,the original NP-hard joint partner selection and powerallocation problem can be decoupled into two separatesubproblems and solved in a tractable manner.

• One main focus of this work is how to establish mutualpreferences from the perspectives of EE. We model aD2D pair (or CU)’s preference over a CU (or D2Dpair) as the maximum achievable EE under the specifiedmatching. The power allocation problem developed forobtaining UE preference is modeled as a noncooperativegame, in which each UE aims at optimizing its indi-vidual EE given the transmission power strategies of itschannel-reusing partner. A nonlinear fractional program-ming and Lagrange dual decomposition based iterativealgorithm is developed for establishing preference profilesof both D2D pairs and CUs [44], [45]. The existenceand uniqueness properties of the Nash equilibrium andits relationships with optimum power allocation strategiesare analyzed theoretically via mathematical proofs.

• Finally, we propose to advocate the Gale-Shapley (GS)algorithm to solve the formulated energy-efficient match-ing problem under established two-sided preferences andcorresponding power allocation strategies. The proposedmatching algorithm is also extended to address scala-bility issues encountered in large-scale networks. De-tailed discussion and in-depth analysis of matching sta-bility, distributed/centralized implementation, and com-putational complexity are presented. Simulation resultsshow that the proposed algorithm can obtain significantEE performance gains and remarkably improve matchingsatisfactions for a wide range of satisfaction thresholdvalues.

The remaining parts of this paper are outlined as follows.Section II provides a detailed description of the objectivefunctions and preference modeling for the formulated resourceallocation problems. Section III develops the proposed energy-efficient matching algorithm. Simulation results are presentedand discussed in section IV. Section V concludes the paperand presents future research directions.


II. ENERGY-EFFICIENT RESOURCE ALLOCATIONPROBLEM FORMULATION

In this section, we firstly provide a detailed descriptionof the system model of D2D communications underlayingcellular networks with UE preferences, and then present theformulation of the energy-efficient resource allocation prob-lem.

A. System Model

We consider uplink spectrum sharing in D2D communica-tions underlaying cellular networks, which is shown in Fig. 1.Uplink spectrum sharing is considered in particularly becausefirstly, uplink spectrum resources are usually under-utilizedcompared to the downlink in frequency division duplexing(FDD) based cellular systems [27]; secondly, co-channel inter-ference caused by D2D UEs can be handled more easily by apowerful base station (BS) than CUs. We assume that each CUis allocated with an orthogonal channel (e.g., an orthogonalresource block in LTE), i.e., K active CUs occupy a total ofK orthogonal channels and there is no co-channel interferenceamong CUs. A pair of D2D transmitter and receiver that meetD2D communication requirements form a D2D pair, and areallowed to reuse at most one CU’s channel for transmission.The scenario that each D2D pair reuses more than one channelis equivalent to a one-to-many matching problem, which is outof the scope of this paper and left to future works.

As a result, all of active D2D transmitters cause co-channel interference to the BS, and a CU causes co-channelinterference to the D2D receiver that operates in the samechannel. QoS requirements are imposed for both CUs and D2Dpairs. In this paper, we assume that the D2D mode and peerselection process has already been finished, and we mainlyfocus on the resource allocation part. The joint optimizationof mode selection, peer selection, and resource allocation is acompletely new problem, which is out of the scope of thispaper and will be treated in our future works. Regardinghow to perform mode selection and peer selection, interestedreaders may refer to [7], [11], [46] and references therein formore details.

There are a total of N D2D pairs and K CUs. Through-out the paper, the index sets of active D2D pairs andCUs are denoted as D = {d1, · · · , di, · · · , dN}, and C={c1, · · · , ck, · · · , cK}, respectively.

Definition 1. The partner selection matrix of D2D pairs isdenoted as XN×K , where the (i, k)-th element xi,k ∈ {0, 1}indicates the selection decision of the D2D-CU partnership(di, ck) for the D2D pair di, ∀di ∈ D, ∀ck ∈ C. If xi,k = 1,di prefers to forming a partnership with ck, and if xi,k = 0,otherwise.

Definition 2. The partner selection matrix of CUs is denotedas YK×N , where the (k, i)-th element yk,i ∈ {0, 1} indicatesthe selection decision of the D2D-CU partnership (ck, di) forthe CU ck, ∀ck ∈ C, ∀di ∈ D. If yk,i = 1, ck prefers toforming a partnership with di, and if yk,i = 0, otherwise.

Remark 1. Due to the individualized and differentiatedpreferences of di and ck, it is very possible to have conflicting

Fig. 1. Practical implementation and resource allocation design for D2Dcommunications underlaying cellular networks with UE preferences.

partner selection decisions, i.e., xi,k 6= yk,i. A D2D-CU part-nership (di, ck) can be formed if and only if xi,k = yk,i = 1.

Regarding channel models, both fast fading and slow fadingwhich are caused by multi-path propagation, shadowing, andpathloss are taken into consideration [9]. The channel gain ofthe interference from ck to di is given by

gck,i = $βck,iζck,id−αk,i , (1)

where $ is the pathloss constant, βck,i is the fast-fading gainwith exponential distribution, ζck,i is the slow-fading gain withlog-normal distribution, α is the pathloss exponent, and dk,iis the transmission distance. In a similar way, we can definethe channel gain of di as gdi , the interference channel gainbetween the transmitter of di and the BS as gdi,B , and definethe channel gain between ck and the BS as gck,B .

The achievable SE (defined as bits/s/Hz) of di is given by

Udi =∑ck∈C

log2

(1 +

xi,kyk,ipdi gdi

N0 + xi,kyk,ipckgck,i

), (2)

where pdi and pck represent the transmission power of di andck, respectively. N0 is the noise power on each channel. Theachievable SE of ck is given by

U ck = log2

(1 +

pckgck,B

N0 +∑di∈D xi,kyk,ip

di gdi,B

). (3)

The total power consumptions of di and ck are given by

Edi =∑ck∈C

1

ηxi,kyk,ip

di + 2pcir, (4)

Eck =1

ηpck + pcir. (5)

pcir is the total circuit power consumption which includesvalues of mixer, frequency synthesizer, digital-to-analog con-verter (DAC)/analog-to-digital converter (ADC), etc. η is thepower amplifier (PA) efficiency, i.e., 0 < η < 1. The power


consumption of the BS is not considered because it is poweredby external grid power.

B. Problem Formulation

Eventually better channel conditions and proper transmis-sion power strategies can improve the EE performance moreefficiently. Therefore, for any D2D pair or CU, the followingquestions need to be answered before reaching a decision:• How to select a partner to form a D2D-CU channel-

reusing pair for optimizing EE performance?• How to perform power allocation for the expected D2D-

CU pair?• How to satisfy various practical resource allocation con-

straints such as maximum transmission power levels, QoSrequirements, and channel-reusing rules, etc?

• How to avoid disruptions from other D2D pairs or CUswhich also wish to be matched with the preferred D2Dpair or CU?

The above questions indicate that the optimizationof EE involves solving a joint partner selectionand power allocation problem. To be more general,denoting xi = {xi,1, · · · , xi,k, · · · , xi,K} andyk = {yk,1, · · · , yk,i, · · · , yk,N} as di’s and ck’s binarypartner selection strategy sets, respectively, and denoting pdiand pck as di’s and ck’s continuous power allocation strategies,respectively, the objective function in terms of EE (bits/J/Hz)is defined as the SE (bits/s/Hz) divided by the total powerconsumption (W) [47]. The EE objective functions of di(including both the transmitter and receiver) and ck are givenby

Udi,EE(xi, pdi ) =

Udi (xi, pdi )

Edi (xi, pdi )

=

∑ck∈C log2

(1 +

xi,kyk,ipdi g

di

N0+xi,kyk,ipckgck,i

)∑ck∈C

1ηxi,kyk,ip

di + 2pcir

, (6)

U ck,EE(yk, pck) =

U ck(yk, pck)

Eck(pck)

=

log2

(1 +

pckgck,B

N0+∑

di∈D xi,kyk,ipdi gdi,B

)1ηp

ck + pcir

. (7)

The joint partner selection and power allocation problemfor di can be formulated as

max(xi,pdi )

Udi,EE(xi, pdi )

s.t. Cdi,1 : 0 ≤ pdi ≤ pdi,max,Cdi,2 : Udi (xi, p

di ) ≥ Udi,min,

Cdi,3 : xi,k = {0, 1},∀ck ∈ C,

Cdi,4 :∑ck∈C

xi,k ≤ 1. (8)

Cdi,1 ensures that the power allocation of di should not exceedthe maximum allowed transmission power pdi,max. Cdi,2 speci-fies the QoS requirement which represents that the minimum

SE should not fall below Udi,min. Cdi,3 and Cdi,4 are the channel-reusing constraints which make sure that at most one channelcan be shared simultaneously by di and one existing CU.

The problem formulation for ck is given by

max(yk,pck)

U ck,EE(yk, pck)

s.t. Cck,1 : 0 ≤ pck ≤ pck,max,Cck,2 : U ck(yk, p

ck) ≥ U ck,min,

Cck,3 : yk,i = {0, 1},∀di ∈ D,

Cck,4 :∑di∈D

yk,i ≤ 1. (9)

Cck,1 and Cck,2 specify the transmission power and QoS con-straints. Cck,3 and Cck,4 ensure that at most one D2D pair canshare the same channel with ck simultaneously.

Remark 2. By observing (8) and (9), we find that thepartner selection problem is coupled with the power allocationproblem. The formulation obtained is an NP-hard MINLPproblem, which involves both binary and continuous variablesfor resource allocation optimization. Thus, the formulationsobtained in neither (8) nor (9) cannot be solved directlyby using either nonlinear fractional programming or integerprogramming. Furthermore, neither of the two programmingapproaches has taken UEs’ preferences and satisfactions intoconsideration, which may lead to unstable and unsatisfiedresource allocation decision.

To solve (8) and (9), we introduce a one-to-one matchingmodel to match D2D pairs with CUs according to their mutualpreferences. In this fashion, the original NP-hard MINLPproblem can be decoupled into two separate subproblems andsolved in a tractable manner. We use the triple (C,D,P) todenote the formulated matching problem, i.e., D, C representthe two finite and distinct sets of D2D pairs and CUs, respec-tively, and P is the set of mutual preferences. Both D2D pairsand CUs seek to form proper channel-reusing partnershipsto maximize EE under constraints of QoS and transmissionpower. The definition of a matching µ is given by [35]:

Definition 3. For the matching problem (C,D,P), µ is apoint-by-point mapping from C∪D onto itself under preferenceP . This is, for any ck ∈ C and di ∈ D, µ(ck) ∈ D∪{ck} andµ(di) ∈ C ∪ {dk}. µ(ck) = di if and only if µ(di) = ck.

If µ(di) = di or µ(ck) = ck, di or ck stays single. Eitherdi or ck can send a request for forming a partnership withits preferred partner based on its preference (which is thepartner selection subproblem), and demonstrate the allocatedtransmission power for the formed partnership (which is thepower allocation subproblem). Both di and ck are assumedto only care about their own matched partners and showlittle concerns to matching results of others. This assumptionis valid because UEs are privately owned and operated byindependent individuals.

III. ENERGY-EFFICIENT STABLE MATCHING FOR D2DCOMMUNICATIONS

In this section, we introduce the proposed energy-efficientstable matching approach. First, we develop an iterative al-gorithm for preference establishment based on noncoopera-tive game theory and nonlinear fractional programming in


Subsection III-A. Then, the derivation of the energy-efficientmatching based on the GS algorithm is presented in SubsectionIII-B. Finally, in-depth discussions and theoretical analysis ofmatching stability, distributed/centralized implementation, andcomputational complexity are provided in Subsection III-C.

A. Preference Establishment

1) Noncooperative Game based Preference Modeling: Theset of UEs’ preferences P is necessary for developing theenergy-efficient matching. We model di’s preference over ckas the maximum achievable EE under the matching µ(di) = ck(xi,k = yk,i = 1). Thus, the partner selection decision of dihas been already fixed, and only the power allocation strat-egy needs to be optimized. The formulated power allocationproblem is given by

maxpdi

Udi,EE(pdi )∣∣∣µ(di)=ck

s.t. Cdi,1, Cdi,2. (10)

The power allocation problem for ck under the matchingµ(ck) = di is given by

maxpck

U ck,EE(pck)∣∣∣µ(ck)=di

s.t. Cck,1, Cck,2. (11)

There are two challenges when solving the above optimiza-tion problems. First, from (6) and (7), Udi,EE and U ck,EE areinter-correlated through the interference terms, i.e., pckg

ck,i and

pdi gdi,B . Second, the problems formulated in (10) and (11) are

still nonconvex due to the fractional form of Udi,EE and U ck,EE .In order to study the inter-connections between D2D pairs

and CUs (to solve the first challenge), we adopt a game-theoretic approach to model the distributed power allocationproblem as a noncooperative game G. UEs are assumed asrational and selfish [48], i.e., each di ∈ D (or ck ∈ C)cares about its individual objective Udi,EE (or U ck,EE), butis not otherwise concerned with Udj,EE ,∀dj ∈ D\{di} (orU cm,EE ,∀cm ∈ C\{ck}). The game G can be described asG = (C,D,A,U), wherein A = {Ad1, · · · , AdN , Ac1, · · · , AdK}is the set of possible strategies that a UE can take, andU = {Ud1,EE , · · · , UdN,EE , U c1,EE , · · · , U cK,EE} is the set ofUEs’ utilities. For example, if Adi = {[0, pdi,max]}, then di isallowed to select pdi from the interval [0, pdi,max].

2) Objective Function Transformation: To overcome thesecond challenge, nonlinear fractional programming is em-ployed to transform the nonconvex problem in the fractionalform to equivalent convex ones. The optimum result of (10)is defined as

qd∗i = maxpdi

Udi,EE(pdi )∣∣∣µ(di)=ck

=Udi (pd∗i )

Edi (pd∗i ), (12)

where pd∗i is the optimum power allocation strategy of di.Based on [44], we have

Theorem 1: qd∗i is achieved if and only if

maxpdi

Udi (pdi )− qd∗i Edi (pdi ) = Udi (pd∗i )− qd∗i Edi (pd∗i ) = 0.

(13)

Fig. 2. The relationship between inner loop and outer loop iterations of theiterative power allocation algorithm.

Theorem 1 reveals that there exists an equivalent trans-formed problem with an objective function in subtractiveform, which leads to the same maximum EE obtained bydirectly solving (10). The equivalent optimization problem insubtractive form is given by

maxpdi

Udi (pdi )− qd∗i Edi (pdi )

s.t. Cdi,1, Cdi,2. (14)

(14) is actually a multi-objective convex optimization problemwhere the variable qd∗i can be regarded as a negative weightof Edi . In the same way, defining qc∗k and pc∗k as the optimumEE and the corresponding strategy of ck, respectively, thetransformed problem that is equivalent to (11) is given by

maxpck

U ck(pck)− qc∗k Eck(pck)

s.t. Cck,1, Cck,2. (15)

3) Distributed Iterative Power Allocation: Both (14) and(15) are standard convex optimization problems and can besolved efficiently. However, the specific values of qd∗i and qc∗kare required to solve (14) and (15), respectively. In order toobtain qd∗i and qc∗k , an iterative algorithm is developed basedon Dinkelbach’s method and is given in Algorithm 1 [44]. Theiterative Algorithm 1 consists of two loops: the outer loop withthe iteration index l represents iterations of the noncooperativegame, and the inner loop with the iteration index n representsiterations of Dinkelbach’s algorithm. The relationship betweeninner loop and outer loop iterations is shown in Fig. 2. Foreach round of the game, the inner loop is executed to findthe corresponding optimum power allocation strategy for eachplayer, which stops if either the iteration stopping criteriaor the maximum loop number Nmax is reached. The gameiteration continues until the achieved power allocation strategyconverges to a Nash equilibrium, i.e., none player is capableof unilaterally achieving better performance by deviating fromit.


Algorithm 1 Distributed Iterative Power Allocation Algorithmfor Obtaining qd∗i and qc∗k

1: Input: gdi , p̂di gdi,B , gck,B , p̂ckg

ck,i, p

di,max, pdk,max, Cdi,min,

Cck,min.2: Output: qd∗i , qc∗k , pd∗i , pc∗k .3: Initialize: qdi , qck, Nmax, ∆g , ∆d, ∆c, p̂ck, p̂di4: while | qd∗i (l)− qd∗i (l − 1) |≤ ∆g ,

& | qc∗k (l)− qc∗k (l − 1) |≤ ∆g do5: while n < Nmax do6: obtain p̂di (n) using (20)7: if Udi [p̂di (n)]− qdi (n)Edi [p̂di (n)] > ∆d then8: Update: qdi (n+ 1) = Udi [p̂di (n)]/Edi [p̂di (n)]9: else

10: pd∗i = p̂di (n), and qd∗i = Udi (pd∗i )/Edi (pd∗i )11: end if12: obtain p̂ck(n) using (23)13: if U ck [p̂ck(n)]− qck(n)Eck[p̂ck(n)] > ∆c then14: Update: qck(n+ 1) = U ck [p̂ck(n)]/Eck[p̂ck(n)]15: else16: pc∗k = p̂ck(n), and qc∗k = U ck(pc∗k )/Eck(pc∗k )17: end if18: Update the Dinkelbach iteration index: n→ n+119: end while20: Update: qdi (l) = qd∗i , qc∗k (l) = qc∗k21: Update the game iteration index : l→ l + 122: end while

At the n-th iteration of the l-th round game, pdi (n) andpck(n) are obtained by solving the following problems withqdi (n) and qck(n) obtained from the (n− 1)-th iteration:

maxpdi

Udi [pdi (n)]− qdi (n)Edi [pdi (n)]

s.t. Cdi,1, Cdi,2. (16)

maxpck

U ck [pck(n)]− qck(n)Eck[pck(n)]

s.t. Cck,1, Cck,2. (17)

The augmented Lagrangian of (16) is given by

LEEi (pdi , δdi , θ

di ) = Udi [pdi (n)]− qdi (n)Edi [pdi (n)]

−δdi (n)[pdi (n)− pdi,max] + θdi (n)(Udi [pdi (n)]− Udi,min

),

(18)

where δdi and θdi are the Lagrange multipliers for constraintsCdi,1 and Cdi,2, respectively. By using Lagrange dual decom-position, (18) is decomposed as [45]

min(δdi , θ

di ≥ 0)

max(pdi )

LEEi (pdi , δdi , θ

di ). (19)

By exploiting Karush-Kuhn-Tucker (KKT) conditions, theoptimal value p̂di (n) corresponding to qdi (n) is given by

p̂di (n) =

[η[1 + θdi (n)] log2 e

qdi (n) + ηδdi (n)−p̂ckg

ck,i(n) +N0

gdi

]+, (20)

Algorithm 2 Iterative Preference Establishment Algorithm forObtaining P

1: Input: the set of CUs and D2D pairs, C, D.2: Output: the set of preference profiles P .3: for di ∈ D do4: for ck ∈ C do5: calculate maximum achievable qd∗i

∣∣∣µ(di)=ck

and

qc∗k

∣∣∣µ(ck)=di

for the D2D-CU pair (di, ck) by em-

ploying Algorithm 1.6: end for7: end for8: for di ∈ D do9: sort all of CUs ck ∈ C in a descending order according

to qd∗i∣∣∣µ(di)=ck

.

10: end for11: for ck ∈ C do12: sort all of D2D pairs di ∈ D in a descending order

according to qc∗k∣∣∣µ(ck)=di

.

13: end for

where [x]+ = max{0, x}. Then, by employing the gradientmethod [49], we update the Lagrange multipliers as

δdi (n, τ + 1) =[δdi (n, τ) + εi,δ(n, τ)

(p̂di (n, τ)− pdi,max

)]+,

(21)

θdi (n, τ + 1) =[θdi (n, τ)− εi,θ(n, τ)

(Udi (n, τ)− Udi,min

)]+,

(22)

where τ is the iteration index of Lagrange multiplier updating,εi,δ and εi,θ are the step sizes. The step sizes should becarefully chosen to guarantee convergence and optimality.

Then, p̂di (n) obtained in (20) is used to update qdi (n+1) forthe (n+1)-th iteration as qdi (n+1) = Udi [p̂di (n)]/Edi [p̂di (n)]. Inthe final iteration of the inner loop, setting pd∗i = p̂di , qd∗i canbe obtained by using (12) and saved as the l-th element of thevector qdi , i.e., qdi (l) = qd∗i . The optimization problem (17) issolved in the same way. The optimal value p̂ck(n) correspondsto qck(n) is given by

p̂ck(n) =

[η[1 + ξck(n)] log2 e

qck(n) + ηρck(n)−p̂di (n)gdi,B +N0

gck,B

]+. (23)

Details for how to obtain qc∗k are omitted due to spacerestrictions. The outer loop stops if the maximum EE (qd∗i , q

c∗k )

obtained in the l-th round of the game varies little from theoptimization result achieved in the previous round, where thecorresponding optimum strategy set (pd∗i , p

c∗k ) has converged

to a Nash equilibrium.4) Preference Profile Establishment: Algorithm 2 presents

how to establish the set of preference profiles P . For everydi ∈ D, the maximum achievable qd∗i under the matchingµ(di) = ck,∀ck ∈ C, is denoted as qd∗i

∣∣∣µ(di)=ck

, and can be

obtained by using Algorithm 1. We write ck �di cm to meandi prefers ck to cm, which is defined as

ck �di cm ⇔ qd∗i

∣∣∣µ(di)=ck

> qd∗i

∣∣∣µ(di)=cm

, (24)


where � is a complete, reflexive, and transitive binary prefer-ence relation [35]. In addition, we write ck �di cm to meandi likes ck at least as well as cm, which is defined as

ck �di cm ⇔ qd∗i

∣∣∣µ(di)=ck

≥ qd∗i∣∣∣µ(di)=cm

. (25)

Similarly, we write di �ck dj to mean ck prefers di to dj ,which is defined as

di �ck dj ⇔ qc∗k

∣∣∣µ(ck)=di

> qc∗k

∣∣∣µ(ck)=dj

. (26)

After obtaining qd∗i

∣∣∣µ(di)=ck

, ∀ck ∈ C, the preference profile

P (di) = {· · · , ck, cm · · · } is obtained by sorting all of CUs ina descending order according to the criteria of qd∗i

∣∣∣µ(di)=ck

,

∀ck ∈ C. The preference profile of ck is denoted as P (ck),which is obtained by sorting all of available D2D pairs accord-ing to qc∗k

∣∣∣µ(ck)=di

,∀di ∈ D. The total set P is constructed as

P = {P (d1), · · · , P (dN ), P (c1), · · · , P (cK)}.

B. Energy-Efficient Stable Matching

After obtaining P (di) and P (ck) for each di ∈ D and ck ∈C, we propose Algorithm 3 to match D2D pairs with CUsby employing the GS algorithm [34]. In the first iteration,every di ∈ D sends a partner request to its most preferred CUmax{qd∗i

∣∣∣µ(di)=ck

,∀ck ∈ C}. Then, every ck ∈ C receives the

request and rejects the D2D pair if it already holds a bettercandidate. Any di ∈ D that is not rejected by the CUs atthis step is held as a candidate. In the next step, any di ∈ Dthat has been already rejected sends a new request to its mostpreferred choice from the set of CUs that have not yet issueda rejection. If a D2D pair is rejected by all of its preferredCUs, it will give up and send no further request. Each ck ∈ Ccompares all of the received requests including the candidatethat was held from previous steps and only accepts the mostpreferred D2D pair. The request sending and rejection processfinishes when every di ∈ D has already found a partner orhas been rejected by all of CUs to which it has sent requests.Algorithm 3 has the property of deferred acceptance due tothe fact that the best candidate kept at any step can be rejectedlater on if a better candidate appears.

C. Discussion and Analysis

In this subsection, we provide an in-depth theoretical anal-ysis for the proposed energy-efficient matching algorithm.

1) Nash Equilibrium Analysis: Theorem 2: Under thematching µ(di) = ck, ∀i ∈ N ,∀k ∈ K, the power allocationstrategy set (pd∗i , p

c∗k ) obtained by the iterative algorithm

constitutes a Nash equilibrium, which exists but is not unique.Furthermore, none individual UE is able to unilaterally getbetter performance by deviating from the Nash equilibrium.

Proof: Please see Appendix A.

Algorithm 3 Energy-Efficient Stable Matching Algorithm1: Input: C,D,P .2: Output: µ.3: Initialize: µ = φ, Φ = D.4: while Φ 6= φ do5: for di ∈ Φ do6: di chooses the CU with the highest ranking from

P (di).7: end for8: for ck ∈ C do9: if ck receives a request from di, and prefers di to its

current candidate dj held from previous steps, i.e.,di �ck dj then

10: di is held as a new candidate, while ck issues arejection to dj , i.e., µ(ck) = di;

11: add dj into Φ, remove di from Φ, and remove ckfrom P (dj).

12: else13: ck issues a rejection to di, and holds dj continually

as its candidate, i.e., µ(ck) = dj .14: remove ck from P (di).15: end if16: end for17: end while

2) Stability and Optimality: The stability and optimalityproperties can be easily proved due to the structures of theGS algorithm. A short version of proofs are provided here forreference and more details can be found in [35], [50].

Assuming that di and ck are not matched with each otherunder µ, i.e., µ(di) 6= ck and µ(ck) 6= di, (di, ck) can form ablocking pair that blocks µ if di �ck µ(ck), and ck �di µ(di).µ is said to be unstable if there exists a blocking pair [35].

Theorem 3: The proposed energy-efficient matching µ isstable for every di ∈ D and ck ∈ C.

Proof: Please see Appendix B.Theorem 4: For every di ∈ D and ck ∈ C under µ(di) =

ck, q̂di∣∣∣µ(di)=ck

and q̂ck

∣∣∣µ(di)=ck

obtained by Algorithm 1

converges to unique qd∗i

∣∣∣µ(di)=ck

and qc∗k

∣∣∣µ(di)=ck

in finite

iterations, respectively [44], [49].Theorem 5: The obtained energy-efficient stable matching

µ is weak Pareto optimal for every di ∈ D.Proof: Please see Appendix C.3) Scalability: Implementing the proposed algorithm in

cellular networks with a large number of UEs will encounterscalability problems. For example, it becomes difficult tohave channel state information (CSI) of every link due tolimited processing capability, increasing signalling overhead,and strict QoS requirement, etc. If di and ck only havelimited information about each other, Algorithm 1 is not beable to produce qd∗i

∣∣∣µ(di)=ck

and qc∗k∣∣∣µ(ck)=di

, which leads to

the matching problem under incomplete preference lists. Toaddress such challenges, we modify Algorithm 3 to handlethis problem by deleting di and ck from P (ck) and P (di),respectively, if they cannot be involved in a stable matching.


P (di) and P (ck) are consistent if deleting ck from P (di)represents that di is also removed from P (ck) [36]. For everydi ∈ D and ck ∈ C, P (di) and P (ck) are assumed to beconsistent. With consistent preference profiles, the modifiedalgorithm is able to proceed in the same fashion as Algorithm3 and obtain a matching in polynomial time. The new matchingmay be partial stable due to the fact that some UEs that aredeleted from preference profiles may be left unmatched.

Another frequent scalability problem is that some D2D pairor CU may have more than one best potential matching part-ners, e.g., a tie exists such that qd∗i

∣∣∣µ(di)=ck

= qd∗i

∣∣∣µ(di)=cm

.

To break the tie, we propose some tie-breaking rules toforce di to choose between ck and cm at any step bycomparing new criteria such as the optimum SE Udi , or thetotal power consumption Edi . The modified algorithm canhandle situations such as qd∗i

∣∣∣µ(di)=ck

= qd∗i

∣∣∣µ(di)=cm

and

qc∗k

∣∣∣µ(ck)=di

= qc∗k

∣∣∣µ(ck)=dj

, ∀di, dj ∈ D, ∀ck, cm ∈ C,

etc. In the following, we assumes that any ck ∈ C doesnot care which D2D pair would reuse its spectrum, thatis, qc∗k

∣∣∣µ(ck)=di

= qc∗k

∣∣∣µ(ck)=dj ,∀dj 6=di

. This scenario can be

considered as a special case of the preference tie scenario,where any potential matching partner is the best candidate.In this case, some tie-breaking rules such as “first come firstserve” can be proposed to force CUs to make a decision at anystep based on the new criteria. Furthermore, the time-breakingrules can be flexibly designed not only to optimize EE orSE, but also to improve miscellaneous performance metricsincluding reliability, security, fairness, and coverage, etc.

4) Implementation: In the distributed resource allocationalgorithm, each D2D pair only needs to estimate the receivedinterference rather than knowing the specific power allocationor partner selection strategies of interferers. The reason is thatthe sufficient information of strategies are contained in theform of interference. CUs also need to know the knowledgeof interference, which can be estimated firstly by centralizedpowerful BSs and then fed back to CUs. Although the energy-efficient matching is designed in a distributed way, it is alsosuitable for centralized implementation to comply with thecentralized architecture of existing cellular networks. A BSequipped with advanced signal processing and computation ca-pability can operate as a matchmaker to organize the matching(C,D,P). The details are given as follows.

First, the BS sends requests to every di ∈ D and ck ∈ Cto obtain required information for building preference profilesP (di) and P (ck). CSI of some links such as gck,B ,∀ck ∈ C,can be directly obtained by performing channel estimation inthe BS using pilot signals. CSI of D2D links such as gdi has tobe estimated by D2D receivers and then is fed back to the BS.CSI of interference links such as gdi,B and gdk,i, ∀di ∈ D,∀ck ∈C is not required. After collecting enough information, the BSestablishes the preference profile P (di) for each di ∈ D andP (ck) for each ck ∈ C based on Algorithm 2. Then Algorithm3 is employed (with D2D pairs sending requests to CUs) toproduce µ under the established preference profiles.

The advantages of centralized implementation can be fullyexploited by future cloud radio access network (C-RAN) based

TABLE ISIMULATION PARAMETERS.

Simulation Parameter ValueCell radius 500 mMax D2D transmission distance ddmax 20 ∼ 100 mPathloss exponent α 4Pathloss constant $ 10−2

Shadowing ζck,i (standard deviation of 8 dBa log-normal distribution)Multi-path fading βc

i,k (the mean of 1an exponential distribution)Max Tx power pdi,max, p

ck,max 23 dBm

Constant circuit power pcir 20 dBmNoise power N0 -114 dBmNumber of D2D pairs N 5 ∼ 50Number of cellular UEs K 5 ∼ 50PA efficiency η 35%QoS requirement Cc

k,min, Cdi,min 0.5 ∼ 1

(uniform distribution) bit/s/Hz

mobile communication systems [51]. Preference establishmentand D2D-CU matching can be realized by the powerful baseband unit (BBU) pool which exploits cell cooperation andcoordination among densely distributed remote radio heads(RRHs).

5) Complexity: For a pair of (di, ck), the computationalcomplexity for establishing P (di) and P (ck) mainly dependson Algorithm 1. qdi and qck produced by Algorithm 1 increasesat each iteration and converges to qd∗i and qc∗k at a super-linear rate [49]. With N D2D pairs and K CUs, the computa-tional complexity of Algorithm 1 isO(NKLloopLdual), whereLloop and Ldual are the numbers of iterations required forconverging to the optimum EE and optimal power allocationstrategies, respectively. In Algorithm 2, sorting N D2D pairsand K CUs in a descending order leads to a complexity ofO(NK log(NK)

). In Algorithm 3, under the rule that every

di ∈ D has only one chance to send requests to CUs in P (di),the matching µ can be obtained with a complexity of O(KN)[36].

IV. NUMERICAL RESULTS

In this section, the proposed energy-efficient matchingalgorithm, labeled as “energy-efficient stable matching”, iscompared with several heuristic algorithms. The first is thepower greedy algorithm, which always allocates the maximumtransmission power pdi,max (or pck,max). The second is therandom power allocation algorithm, which allocates poweruniformly distributed in the range [0, pdi,max] (or [0, pck,max]).The third is the spectrum-efficient algorithm based on thewater-filling power allocation (SINR maximization) [32], [52],[53]. The first two heuristic algorithms employ random match-ing which matches D2D pairs with CUs in a random way,while the third one adopts maximum-SINR based association.The values of simulation parameters are based on [9], [23],[32], and are summarized in Table I. A single cellular networkis considered and the cell radius is 500 m. In each time ofsimulation, locations of CUs and D2D pairs are generated ina random way as shown in Fig. 3. QoS requirements in terms


−400 −200 0 200 400 600−500

−400

−300

−200

−100

0

100

200

300

400

500

600

Location in x (m)

Lo

catio

n in

y (

m)

D2D Tx

D2D Rx

CU

BS

Fig. 3. A snapshot of locations of K CUs and N D2D pairs in a singlecellular network (K = N = 50, the cell radius is 500 m).

20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Maximum D2D Transmission Distance

Ave

rag

e E

ne

rgy E

ffic

ien

cy (

bits/H

z/J

)

energy−efficient stable matching

random power, random matching

spectrum−efficient, max−SINR matching

power greedy, random matching

Fig. 4. Average EE of D2D pairs versus maximum D2D transmission distance(K = N = 5, ddmax = 20 ∼ 100 m).

of SE are generated randomly from a uniform distribution inthe range [0.5, 1] bit/s/Hz. We average the simulation resultsover 103 times.

The average EE performance and UE satisfactions of theobtained energy-efficient stable matching is evaluated andverified. We adopt a statistical model to define a UE’s satis-faction as the cumulative distribution functions (CDFs) of thematching result that is higher than its satisfaction threshold.For example, defining di’s satisfaction threshold as cm, thematching result µ(di) is compared with the threshold cm toevaluate whether di is satisfied with µ(di). di is said to besatisfied with µ(di) if di prefers µ(di) at least as well as cm,i.e., µ(di) �di cm. Otherwise, di is said to be unsatisfiedwith µ(di) if it is matched to a partner that is less preferredto the threshold, i.e., cm �di µ(di). The CDF is denoted asPr{µ(di) �di cm}, which is the probability that di is matchedwith a partner that is more preferred to the threshold cm.

5 10 150

100

200

300

400

500

600

700

Number of D2D Pairs (CUs)

Ave

rag

e E

ne

rgy E

ffic

ien

cy (

bits/H

z/J

)

energy−efficient stable matching

random power, random matching

spectrum−efficient, max−SINR matching

power greedy, random matching

Fig. 5. Average EE of D2D pairs versus numbers of active D2D pairs andCUs (ddmax = 20 m, K = N = 5 ∼ 15).

5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Satisfaction Threshold

CD

F o

f S

atisfa

ctio

n

energy−efficient stable matching, K=M=20

energy−efficient stable matching, K=M=50

random matching, K=M=20

random matching, K=M=50

Fig. 6. CDF of D2D pairs’ satisfactions versus satisfaction threshold (N =K = 20, 50, ddmax = 20 m).

Fig. 4 shows the average EE performance of D2D pairsversus the maximum D2D transmission distance ddmax withK = 5 CUs and N = 5 D2D pairs. Simulation resultsdemonstrate that the proposed algorithm achieves the best EEperformance in the whole regime. The proposed algorithmoutperforms the random power allocation algorithm, the powergreedy algorithm, and the spectrum-efficient algorithm by132%, 206%, and 248% for ddmax = 20 m, respectively.Random allocation achieves the second best performance sincethere is a large probability to have a higher EE than thespectrum-efficient and power greedy algorithms which alwaystake full advantage of any available power. It is clear thatthe SE gain achieved by increasing transmission power isnot able to compensate for the corresponding EE loss. Thepower greedy algorithm has the worst EE performance amongthe four due to two reasons. First, power consumption iscompletely ignored in the resource allocation process. Second,


1 2 3 4 5 6 7 8 9 10200

300

400

500

600

700

800

Number of Game Iterations

Ave

rag

e E

ne

rgy E

ffic

ien

cy (

bits/H

z/J

)

K=N=5

K=N=10

K=N=15

Fig. 7. Average EE versus the number of game iterations (ddmax = 20 m,K = N = 5, 10, 15).

increasing transmission power beyond the point for optimumSE not only brings no SE improvement in an interference-limited environment but also causes significant EE loss.Note that, as the D2D transmission distance increases, theEE performance of all algorithms decreases because highertransmission power is required to maintain the same QoSperformance than in the scenario of short distance.

Fig. 5 shows the average EE performance of D2D pairsversus the number of active CUs K and D2D pairs N withddmax = 20 m. The average EE performance of all algorithmsincreases linearly as the active number of CUs and D2D pairsincreases. The reason is that as the number of CUs increases,not only the total number of available orthogonal channelsincreases, but also each D2D pair has a wider variety of choicein the expanded matching market than in the original one.The probability for a D2D pair to be matched with a betterpartner becomes higher in the expanded matching market. Theproposed algorithm has the steepest slope among the four,which indicates that it can exploit more benefits from thediversity of choices than the heuristic algorithms could. Boththe spectrum-efficient and the power greedy algorithms havethe flattest slope since the value of choice diversity is notfully exploited and power consumption is also ignored in theresource allocation process.

Fig. 6 shows the CDF of D2D pairs’ satisfactions versusvarious satisfaction thresholds with K = N = 20, 50 andddmax = 20 m. We adopt the Monte-Carlo method to calculatethe CDF that uses repeated matching results (104 times) toobtain the numerical results. In the case of K = N = 20,the probability of being matched to the first three choices forD2D pairs is 66.4%. In contrast, the corresponding probabilityunder random matching is only 15.4%. When the number ofD2D pairs and CUs is increased from 20 to 50, there is stillas high as 56.9% of D2D pairs that have been matched to thefirst three choices, while the corresponding probability underrandom matching is decreased dramatically from 15.4% toonly 6.4%. Significant UE satisfaction gains can be achieved

by the proposed algorithm compared to the random matching.In addition, the simulation results also reveal the fact that theproposed algorithm is able to outperform the random matchingfor a wide range of satisfaction values.

Fig. 7 shows the convergence of the iterative algorithm(Algorithm 1) versus the number of game iterations. It isshown that the proposed algorithm only requires 3 ∼ 4iterations to converge to the equilibrium. In the first gameiteration, higher EE performance can be achieved because CUsare not aware of the D2D pairs and transmit in lower powerlevels. With the entry of D2D pairs into the game, CUs have toincrease their transmission power to satisfy QoS requirements,which in turn causes co-channel interference to D2D pairsand reduce their achievable EE performance until they reacha Nash equilibrium.

V. CONCLUSIONS AND FUTURE WORK

In this paper, an energy-efficient stable matching algorithmwas proposed for the resource allocation problem in D2Dcommunications. Taking into account UEs’ preferences andsatisfactions, the joint partner selection and power allocationproblem was formulated to maximize the achievable EE undermaximum transmission power and QoS constraints. The re-sulting problem is nonconvex and computationally intractable.Inspired by the matching theory and game theory, the NP-hardproblem was transformed into a one-to-one matching problemwith UEs’ preferences modeled as the optimum EE undera specific matching. A noncooperative game based iterativealgorithm was proposed to establish mutual preferences byexploiting nonlinear fractional programming. The proposedmatching algorithm was proved to be stable and weak Paretooptimal. Extensive simulation results validate the effectivenessand superiority of the proposed algorithm. The present match-ing approach sheds new light on the research directions forresource allocation problems in green D2D communications.Potential future works include the extension of one-to-manymatching, the modeling of UE preference from a big-dataperspective, and the consideration of context-aware contentcaching, etc.

APPENDIX AProof of Theorem 2

Proof: First, if the strategy pd∗i obtained by the iterativealgorithm is not the Nash equilibrium, the D2D transmittercan choose the Nash equilibrium p̂di to obtain the maximumEE qd∗im . However, by Theorem 1 and Theorem 4, qd∗i can alsobe achieved by choosing pd∗i , and qd∗i is unique. As a result,pd∗i is also part of a Nash equilibrium. A similar proof holdsfor pd∗k .

Second, according to [48], a Nash equilibrium exists ifthe utility function is continuous and quasiconcave, and theset of strategies is a nonempty compact convex subset of aEuclidean space. Taking the EE objection function definedin (6) as an example, under the matching µ(di) = ck, thenumerator Udi defined in (2) is a concave function of pdi ,∀i ∈ N . The denominator defined in (4) is an affine functionof pdi . Therefore, Udi,EE is quasiconcave (Problem 4.7 in [45]).


The set of the strategies {[0, pdi,max]} is a nonempty compactconvex subset of the Euclidean space. Similarly, it is easilyproved that the above conditions also hold for the cellularUE. Therefore, a Nash equilibrium exists in the noncooperativegame.

Third, there may be multiple equilibria that satisfy theoptimality condition. This conclusion can be directly drawnfrom the properties of nonlinear fractional programming asshown in page 4 of [44], which shows that the solutionspd∗i of the equation Udi (pd∗i ) − qd∗i Edi (pd∗i ) = 0, and pc∗k ofthe equation U ck(pc∗k ) − qc∗k Eck(pc∗k ) = 0 may not be unique.Therefore, if there exists multiple solutions, the combinationof these solutions may generate multiple Nash equilibria.However, despite that there may be multiple equilibria, themaximum EE obtained by the iterative algorithm is unique.The proof is similar to the proof of Lemma 4 in [44], whichproves that the optimum result obtained by nonlinear fractionalprogramming is unique.

Finally, given a Nash equilibrium {pd∗i , pc∗k }, the corre-sponding EE {qd∗i , qc∗k } produced by Algorithm 1 is uniqueaccording to Theorem 1 and Theorem 4. If di chooses p̃dirather than pd∗i (p̃di 6= pd∗i ) as its transmission strategy, thecorresponding CU ck will also choose p̃ck rather than pc∗k(p̃ck 6= pc∗k ) to improve its individual EE. The iteration processin Algorithm 1 will continue until the strategy set {p̃di , p̃ck}converges to a new Nash equilibrium, and the correspondingEE is denoted as {q̃di , q̃ck}. Then, we must have q̃di = qd∗i , andq̃ck = qc∗k , which otherwise contradicts with Theorem 1 andTheorem 4. Therefore, this proves that either di or ck is ableto unilaterally achieve better performance by deviating fromthe Nash equilibrium. This completes the proof.

APPENDIX BProof of Theorem 3

Proof: For any di ∈ D and any ck ∈ C that are not matchedwith each other, µ is stable if (di, ck) do not form a blockingpair, i.e., di �ck µ(ck), ck �di µ(di). We prove the theoremby showing that the two necessary conditions di �ck µ(ck)and ck �di µ(di) cannot hold at the same time.

Let us begin from the assumption that ck �di µ(di), then arequest must have already been sent by di to µ(di) based onthe defined matching rules. With the matching result µ(di) 6=ck, it means that di is less preferred by ck compared to µ(ck),i.e., µ(ck) �ck di. Thus, although ck is more preferred by dithan µ(di), ck has no incentive to be matched with di, i.e.,the condition di �ck µ(ck) does not hold. The same provingprocess can be repeated with minor revision to show that thecondition ck �di µ(di) does not hold if di �ck µ(ck). Asa result, di and ck do not form a blocking pair for µ sincedi �ck µ(ck) and ck �di µ(di) cannot hold at the same time,which proves that µ is stable.

APPENDIX CProof of Theorem 5

Proof: ∀di ∈ D, we assume that there is a better matchingµ

′such that µ

′(di) �di µ(di). In other words, every di ∈ D

is matched to a better partner under µ′

compared to µ, that

is, every di ∈ D is matched to some CU under µ′

whichhas rejected its request under µ. Thus, every CU in the set ofµ

′(D) must have issued a rejection under µ. However, any CU

which receives a request in the final step of Algorithm 3 hasnot issued a rejection to any D2D pair due to the matchingrule. Otherwise, at least one more iteration is required to matchthe rejected D2D pairs with some CU. This contradicts theassumption of existing µ

′and proves that µ is weak Pareto

optimal for D2D pairs.

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Zhenyu Zhou (S’06-M’11) received his M.E. andPh.D degree from Waseda University, Tokyo, Japanin 2008 and 2011 respectively. From April 2012 toMarch 2013, he was the chief researcher at Depart-ment of Technology, KDDI, Tokyo, Japan. FromMarch 2013 to now, he is an Associate Professorat School of Electrical and Electronic Engineering,North China Electric Power University, China. He isalso a visiting scholar with Tsinghua-Hitachi JointLab on Environment-Harmonious ICT at Universityof Tsinghua, Beijing from 2014 to now. He served

as workshop co-chair for IEEE ISADS 2015, and TPC member for IEEE VTC2017, IEEE VTC 2016, IEEE ICC 2016, IEEE ICC 2015, Globecom 2015,ACM Mobimedia 2015, IEEE Africon 2015, etc. He received the ”YoungResearcher Encouragement Award” from IEEE Vehicular Technology Societyin 2009. His research interests include green communications and smart grid.He is a member of IEEE, IEICE, and CSEE.


Kaoru Ota (M’12) was born in Aizu Wakamatsu,Japan. She received M.S. degree in Computer Sci-ence from Oklahoma State University, USA in 2008,B.S. and Ph.D. degrees in Computer Science andEngineering from The University of Aizu, Japanin 2006, 2012, respectively. She is currently anAssistant Professor with Department of Informationand Electronic Engineering, Muroran Institute ofTechnology, Japan. From March 2010 to March2011, she was a visiting scholar at University ofWaterloo, Canada. Also she was a Japan Society of

the Promotion of Science (JSPS) research fellow with Kato-Nishiyama Labat Graduate School of Information Sciences at Tohoku University, Japan fromApril 2012 to April 2013. Her research interests include Wireless Networks,Cloud Computing, and Cyber-physical Systems. She has received best paperawards from ICA3PP 2014, GPC 2015, IEEE DASC 2015 and IEEE VTC2016. She serves as an editor for Peer-to-Peer Networking and Applications(Springer), Ad Hoc & Sensor Wireless Networks, International Journal ofEmbedded Systems (Inderscience), as well as a guest editor for IEEE WirelessCommunications, IEICE Transactions on Information and Systems. She iscurrently a research scientist with A3 Foresight Program (2011-2016) fundedby Japan Society for the Promotion of Sciences (JSPS), NSFC of China, andNRF of Korea.

Mianxiong Dong (M’13) received B.S., M.S. andPh.D. in Computer Science and Engineering fromThe University of Aizu, Japan. He is currently an As-sociate Professor in the Department of Informationand Electronic Engineering at the Muroran Instituteof Technology, Japan. Prior to joining Muroran-IT, he was a Researcher at the National Instituteof Information and Communications Technology(NICT), Japan. He was a JSPS Research Fellowwith School of Computer Science and Engineering,The University of Aizu, Japan and was a visiting

scholar with BBCR group at University of Waterloo, Canada supported byJSPS Excellent Young Researcher Overseas Visit Program from April 2010to August 2011. Dr. Dong was selected as a Foreigner Research Fellow (atotal of 3 recipients all over Japan) by NEC C&C Foundation in 2011. Hisresearch interests include Wireless Networks, Cloud Computing, and Cyber-physical Systems. He has received best paper awards from IEEE HPCC 2008,IEEE ICESS 2008, ICA3PP 2014, GPC 2015, IEEE DASC 2015 and IEEEVTC 2016. Dr. Dong serves as an Editor for IEEE Communications Sur-veys and Tutorials, IEEE Network, IEEE Wireless Communications Letters,IEEE Cloud Computing, IEEE Access, and Cyber-Physical Systems (Taylor& Francis), as well as a leading guest editor for ACM Transactions onMultimedia Computing, Communications and Applications (TOMM), IEEETransactions on Emerging Topics in Computing (TETC), IEEE Transactionson Computational Social Systems (TCSS). He has been serving as SymposiumChair of IEEE GLOBECOM 2016, 2017. Dr. Dong is currently a researchscientist with A3 Foresight Program (2011-2016) funded by Japan Society forthe Promotion of Sciences (JSPS), NSFC of China, and NRF of Korea.

Chen Xu (S’12-M’15) received the B.S. degreefrom Beijing University of Posts and Telecom-munications, in 2010, and the Ph.D. degree fromPeking University, in 2015. She is now a lecturerin School of Electrical and Electronic Engineering,North China Electric Power University, China. Herresearch interests mainly include wireless resourceallocation and management, game theory, optimiza-tion theory, heterogeneous networks, and smart gridcommunication. She received the best paper awardin International Conference on Wireless Communi-

cations and Signal Processing (WCSP 2012), and she is the winner of IEEELeonard G. Abraham Prize 2016.

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Energy-Efficient Matching for Resource Allocation in D2D ...source allocation problems for D2D communications are sum-marized and compared as follows. In [40], matching theory was