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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. XX, NO. XX, XXX 2019 1

Energy-Efficient Resource Allocation forHigh-Rate Underlay D2D Communications with

Statistical CSI: A One-to-Many StrategyRunzhou Li, Student Member, IEEE, Peilin Hong, Kaiping Xue, Senior Member, IEEE, Ming Zhang, Te Yang

Abstract—For the prominent superiorities in energy andspectrum efficiency, Device-to-Device (D2D) communication hasbecome a hot topic among the 5th generation (5G) technologies.However, the widely adopted mode that one or more D2Dlinks reuse one cellular user’s uplink spectrum may lead tosevere limitation on D2D communication rate, especially whencellular user’s uplink spectrum is very limited. In this paper,we will face the high rate requirements of D2D pairs (DPs),with the help of carrier aggregation technology, each DP canreuse the uplink spectrum of multiple cellular users when needed.More practically, we consider that only statistical channel stateinformation (CSI) of certain communication links is availablehere. Then, we formulate the problem to minimize the totalpower consumption of the mobile devices to obtain an energy-efficient resource allocation result, including spectrum and powerallocation. Meanwhile, the quality of service (QoS) requirementsof cellular and high-rate D2D communications are both ensured.The formulated problem is mathematically a non-convex mixed-integer nonlinear programming (MINLP) problem, which is NP-hard. To solve it, we first tighten the constraints and deploytransformation methods on it to make it convex. Then, wepropose a two-layer algorithm, the independent-power greedy-based outer approximation (IPGOA), to solve the transformedproblem. Besides, to handle the involved uniform power alloca-tion circumstance in the carrier aggregation process, a uniform-power GOA (UPGOA) algorithm, which can be regarded as asimplified version of IPGOA, is also proposed. The simulationresults show that in different scenarios, our proposed scheme isapproximately optimal.

Index Terms—D2D communications, spectrum and power al-location, high-rate, carrier aggregation, outage probability, non-convex MINLP.

I. INTRODUCTION

THE upcoming 5th generation (5G) communication hasput forward higher requirements on cellular systems,

especially in the aspects of transmission rate, system capacity,etc [1], and these new requirements have inspired researchersto develop more promising communication paradigms. Device-to-Device (D2D) communication, as a communication patternthat enables direct communication between user equipments(UEs), has attracted much attention for the benefit of helping

Copyright (c) 2015 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].

R. Li, P. Hong, K. Xue, M. Zhang and T. Yang are with the KeyLaboratory of Wireless-Optical Communications, School of InformationScience and Technology, University of Science and Technology of Chi-na, Hefei 230026, China (e-mail: [email protected], {plhong, kpx-ue}@ustc.edu.cn, {mzhang95, yangte}@mail.ustc.edu.cn).

K. Xue is also with the school of Cyber Security, University of Scienceand Technology of China, Hefei 230026, China.

cellular networks to achieve higher energy and spectrumefficiency, etc [2, 3]. With D2D communications, the UEs thatare within certain proximity to each other do not need to passthrough the base station (BS) to share contents. Moreover,D2D communications are also enabled to reuse the spectrumallocated to cellular users (CUs), which is effective for allevi-ating the scarcity of licensed spectrum and has attracted muchattention [4].

In D2D communications, the underlay D2D communication[2, 5] is mostly investigated for the feature of reusing some ofthe spectrum allocated to cellular communications. In underlaymode, one crucial issue is the interference coordination be-tween CUs and D2D pairs (DPs) [6–10]. Within the spectrumreusing modes in underlay D2D communication, reusing theuplink spectrum of cellular users is usually preferred, becausethe BS may cause serious interference to D2D communicationsduring downlink transmission if D2D links reuse the downlinkspectrum of CUs [5]. Most published literatures adopted theone-to-one (one DP reuses one CU’s spectrum) or many-to-one (many DPs reuse one CU’s spectrum) spectrum reusingscheme for various purposes, like simplicity of deploymentor purely for higher spectrum efficiency, etc [6–14]. Whilethe popularity of mobile social media sharing has put forwardunprecedented demand for higher transmission rate to ensureusers’ Quality of Experience (QoE), but the cruel reality is thatthe cellular uplink spectrum is usually scarce, and the uplinkspectrum allocated to each CU is usually narrow [15]. Thusboth schemes above come with an inherent drawback, i.e., aDP is generally not able or allowed to transmit with high datarate due to the limited bandwidth of a CU, simply increasingthe transmit power cannot bring significant improvementsunder the interference and power limits. Therefore, it is exigentto investigate how to meet the increasing demand of highertransmission rate in D2D communications.

The technology focusing on aggregating multiple spectrumis called carrier aggregation (CA), and it has been developedin the past years and implemented in cellular networks likethe Long Term Evolution-Advanced (LTE-A) systems [16–18].CA was also proposed to integrated into D2D communicationsystems [19]. With the aid of CA techniques, a DP will beable to reuse the uplink spectrum of multiple CUs, thus thisprovides us a new inspiration to deal with the above problem.Moreover, the increasing demand on high-rate communicationmay also lead to the increase of power consumption, whileonly little progress has been made in battery technologies.Thus, energy-efficient methods should be deployed to make

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full use of the limited energy in UEs while guaranteeing theQoS of communications.

In this paper, we propose to combine the CA technologyto meet the high rate requirements of DPs, where the one-to-many strategy is considered, i.e., a DP can reuse no less thanone CU’s uplink spectrum. However, full channel state infor-mation (CSI) is generally unavailable due to the time-varyingcharacteristic of channels [20, 21]. Thus, we consider that onlypartial instantaneous CSI of links is available here [20], as forthe D2D and interference links, only statistical CSI is avail-able. In order to obtain an energy-efficient resource allocationresult, we aim to minimize the total power consumption ofCUs and DPs while meeting their QoS requirements with onlystatistical (partial) CSI. Therefore, the major contributions canbe summarized as follows:

1) We propose to use the CA technology in D2D communi-cations to meet the high rate requirements of DPs, and weaim to accomplish the spectrum and power allocation tominimize the total power consumption of mobile devicesunder only statistical (partial) CSI. Especially, the QoSrequirements of DPs, where each DP may reuse the up-link spectrum of multiple CUs, are ensured in the form ofoutage probability from the entire DP’s perspective, ratherthan separately from each reused spectrum’s perspective.To the best of our knowledge, it is the first time thatthis synthesis issue is considered. And the formulatedproblem is mathematically a MINLP problem which hasbeen proven to be NP-hard.

2) To address the formulated problem, due to the difficultyof directly solving it, we first deploy transformationand variable substitution methods to make the problemmore tractable. After that, the resource allocation ofCUs is coupled into the resource allocation of DPs, andthe problem becomes concave under specific conditions.Then we propose a two-layer algorithm, the independent-power greedy-based outer approximation (IPGOA), tosolve the transformed problem. Furthermore, we have al-so considered the uniform power allocation circumstancein CA technology, and then propose the uniform-powerGOA (UPGOA) algorithm.

3) At last, simulation results are presented to assess theeffectiveness of the proposed methods. The proposedIPGOA algorithm can closely approach the optimal result,and the UPGOA can also obtain distinguished results insome circumstances. Further more, deployment scenariosare also distinguished in the comparison results.

The remainder of this paper is organized as follows. Somerelated works are presented in section II. In section III,the system model is depicted in the aspect of network andspectrum reusing model, channel model, problem formulationrespectively. Section IV focuses on the analysis and transfor-mation of the mathematical problem. In section V, the twoproposed methods IPGOA and UPGOA as well as a spacecompression method for spectrum allocation are elaboratedin detail. Numerical simulations and comparison results areshown in section VI. Finally, section VII concludes this paper.

II. RELATED WORKS

Researchers have been working on the spectrum and powerallocation issue in D2D communications [6–9], where maxi-mizing the communication rates has received much attention.Hoang et al. [6] took the proportional fairness of commu-nication rate into consideration during resource allocation.Luo et al. [7] proposed an iterative algorithm as well as aheuristic one with lower complexity to maximize the systemsum rate in energy harvesting scenario, but a single D2Dlink can only reuse one CU’s uplink spectrum. Authors in [8]proposed to maximize the sum rate of D2D communicationswith optimization and game theoretical methods in centralizedand distributed manner respectively. Sun et al. [9] used acheating strategy in Gale-Shapley algorithm to maximize thesum rate of D2D communications. In terms of the workcombining D2D communication with CA technology, only fewliteratures have considered the resource allocation issue. Forexample, the authors [19] considered a balance between powerminimization and UE target rate while spectrum allocation andmode selection in D2D communications, where CA is onlyconsidered in the BS. Nevertheless, the above literatures are allbased on the technically impractical assumption that full CSIis available to devices. Besides, few of the schemes in theseliteratures are suitable for high-rate D2D communications,one-to-one or many-to-one modes are mostly adopted therein.

In addition, the energy efficiency issue in D2D commu-nications has also received considerable attention, especiallyunder the background that mobile devices still suffer from thepoor capacity and slow development of batteries. The energyefficiency in D2D communications is usually investigated intwo ways, including the energy efficiency of links and theenergy consumption of links [10–14, 22]. Wang et al. [10]proposed to maximize the energy efficiency of D2D links inenergy harvesting scenario, an iterative method is presentedto solve the resource allocation problem. Similarly, the energyefficiency maximization problem is also investigated in [11–13]. Different from the above optimization objectives, someother researchers focus on minimizing the power consumptionof D2D communications or the whole system [14, 22], andthis is also the objective of our work in this paper. Ying etal. [14] took the social relationships into consideration andthen came up with a power efficient method to accomplishrelay selection in multi-hop D2D communication. In [22], theauthors proposed to decompose the joint subcarrier and powerallocation problem into two subproblems, through solvingtwo subproblems, they obtained an energy-efficient allocationresult.

With the aim of achieving higher energy efficiency in D2Dcommunications, the above literatures mostly assume that theBS or DPs know the full CSI of communication links [8–14, 22]. However, in real communication scenarios, full CSIis usually impractical and only partial CSI is available. Somework has been done on utilizing partial CSI to accomplishresource allocation [20, 21, 23, 24]. Wang et al. [20] investi-gated the expected transmission rate maximization problemin Rayleigh fading channels with partial CSI, and the socalled “one-to-many” case is simplified as independent one-





to-one cases. In [21], authors proposed an optimal dynamicprogramming algorithm to finish the channel assignment underpartial CSI, where the BS is assumed to be able to acquirepartial CSI of cellular, D2D and interference links. Sun et al.[23] aimed to minimize the D2D transmit power to reducethe interference caused by D2D communications, as wellas optimize the selection of access threshold with statistical(partial) CSI. However, none of them considered or are suitablefor the resource allocation with high rate requirements of DPs,the D2D transmission were only considered independently onreused spectrum instead of from an entire DP’s perspective,and the total power minimization issue was not consideredeither.

III. SYSTEM MODEL

In this section, we depict the network and spectrum reusingmodel in the first part. Then, we give explanations of thestatistical CSI model and the channel model. At the end,communication QoS requirements are expressed as the outageprobability constraints, and the mathematical model of theenergy-efficient spectrum and power allocation problem isformulated.

A. Network and Spectrum Reusing Model

In our model, we consider a cellular area illustrated in Fig.1 where the CUs and DPs coexist in the area and the numberof CUs is much more than DPs. The BS is in the center of theround cellular area with omnidirectional antennas, and CUsare uniformly distributed within the cellular area. The set ofall the CUs is denoted by ΦC = {c|c = 1, 2, ..., NC}, whereNC is the number of CUs. And the DPs exist near the edgeof the cellular area, here we denote the set of DPs by ΦD ={d|d = 1, 2, ..., ND}, where ND is the number of DPs.

D2D linkCU1

CU2

CU3

BS

Communication Signal Interference Signal

DP1

DP2

Sublinks

DP2

Fig. 1: System model of underlay DPs reusing the uplinkspectrum of CUs in a cell. Each CU transmits in uplink mode,DP2 activates two sublinks to transmit with higher rate.

In this scenario, each CU is allocated a chunk of spectrumwith bandwidth B for uplink transmission, where each CU’s

spectrum is called a component carrier (CC), and those CCsare orthogonal to each other. For the DPs, each DP is equippedwith CA techniques to aggregate multiple CCs for a broaderbandwidth [16]. Here, each DP can activate up to M sublinksto aggregating M CCs for higher transmission rate. In a DP,each sublink can be activated to reuse the uplink spectrumof a CU, or simply shut down if not needed. Moreover, weintroduce a binary variable xc,d,k = {0, 1} to indicate thespectrum reusing relationship between CU c and DP d’s k-thsublink. When xc,d,k = 1, it means the k-th sublink of DPd is activated to reuse the uplink spectrum of CU c, and thevalue “0” means that the reusing relationship does not exist.Besides, if

∑cxc,d,k = 0, it means the k-th sublink of DP d

is shut down. Here, we assume that each sublink can eithertransmit by reusing one CU’s uplink spectrum or simply shutdown, and the uplink spectrum of a CU can only be reusedby up to one sublink to avoid multiple interference sources.Thus the above assumptions can be expressed by the followingconstraints on xc,d,k:∑

d

∑k

xc,d,k ≤ 1, (1a)∑c

xc,d,k ≤ 1, (1b)∑c

∑k

xc,d,k=Md, (Md ≤M), (1c)

where Md is introduced to denote the number of the activatedsublinks in DP d, and it is also called as DP d’s reusingnumber. Specifically, Md is a variable here, which can be atmost M because of the physical limits in DPs [18].

In addition, we denote the transmit power of CU c by pc,and pd,k denotes the transmit power of the k-th sublink ofDP d, where the power consumption of the receiver is notconsidered because of the low-power property of the decodeprocess. For a DP d, the transmit power on its sublinks canbe independent or uniform on the reused CCs [16, 18], wechoose to consider the independent-power circumstance first,then the uniform-power scenario will be referred at the endingpart.

B. Statistical CSI and Channel Model

In this paper, we conduct our work under the Rayleigh fad-ing channels, which can characterize a large class of wirelesscommunication channels and have been widely adopted inliteratures [25, 26]. The Rayleigh fading channel gain obeysan exponential distribution [27–29], whose probability densityfunction (PDF) is depicted as:

f(x) = λe−λxu(x), (2)

where the parameter λ is the distribution coefficient, and xis the random variable. u(x) is the unit step function, whichtakes 1 when x > 0, or 0 otherwise.

Assume that BS can only obtain statistical information ofcertain links, including the channel gain of the interferencelinks and the communication links that are not directly con-nected to the BS. To be specific, the BS has statistical informa-tion hd ∼ exp(λd), hc,d ∼ exp(λc,d) and hd,c ∼ exp(λd,c),





which represent the channel gain between DP d’s tranceiver,the interference channel gain from DP d to CU c and fromCU c to DP d respectively.

For the channel gain of cellular links, they are assumed to betimely and perfectly estimated by the BS within the coherenttime [27]. Here we denote the uplink channel gain of CU cby hc, it is perfectly estimated by the BS and supposed to befixed during a time slot [30, 31]. For hc, it can be expressedas hc = L−αc · |h0|2, where Lc is the distance between CU cand the BS, α is the path loss coefficient, and h0 ∼ CN (0, 1)is a zero mean Gaussian random variable with unit variance.

Thus, based on the above conditions, the achievable com-munication rate of cellular links and D2D links are separatelyformulated according to the Shannon formula as:

rc = B · log(1 +pchc

n+∑k

∑d

xc,d,kpd,khc,d), (∀c ∈ ΦC), (3)

rd = B ·∑k

log(1 +pd,khd

n+∑cxc,d,kpchd,c

), (∀d ∈ ΦD), (4)

where n is the noise power which equals to B ·ρ, and ρ is thedensity of noise.

Based on (4), it is easy to find that the integer variable xc,d,kcan be taken out of the log function and therefore becomes amultiplier. Thus the achievable communication rate of d canbe transformed as follows:

rd,k =∑c

xc,d,k ·B log(1 +pd,khd

n+ pchd,c), (5a)

rd =∑k

rd,k, (∀d ∈ ΦD), (5b)

where rd,k is the achievable communication rate of the k-thsublink of DP d, and it will be zero when

∑cxc,d,k = 0,

which means the k-th sublink of DP d is not activated totransmit data. Recall the beginning of this section, the aboveresults are calculated with random variables, thus it is obviousthat the achievable communication rate of DP d is also arandom variable. Similarly, the communication rate of CUc in (3) is also a random variable when CU c suffers frominterference from any DP, while it will be an exact value whenthe interference term values 0.

C. Problem FormulationIn this paper, we aim to obtain an energy-efficient spectrum

and power allocation result while guaranteeing the QoS re-quirements of CUs and DPs. The total power consumption ofmobile devices is used to reflect the energy efficiency of thesystem.

Based on statistical CSI, the QoS requirements are rep-resented as outage probability constraints on communicationlinks that transmit with certain rates. For DP d, the requiredtransmission rate is denoted by rRd , thus the outage probabilityis the probability that the achievable D2D communication rateis lower than the required transmission rate. Therefore, if DPd transmits with a required transmission rate of rRd , its QoSrequirement can be expressed as

Pr{Od}, Pr{rd ≤ rRd } ≤ ηd, (∀d ∈ ΦD), (6)

where ηd is the required outage probability threshold. Thecomplicated data stream allocation issue, which decides howmuch data to be allocated on each sublink inside DP’s trans-mitter, is out of the scope of the resource allocation issue hereand is simplified. Similar to [32], we allocate the requireddata transmission rate evenly on the activated sublinks in aDP. Since DP d requires a transmission rate of rRd , with Md

activated sublinks, the data transmission rate allocated on eachsublink is rRd,sub, obtained by

rRd,sub =rRdMd

, (∀d ∈ Φd). (7)

Similarly, the QoS requirements on cellular links can beexpressed as

Pr{Oc} , Pr{rc ≤ rRc } ≤ ηc, (∀c ∈ ΦC), (8)

where rRc is the required cellular transmission rate and ηc isthe outage probability threshold.

Therefore, with the objective of obtaining an energy-efficient spectrum and power allocation result, the mathemat-ical model can be formulated as follows:

P1 : minPC ,PD,X

∑c

∑d

∑k

xc,d,k · (pd,k+psub0 )+∑c

(pc + pc0),

(9a)s.t. (1a), (1b), (1c), (9b)

(6), (8), (9c)0 ≤ pc ≤ pmax

c (∀c ∈ ΦC), (9d)0 ≤ pd,k ≤ pmax

sub , (∀c ∈ ΦC ,∀d ∈ ΦD), (9e)∑k

∑c

xc,d,k ·rRd,sub = rRd , (∀d ∈ ΦD), (9f)

rRd,sub ≥ rth0 , (∀d ∈ ΦD), (9g)

where pc0, psub0 are the static power of a CU and an activatedD2D sublink respectively, and they are generally constantvalues [17]. pmax

c , pmaxsub are the maximum transmit power of

CU c and DP’s each sublink respectively. PC , PD and Xstand for the multiplex indicator parameter set of pc, pd,k andxc,d,k respectively, for example, X = {xc,d,k, (c ∈ ΦC , d ∈ΦD, k = 1, ...,M)}. Similarly, the uppercase form of variablesin the rest of this paper follows the same rule. Moreover, (9f)makes sure that, in a DP, the sum data transmission rate of allactivated sublinks is equal to the DP’s required transmissionrate. Besides, as (9g) shows, the allocated transmission rateon each activated sublink in DP d should not be lower thanthe threshold.

As shown above, the mathematical model is formulated withthe objective of minimizing the total power consumption ofCUs and DPs, it is obvious that the objective function is inlinear form and is linear with respect to the integer variablexc,d,k. Besides, the constraint (9c) is apparently non-concave.As a result, the problem P1 is a typical non-convex MINLPproblem which has been proven to be NP-hard [33].

IV. PROBLEM ANALYSIS AND TRANSFORMATION

To analyze the problem more explicitly, the expanded outageprobability expressions of the cellular and D2D links are





derived in this section. Moreover, due to the difficulty ofsolving the original problem P1, transformation operationsare employed on it, finally, an equivalent problem in a moretractable form is obtained.

A. Outage Analysis and Expression

In section III, the outage probability expressions of thecellular and D2D links are formulated only according tothe definitions. However, those expressions are intuitivelyhard to understand, and not convenient for problem solvingeither. Thus, we will first derive elaborated outage probabilityexpressions in the following.

1) Outage Analysis of Cellular Links: Firstly, analysis onthe outage probability of cellular links is introduced. For acellular user, whether its uplink spectrum is reused by a DPwill lead to different outage expressions of the cellular uplink.

Let zc = ercB − 1 and zmin

c = erRcB − 1, thus (3) can be

transformed into:

pchcn+

∑k

∑d

xc,d,kpd,khc,d= zc. (10)

For variable xc,d,k, assume that ∃xc,d,k = 1, i.e., the uplinkspectrum of CU c is reused up a DP’s sublink, then (10) canbe simplified as

pchcn+ pd,khc,d

= zc, (xc,d,k = 1), (11)

based on which the outage probability can be formulated as

Pr{Oc} = Pr{zc ≤ zminc }

= Pr{ pchcn+ pd,khc,d

≤ zminc }

= Pr{hc,d ≥pchczminc− n

pd,k}, (xc,d,k = 1).

(12)

Thus, based on the PDF of hc,d, the outage probability canbe calculated through the integral operation, which gives

Pr{Oc} =

+∞∫( pchczminc−n)/pd,k

λc,de−λc,dxdx

= e−λc,dpd,k

(pc·hczminc−n

), (xc,d,k = 1).

(13)

However, we notice that the above outage probability expres-sion only covers the scenario where the uplink spectrum of CUc is reused. Considering another circumstance, where there isno interference caused onto the cellular link of CU c, i.e.,∑d

∑k

xc,d,k = 0, the outage probability can be formulated in

a simpler form:

Pr{Oc} = Pr{zc ≤ zminc }

= Pr{pchcn≤ zmin

c }

= Pr{pc ≤nzmin

c

hc}, (∑d

∑k

xc,d,k = 0).

(14)

Based on (13) and (14), the outage probability of CU c canbe jointly formulated as

Pr{Oc} =

e−λc,dpd,k

(pc·hczminc−n

), (∃xc,d,k = 1)

u(pc <

nzminc

hc

), (∑d

∑k

xc,d,k = 0), (15)

where the step function u(·) takes boolean operation on theinput, and we omit recognizing the step point (i.e., pc =

nzminc

hc)

as outage.2) Outage Analysis of D2D Links: With respect to the

outage probability of DP d, it can be formulated accordingto the following theorem:

Theorem 1. Based on (6) and the not outage probability ofeach activated sublink, here is the expanded form of the outageprobability expression of DP d:

Pr{Od} = 1−∏

c,k:∀xc,d,k=1

Pr{rd,k ≥ rRd,sub}. (16)

Proof. Based on the definition in (6), the outage probabilityof DP d can be calculated based on the not outage probability,shown as:

Pr{Od} = 1− Pr{rd ≥ rRd , rd,k ≥ rRd,sub, ..., (∀xc,d,k = 1)}.

For DP d, the activated sublinks separately transmit with adata rate of rRd,sub. However, from the DP’s perspective, ifoutage happens on any of the activated sublinks, data will notbe completely or successfully transmitted, thus leading to theoutage of DP d. And based on Bayes formula, we have

Pr{rd ≥ rRd , rd,k ≥ rRd,sub, ..., (∀xc,d,k = 1)}= Pr{rd,k1

≥ rRd,sub} · Pr{rd,k2≥ rRd,sub|rd,k1

≥ rRd,sub}· ... · Pr{rd ≥ rRd |rd,k1

≥ rRd,sub, ..., rd,kMd ≥ rRd,sub},

where rd,k1, ..., rd,kMd are the achievable rates of DP d’s sub-

links. In the above equation, each condition in the conditionalprobability term is independent, thus (16) is obtained.

With Theorem 1, it is obvious that only if any sublink in DPd does not experience an outage, the DP will not experiencean outage. Therefore, the outage probability of DP d can beobtained by calculating the product terms in (16). It is easyto notice that each product term in (16) corresponds to thenot outage probability of each sublink, thus we denote eachproduct term by Pr{Od,k}

∆= Pr{rd,k ≥ rRd,sub}. Then, for a

certain xc,d,k = 1, let zd,k = erd,kB −1 and zmin

d,k = erRd,subB −1,

thus the not outage probability of the k-th sublink of DP d is

Pr{Od,k}=Pr{zd,k ≥ zmind,k }=Pr{ pd,khd

n+ pchd,c≥ zmin

d,k }. (17)

Let us denote U = pd,khd and V = n+ pchd,c, thus thenot outage probability in (17) can be obtained through thefollowing integral operation:

Pr{Od,k}= 1− Pr{Od,k}

= 1−+∞∫n

v·zmind,k∫

0

[λdpd,k

e− λdpd,k

u]·[λd,cpc

eλd,cpc

(n−v)]dudv.

(18)





Thus the integral result gives

Pr{Od,k} = ψd,k(pc, pd,k, rRd,sub(X))

=Td,k

Td,k + T cd · zmind,k

e−zmind,kTd,k ,

(19)

where Td,k = 1λd

pd,kn = 1

λdSNRd,k, T cd = 1

λd,c

pcn =

1λd,c

SNRc, and rRd,sub(X) means rRd,sub is related to the valueof X .

Therefore, the outage probability of DP d can be representedaccording to the following corollary.

Corollary 1. The outage probability of DP d can be expressedas

Pr{Od}=1−∏

c,k:∀xc,d,k=1

ψd,k(pc, pd,k, rRd,sub(X)). (20)

B. Problem Transformation

Recall the previous section, the expressions in (15) and (20)are piecewise and non-convex respectively, which makes theoriginal problem P1 very difficult to solve. In this section,we implement some transformation operations on the originalproblem to obtain a more tractable equivalent problem.

As depicted in P1, we aim to obtain an energy-efficientspectrum and power allocation result to minimize the totalpower consumption of CUs and DPs. With this objective, hereis a theorem:

Theorem 2. The constraints on the outage probability ofcellular links are always binding in the process of searchingfor the optimal solution of P1, as well as when the solutionis obtained. And under this circumstance, the boundary valueof pc is always within feasible range and thus is reachable.

Proof. In P1, the goal is to consume as less power as possiblewhile meeting the communication requirements within thefeasible range of the problem. Assume that in the solvingprocess, under any feasible value of the spectrum reusingvariable X , which is denoted by X† here, the transmit powerof CU c and the k-th sublink in DP d are p†c and p†d,k,respectively.

For any feasible p†c and p†d,k, the constraints (9b)∼(9g) areall satisfied. If Pr{Oc} ≤ ηc, there must exist p′c ≤ p†c andp′d,k ≤ p

†d,k making the objective power consumption become

less or remain unchanged without breaking the constraints.Regarding to the above conclusion, we have to mention thatPr{Oc} monotonically decreases with the increase of pc whenwe fix other variables. Similarly, Pr{Od} is monotonicallynon-increasing when pc decreases. These characteristics makesure that the constraints still hold when decreasing pc. Thus,as long as we ensure that Pr{Oc} ≤ ηc holds, a solution withlower power consumption can be obtained when the boundaryvalue of pc is reached. Therefore, the proof is completed.

Based on Theorem 2, let us consider (15) and (20). In(15), for CU c when ∃xc,d,k = 1, i.e., the uplink spectrumof CU c is reused by the k-th sublink of DP d, the outage

probability constraint e−λc,dpd,k

(pc·hczminc−n

)≤ ηc holds. Thus it

can be transformed into a constraint on pc as

pc ≥(− log ηcnλc,d

pd,k + 1

)nzmin

c

hc. (21)

Besides, assume that for a CU c,∑d

∑k

xc,d,k = 0 holds, i.e.,

the uplink spectrum (a CC) is not reused by any DP. Similarly,to ensure that u

(pc <

nzminc

hc

)≤ ηc holds, we have to make

sure

pc ≥nzmin

c

hc. (22)

Considering the binding characteristic of the outage proba-bility constraints on cellular links referred in Theorem 2, hereis a corollary.

Corollary 2. pc can be jointly expressed as

pc =

(∑d

∑k

xc,d,k ·− log ηcnλc,d

pd,k + 1

)nzmin

c

hc, (23)

Proof. According to Theorem 2, (21) and (22) are binding,thus the above equation can be obtained.

Therefore, the outage expression of cellular links can betightened as (23), by taking (23) into P1 to replace pc, thepiecewise cellular outage constraints are well handled.

Then, by introducing variable substitution on pd,k by pd,k =e−pd,k , the objective function of P1 becomes convex, shownas below:

f0(PD, X)

=∑c

∑d

∑k

xc,d,k

1 +− ln ηcλc,d

(erRcB − 1)

hc

e−pd,k+psub0

+∑c

(n(e

rRcB − 1)

hc+ pc0)

=f(PD, X)+∑c

(n(e

rRcB − 1)

hc+ pc0),

(24)where the summation term at the end is independent from thevariables, thus it can be omitted in the solving process. Forbrevity, we refer to f(PD, X) as the objective function in therest of this paper.

For the outage probability of D2D links, after substitutingthe variables in (20) with the same substitution approachabove, we exchange the product term with the right side in(20) and take the logarithm of both sides. The expression ofthe outage probability of DP d is finally obtained, as shownin (25) at the bottom of this page.

Based on the above analysis and transformation results, thetransformed problem can be formulated as follows:

P1’ : maxPD,X

−f(PD, X) (26a)

s.t. (9b), (9e), (9f), (9g), (26b)(25), (26c)− log(min{pmax

sub , xc,d,k ·φc,d(pmaxc )}) ≤ pd,k, (26d)





where the function φc,d(pmaxc ) =

nλc,d− ln ηc

(pmaxc ·hc

n(erRc /B−1)

− 1)

issimilar to the inverse function of (23), it makes sure that thetransmit power constraints in P1 are still effective in P1’.

Through the above transformations, it is obvious that theobjective function of problem P1’ is concave, constraint (26c)is concave when we fix Md, and the other constraints arelinear. Thus, based on the structure of P1’, the solving methodwill be put forward in the following section.

V. A TWO-LAYER GREEDY-BASED OUTERAPPROXIMATION METHOD AND SOME SUPPLEMENTS

Several main issues that remain to be solved in the first placecan be listed as follows: Which CU’s uplink spectrum shouldthe DPs reuse; How much power should UEs transmit with. Tosolve the transformed problem, we first propose a two-layeriterative algorithm IPGOA in the first subsection. Furthermore,to eliminate unnecessary computations in spectrum allocation,we propose a supplemental method for compressing the spec-trum allocation space. And at the end of this section, wepropose the UPGOA algorithm for the scenario where uniformpower is allocated on the activated sublinks in a DP.

A. A two-layer Greedy-based Outer Approximation Method

Before introducing the solving method, we have to mentionthat as described in (7), the allocated data transmission raterequirement on each activated sublink rRd,sub is related to thereusing number of the uplink spectrum of CUs, which makesrRd,sub a variable rather than a constant. Thus, to solve thisissue, we develop a two-layer algorithm:• In the outer layer, by greedily searching the reusing

number of the uplink spectrum of CUs for each DP, thenear-optimal reusing number can be obtained.

• In the inner layer, with the fixed reusing number passedfrom the outer layer, problem P1’ becomes a concaveMINLP problem which can be solved by employing theouter approximation (OA) method.

1) The Outer-layer Greedy Searching Method: In this layer,we aim at searching for near-optimal reusing numbers initerations.

Firstly, initialize by setting the reusing number Md =min{M, rRd /r

th0 } for each DP d ∈ ΦD and then

(a) Sequentially choose one DP d (d ∈ ΦD) and continue tothe next step.

(b) For DP d, monotonically decrease Md by 1 (i.e., M ′d =Md − 1) each time and calculate the objective powerconsumption under (M ′d,MΦD\d), where MΦD\d standsfor the value vector of Md′(d

′ ∈ ΦD \ d), the calculation

method will be discussed in section V-A2. In addition, ifthe problem becomes infeasible, refer to operations in (d1)and (d2) until the problem becomes feasible.

(c) Compare the objective power consumption under reusingnumber (M

′

d,MΦD\d) with the previous one with(Md,MΦD\d). If the total power decreases, let Md = M ′d,otherwise, set DP d’s reusing number as M∗d . If there areleft DPs to be traversed, return to (a), otherwise continueto the next step.

(d) Traverse DPs in ΦD one by one to slightly adjust Md.For each d ∈ ΦD, keep traversing according the followingsteps until the objective power consumption stop decreas-ing.(d1) Increase M∗d by 1, compare the objective power

consumption with the previous one. If it decreases,set M∗d = M∗d + 1 and continue increasing M∗d untilthe objective power consumption stops decreasing,else go to next step.

(d2) Decrease M∗d by 1, compare the objective powerconsumption with the previous one. If it decreases,set M∗d = M∗d −1 and continue decreasing M∗d untilthe objective power consumption stops decreasing.Then, start from another d and repeat the aboveoperation. If the total power does not decrease ina full traversal, quit the loop.

Therefore, based on the above descriptions, the outer-layeralgorithm is described in Algorithm 1. Through the analysisbelow, we can prove that the outer-layer algorithm can findnear-optimal reusing numbers.

Theorem 3. For a DP d in DP set ΦD, there must exist aunique optimal reusing number of Md with which the minimumobjective power consumption under MΦD\d is obtained.

Proof. Let us omit the solving procedures in the inner layer,within the finite value range of Md, it is obvious that thereexists a value of Md that outputs the minimum objective powerconsumption under MΦD\d.

Proposition 1. The above greedy searching method can findnear-optimal allocation of reusing numbers for DPs.

Proof. Let us omit the solving procedures in the inner layer.With the decrease of Md, less sublinks are activated fortransmission, thus the required data transmission rate on eachsublink (i.e., rRd,sub) increases, while the total static power con-sumption of sublinks decreases. For each sublink, they musttransmit with higher power to meet the increasing transmissionrate requirement. However, if the transmission rate on eachsublink keeps increasing, the transmit power increases faster

ψd(PD,X) , ln∏

c,k:∀xc,d,k=1

ψd,k(pc, pd,k, rRd,sub(X))

=∑k

∑c

−xc,d,k

ln

1 +λdn(e

rRcB −1)

λd,chc

(− ln ηcnλc,d

+epd,k)(

erRd,sub(X)

B − 1

)+epd,kλdn(erRd,sub(X)

B −1)

≥ ln(1−ηd)(25)





Algorithm 1: Greedy-based Outer-layer Iterative Method

1 Initialization: Set Md for each DP d ∈ ΦD byMd = min{M, rRd /r

th0 }(d ∈ ΦD), fc =∞, δ = 10−3 is

the computational error tolerance, f |(Md,MΦD\d) standsfor the objective power consumption under (Md,MΦD\d);

2 for each d ∈ ΦD do3 while f |(Md,MΦD\d) ≤ fc do4 Let M ′d = Md − 1, calculate f |(M ′d,MΦD\d);5 if fc − f |(M ′d,MΦD\d) ≥ δ and M ′d ≥ 1 then6 Md = M ′d, fc = f |(M ′d,MΦD\d);7 else8 M∗d = Md; Break;9 end

10 end11 end12 do13 for d ∈ ΦD do14 if fc − f |(M∗d−1,MΦD\d) ≥ δ and M∗d > 1 then15 M∗d = M∗d − 1, fc = f |(M∗d ,MΦD\d);16 else if fc − f |(M∗d+1,MΦD\d) ≥ δ and

M∗d < min{M, rRd /rth0 } then

17 M∗d = M∗d + 1, fc = f |(M∗d ,MΦD\d);18 else19 Proceed to another DP;20 end21 end22 while Any Md is changed in a loop;

and faster. Once the Md reaches a certain value, the objectivepower consumption no more decreases.

Due to the traversing characteristic on each DP wherethe reusing number of only one DP is chosen to change inone iteration. The circumstance that multiple DPs’ reusingnumbers are simultaneously changed is not considered here.However, the difference between changing multiple reusingnumbers one time and changing one Md in an iteration formultiple times is generally small. Firstly, Algorithm 1 makessure that, for DP d, any Md > M∗d or Md < M∗d will leadto an increase of the objective power consumption under thevalue of MΦD\d. Secondly, in our model, the number of CUs ismuch more than DPs, DPs can make spectrum reusing choicesfrom numerous CUs. Thus even under an optimal MΦD\d, aDP’s reusing number usually stays the same, and the objectivepower consumption may not or only slightly change. Thus thegap between the above algorithm and the global searchingwill be pretty small, which is also reflected in the simulationresults.

2) The Inner-layer Outer Approximation Algorithm: Withrespect to the inner-layer algorithm, it aims to calculate theobjective power consumption with a certain MΦD referredin the previous section, and the following discussions are allbased on the precondition that MΦD is a constant passed fromthe outer layer. And based on the formulated problem P1’ insection IV-B, some propositions are proposed.

Under a certain MΦD determined in the outer-layer iteration,problem P1’ has the properties described as below:

Proposition 2. In P1’, PD is a non-empty and convex set; theobjective function of (26a) is obviously concave; constraint(25) is obviously concave due to the form of log−exp−sumfunction.

Proposition 3. The objective function (26a) and constraint(26c) are first order continuous differentiable.

Proposition 4. A constraint qualification holds at the solu-tion of each concave nonlinear programming (NLP) problemobtained by fixing the values of X; the concave NLP problemobtained by fixing X can be solved by convex programmingalgorithms exactly.

Based on the above propositions, the inner-layer jointspectrum and power allocation problem P1’ under certainMΦD becomes a concave MINLP problem, which can bedecomposed into two subproblems, an NLP primal problemand a mixed integer linear (MILP) master problem [34]. Byfixing the integer variable X in P1’, a concave NLP problemis obtained. After solving the continuous variable PD, thesolution to the NLP problem provides a lower bound of thesolution to the concave MINLP problem P1’. And based onthe solution of the NLP problem, the MILP master problempredicts a new upper bound of the MINLP problem as wellas a new value of the integer variable in each iteration. Byiteratively solving the sequence of the NLP primal problemand the MILP master problem to obtain the non-increasingupper bound and non-decreasing lower bound, the algorithmfinally reaches an ε-convergence capacity.

Detailedly, in order to obtain the NLP primal problem, theinteger variable is fixed with X†, thus the NLP problem canbe formulated as the following:

P2.1 : maxPD

−f(PD, X†), (27a)

s.t. (25), (27b)

− log(min{pmaxsub , x

†c,d,k ·φc,d(p

maxc )}) ≤ pd,k. (27c)

It is obvious that P2.1 is a concave maximization problemthat can be rapidly solved by using optimization tools, likeCVX [35]. We denote P †D as the solution to problem P2.1,and the objective function value is f lowNLP .

The solution of the above NLP primal problem will be usedfor the master problem shown as follows:

P2.2 : maxX†

(maxPD

−f(PD, X†)

), (28a)

s.t.(26b) ∼ (27c), (28b)

which is the projection of problem P1’ on X . Besides, wenote that constraint (27c) can be decomposed into pd,k ≥− log(pmax

sub ) and log(φc,d(pmaxc /pmaxsub ))xc,d,k + log(pmax

sub ) +pd,k ≥ 0, where the second equation is denoted byϕc,d,k(pd,k, xc,d,k) ≥ 0 for brevity.





Based on Proposition 4 and the solution P †D of the primalproblem, the projection problem has the same solution as theone shown as below [36]:

P2.2’ : maxξ,PD,X

ξ, (29a)

s.t. −f(P †D, X†)−

(∇f(PD, X)

)T ( PD − P †DX −X†

)− ξ ≥ 0,

(29b)

ψd(P†D, X

†) +(∇ψd(PD, X)

)T(PD − P †DX −X†

)−ln(1− ηd) ≥ 0,

(29c)

ϕc,d,k(P †D, X†) +

(∇ϕc,d,k(PD, X)

)T(PD − P †DX −X†

)+log(pmax

sub ) ≥ 0,(29d)

pd,k ≥ − log(pmaxsub ), (29e)

(1a) ∼ (1c), (29f)

where the product term with Hamiltonian (∇) completesthe operation of summing the multiplication results of twoparts, including the derivation result on each variable and thecorresponding subtraction result. Through solving the masterproblem P2.2’, an upper bound of the original problem isobtained, which is denoted by fupMILP , and the solution toP2.2’ is denoted by P ∗D and X∗. As iterations proceed, theupper bound and lower bound gradually approach to eachother, an ε-convergence is reached when the gap between twobounds is within ε. The algorithm is shown in Algorithm 2.

Algorithm 2: Inner-layer Outer Approximation Algorithm

1 initialize: for a given MΦD in Algorithm 1, randomlychoose a feasible X†, ε = 10−3, ∆f =∞;

2 while ∆f ≥ ε do3 Solving the NLP problem (27a), get the solution P †D,

thus the objective function value of (27a) isf lowNLP = −f(P †D, X

†);4 Solving the MILP problem (29a), get the solution

P ∗D, X∗, the objective function value of (29a) is

fupMILP = −f(P ∗D, X∗);

5 ∆f = fupMILP − f lowNLP ;6 if ∆f ≥ ε then7 X† = X∗;8 else9 Solution (P †D, X

†) is obtained, and the value off |MΦD

can be calculated;10 Break;11 end12 end

B. A Method for Compressing Spectrum Allocation Space

In our model, the outage probability of communication linksshould be ensured to be under a certain level, thus some ofthe potential matching pairs of DP and CU can be efficientlyexcluded before solving the inner-layer problem in sectionV-A2. Based on (20), we have the following proposition:

Proposition 5. For a DP d satisfying (20), for each activatedsublink of DP d, the DP-CU matching pair must satisfy

1− ηd < ψd,k(pc, pd,k, rRd,sub(X)), (∀xc,d,k = 1). (30)

Based on the above proposition, here is an efficientcompression method for excluding some impossible DP-CUmatching case:

(a) Calculate the required transmit power prc when CU c isnot interfered.

(b) For DP d, take the required power prc into the place of pcin (30) and determine if the inequality can be true withinfeasible range. If the equation does not hold, exclude thecase that DP d reuses the uplink spectrum of CU c underthe current and bigger Md.

By using the above method before solving the inner-layerproblem in section V-A2, the feasible range of the variable Xcan be greatly compressed.

C. A Uniform-Power Greedy-based Outer Approximation

In CA technology, uniform power can be allocated on theactivated sublinks of a DP for aggregating continuous CCs. Ifthe BS allocates continuous CCs to the CUs whose spectrumare chosen to be reused by a DP, aggregating continuous CCsbecomes feasible in the DP’s side. When the aggregated CCsare continuous, the physical layer can be simplified and itis easier to implement resource allocation and managementalgorithms [18], where the transmit power on a DP’s sublinksare equal. Thus, the spectrum and power allocation problem,where uniform transmit power is allocated on each sublink ina DP, can be regarded as a simplified version of the problemP1, here, the pd,k are equal for the same DP d. Thus, UPGOAis naturally obtained based on IPGOA.

VI. NUMERICAL SIMULATIONS

In this section, we present the numerical simulation resultsof our proposed algorithms in terms of the objective powerconsumption f(PD, X), the reusing number of CUs of theDPs, etc. For comparison, the traditional one-to-one reusingmethod and two other methods are introduced here. One isthe optimal algorithm which is realized by using exhaustivesearching method, the other algorithm is based on the bipartitegraph method which is widely involved in literatures [20, 37,38]. However, due to the difference in system model and targetproblem, the bipartite graph algorithm should be modified andthe details will be described in the following.

In the modified bipartite graph method, the weight of thelinks between each DP d and CUs is set as the requiredtransmit power when the DP reuses the uplink spectrum of theCU, with the constraint that the not outage probability be noless than Md

√1− ηd, where Md has the same meaning with

the one in our methods. Similarly, the value of Md is alsodetermined in the outer layer of the bipartite graph methodthrough exhaustive enumeration.





A. Simulation Settings

As depicted in Fig. 1, the BS is in the center of the roundcellular area, the CUs and DPs are randomly and uniformlydistributed in the area. The radius of cell is set as 300 meters,and the distance between D2D transceiver is between 20 to50 meters. With respect to the available CSI to the BS, it isassumed to be able to timely obtain cellular uplink channelgain by estimation, which is mentioned in section III-B. Andthe BS can also obtain statistical CSI of D2D links as well asinterference links. In order to study the effect of the scale ofCUs on the performance of different algorithms, the numberof DPs is set as 5, and the number of CUs ranges from40 to 110. To eliminate the randomness in simulations, thenetwork is generated 100 times and the results are obtainedby computing the average. Detailed simulation parameters andthe description of each are demonstrated by Table I.

TABLE I: Simulation Parameters

Parameter Description ValueR Cell radius 300 mB Bandwidth of a CC 2 MHzNC Number of CUs [40:10:110]ND Number of DPs 5α Path-loss exponent 4ρ Noise density -174 dBm/HzM Maximum number of aggregated CCs 8rRc Cellular transmission rate requirement 2 MbpsrRd D2D transmission rate requirement [10:110] Mbpsrth0 D2D sublink transmission rate threshold 2 Mbpspmaxc Maximum cellular transmit power 26 dBmpmaxsub Maximum D2D sublink transmit power 23 dBmpsub0 Static power of a D2D sublink 50 mWdistd2d D2D communication distance [20:50] mηc Cellular outage probability threshold 0.1ηd D2D outage probability threshold 0.1

B. Numerical Results

6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 21 . 0 81 . 1 01 . 1 21 . 1 41 . 1 61 . 1 81 . 2 01 . 2 21 . 2 41 . 2 6

Objec

tive po

wer c

onsu

mptio

n (Wa

tt)

T h e r a t i o o f t h e C U a n d D P n u m b e r s

B i p a r t i t e G r a p h M e t h o d U P G O A I P G O A O p t i m a l A l l o c a t i o n

Fig. 2: The objective power consumption versus the ratio ofthe CU and DP numbers

To investigate the relationship between the objective powerconsumption and the ratio of the CU and DP numbers, we setthe D2D transmission rate requirement as 50 Mbps. This is an

2 0 4 0 6 0 8 0 1 0 0 1 2 00 . 0

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5

3 . 0

3 . 5

4 . 0

1 0 1 2 1 4 1 6 1 8 2 0 2 2

0 . 3

0 . 4

Objec

tive po

wer c

onsu

mptio

n (Wa

tt)

D 2 D t r a n s m i s s i o n r a t e r e q u i r e m e n t ( M b p s )

B i p a r t i t e G r a p h M e t h o d U P G O A I P G O A O p t i m a l A l l o c a t i o n T r a d i t i o n a l O n e - t o - O n e R e u s i n g

Fig. 3: The objective power consumption versus the D2Dtransmission rate requirement

eclectic value that makes sure that even if when the ratio of thenumber of CUs and DPs is relatively small (i.e., the numberof CUs are not that many), like 8 indicated on the x-axis,the DPs are still able to reuse the uplink spectrum of someCUs to meet their communication requirements. As shown inFig. 2, the objective power consumption decreases with theincrease of the ratio of the CU and DP numbers, and so doesthe rate of descent. It is obvious that the proposed IPGOA ispretty close to the optimal allocation result, while the objectivepower consumption of UPGOA is between IPGOA and thebipartite graph method. Because when the ratio of the CU andDP numbers is small, the bipartite graph method must makesure that the not outage probability of each activated sublink inDP d is no less than Md

√1− ηd. Thus the DPs must transmit

with higher power, and some CUs not satisfying the aboveconstraint are excluded from the feasible matching candidate,which makes the competition between DPs for suitable CUsmore intense. However, IPGOA can handle the above situationwell. Besides, we notice that the performance of UPGOA ispretty close to IPGOA when the ratio of the CUs and DPsnumbers is more than about 14, while UPGOA costs lesscomputation.

In Fig. 3, we introduce the traditional one-to-one reusingmethod and compare its objective power consumption withthe another four schemes by increasing the D2D transmissionrate requirement rRd . With the increase of rRd , the objectivepower consumption grows faster in all schemes. As we can seein the subgraph, the top line shows how the objective powerconsumption grows with rRd in one-to-ne reusing method, andthe power gap between it and the other schemes becomes big-ger until it cannot meet the transmission requirement aroundthe dash line. Not to mention the many-to-one circumstance,where additional interference will degrades the performancemore seriously. Therefore, the results have clearly shown thatthe traditional one-to-one reusing method has non-negligibledrawbacks in meeting the requirement of high-rate D2D





0 2 0 4 0 6 0 8 0 1 0 0 1 2 012345678

Reus

ing nu

mber


B i p a r t i t e G r a p h M e t h o d U P G O A I P G O A O p t i m a l A l l o c a t i o n B i p a r t i t e G r a p h M e t h o d U P G O A I P G O A O p t i m a l A l l o c a t i o n

N C / N D = 1 0

N C / N D = 2 0

Fig. 4: The average reusing number of CUs of a DP versusD2D transmission rate requirement

communications. As rRd increases to much higher value, thegap between the bipartite graph method and our proposedalgorithms become bigger, however, the proposed IPGOA isalways close to the optimal allocation result.

As we can see from Fig. 4, with the increase of rRd ,the reusing number reveals an increasing trend. It growsfaster before approaching around the upper limit M , whichis set as 8, while when the reusing number is near M , thegrowth begins to slow down and the reusing number graduallyapproaches the upper limit M . Besides, it is obvious thatour proposed methods interfere with less CUs compared withthe bipartite graph method, regardless of how much the ratioof the CU and DP numbers is. We can also notice that theperformance gap of IPGOA and the bipartite graph methodis much bigger when the ratio of the CU and DP numbers issmaller, which has shown that IPGOA can averagely performbetter even though the amount of CUs is relatively small.

And Fig. 5 shows the performance of different methodsin the aspect of the statistical matching probability vs. D2Dtransmission rate requirement rRd . Here, the statistical match-ing probability is calculated by averaging the ratio of thenumber of successfully matched DPs and all the DPs. Forthe infeasible circumstances, we simply exclude the infeasibleDPs and then solve the problem with the remaining DPs. AsrRd grows, the statistical matching probability decreases andthe rate of descent becomes faster. As shown in the figure,the proposed IPGOA is always pretty close to the optimalallocation result, while UPGOA and the bipartite graph methodare inferior to the above two methods. This is because thebipartite graph method tries to ensure that the not outageprobability of each sublink equals to the threshold Md

√1− ηd

and UPGOA demands that the transmit power of each sublinkin a DP be uniform, both of which lead to the reduction ofthe scale of the available uplink spectrum for reusing.

In Fig. 6, the average sum D2D transmission rate is used toindicate the statistical average of the sum rate of the successfultransmissions in DPs, and the transmission rate requirement

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

The m

atchin

g prob

ability



Fig. 5: The matching probability versus D2D transmission raterequirement

6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 46 08 0

1 0 01 2 01 4 01 6 01 8 02 0 02 2 02 4 0Th

e ave

rage s

um D2

D tran

smiss

ion ra

te (M

bps)

T h e r a t i o o f t h e C U a n d D P n u m b e r s


Fig. 6: The average sum D2D transmission rate versus theratio of the CU and DP numbers

of a DP is set as 50Mbps. As the ratio of the CU and DPnumbers increases, the average sum D2D transmission rateincreases, while the growth rate slows down. It is obviousthat the proposed IPGOA is superior to the bipartite graphmethod and pretty close to the optimal result. This is becausewith certain ratio of the CU and DP numbers, especiallywhen the ratio is low, the bipartite graph method will excludemany CUs from the feasible CU set for not satisfying theoutage probability constraints of sublinks, thus some DPs arenot allowed to transmit with the required transmission rate.Obviously, UPGOA is inferior to IPGOA, because it may alsoexclude some CUs from the matching candidate set due tothe uniform power allocation strategy on the sublinks in a DP.Some CUs may not be able to match with the sublinks if thosesublinks transmit with the same power. And similarly, theaverage sum D2D transmission rate is also investigated withrespect to the D2D transmission rate requirement in Fig. 7. Inthe beginning, the average rate increases with the increase of





0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0

3 5 0

The a

verag

e sum

D2D t

ransm

ission

rate

(Mbp

s)



Fig. 7: The average sum D2D transmission rate versus D2Dtransmission rate requirement

D2D communication transmission requirement, but when theD2D transmission rate requirement reaches certain values, theaverage rate begins to decrease in the bipartite graph methodand UPGOA, while IPGOA can always approach the optimalallocation result and still keeps a relatively high average rate.

VII. CONCLUSION

In this paper, we studied the spectrum and power allocationproblem of high-rate D2D underlaid cellular networks underthe condition of only statistical (partial) CSI. We proposedto use the CA technology to enable the DPs to aggregatemultiple CCs, however, this makes ensuring the QoS of D2Dcommunications more complicated from DP’s perspective. Toaccomplish the resource allocation problem, we aimed toobtain an energy-efficient result to minimize the total powerconsumption while ensuring the QoS requirements of com-munication links. Due to the difficulty of directly solving theoriginal problem, we conducted transformations and variablesubstitutions on the original problem to convert it into a moretractable form. In the following, we proposed an iterativealgorithm IPGOA to solve it. Further more, we also consideredthe scenario where uniform power is allocated for each sublinkin a DP, which can be regarded as a simplified version of theoriginal problem, thus UPGOA is proposed. In the simulationpart, the numerical results showed that our proposed methodsoutperform the existing algorithm and IPGOA is pretty closeto the optimal allocation result.

ACKNOWLEDGMENT

This work is supported in part by the National NaturalScience Foundation of China under Grant No. 61671420and Youth Innovation Promotion Association of the ChineseAcademy of Sciences (CAS) under Grant No. 2016394.

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Runzhou Li (S’19) received the B.S. degree in 2018from the Department of Electronic Engineering andInformation Science (EEIS), University of Scienceand Technology of China (USTC), Hefei, China.He is currently pursuing the M.S. degree within thedepartment of EEIS, USTC. His research interestsinclude resource management in D2D communica-tions and multiple access networks.

Peilin Hong received her B.S. and M.S. degreesfrom the Department of Electronic Engineering andInformation Science (EEIS), University of Scienceand Technology of China (USTC), in 1983 and 1986.Currently, she is a Professor and Advisor for Ph.D.candidates in the Department of EEIS, USTC. Herresearch interests include next-generation Internet,policy control, IP QoS, and information security. Shehas published 2 books and over 100 academic papersin several journals and conference proceedings.

Kaiping Xue (M’09-SM’15) received his bachelor’sdegree from the Department of Information Security,University of Science and Technology of China(USTC), in 2003 and received his Ph.D. degreefrom the Department of Electronic Engineering andInformation Science (EEIS), USTC, in 2007. FromMay 2012 to May 2013, he was a postdoctoralresearcher with the Department of Electrical andComputer Engineering, University of Florida. Cur-rently, he is an Associate Professor in the School ofCyber Security and the Department of EEIS, USTC.

His research interests include next-generation Internet, distributed networksand network security. Dr. Xue has authored and co-authored more than80 technical papers in the areas of communication networks and networksecurity. His work won best paper awards in IEEE MSN 2017, IEEE HotICN2019, and best paper runner-up award in IEEE MASS 2018. He serves onthe Editorial Board of several journals, including the IEEE Transactions onWireless Communications (TWC), the IEEE Transactions on Network andService Management (TNSM), Ad Hoc Networks, IEEE Access and ChinaCommunications. He has also served as a guest editor of IEEE Journal onSelected Areas in Communications (JSAC) and a lead guest editor of IEEECommunications Magazine. He is serving as the Program Co-Chair for IEEEIWCMC 2020 and SIGSAC@TURC 2020. He is an IET Fellow and an IEEESenior Member.

Ming Zhang (S’18) received the B.S. degree in 2017from the HeFei University of Technology (HFUT),Hefei, China, where he is currently pursuing thePh.D. degree in communication and informationsystems with the Department of Electronic Engineer-ing and Information Science (EEIS), University ofScience and Technology of China (USTC), Hefei,China. His research interests include wireless videotransmission and multiple access networks.

Te Yang received the B.S. degree in 2018 from theDalian University of Technology (DLUT), Dalian,China. He is currently pursuing the M.S. degreein communication and information systems with-in the Department of Electronic Engineering andInformation Science (EEIS), University of Scienceand Technology of China (USTC), Hefei, China.His research interests includes resource allocationin device-to-device communication networks.


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