An Overlapping Coalitional Game forCooperative Spectrum Sensing and Access in

Cognitive Radio NetworksZhiyu Dai, Zehua Wang, Student Member, IEEE, and Vincent W.S. Wong, Fellow, IEEE

Abstract—Cognitive radio networks (CRNs) provide an effec-tive solution to address the increasing demand for spectrum re-sources. The cooperation among secondary users (SUs) improvesthe sensing performance and spectrum efficiency. In this work,we study the traffic demand-based cooperation strategy of SUs inmultichannel CRNs, in which each SU makes its own cooperativesensing decision according to its traffic demand. When an SU hasa high traffic demand, it can choose to sense multiple channelsin the sensing period and obtain more chances to use spectrumresources. If an SU has no data to transmit, it can choose notto perform spectrum sensing. We formulate this problem as anon-transferable utility (NTU) overlapping coalitional game. Inthis game, each SU implements a cooperation strategy accordingto its expected payoff, which takes into account the expectedthroughput and energy efficiency. We propose two differentalgorithms, namely an overlapping coalition formation (OCF)algorithm and a sequential coalition formation (SCF) algorithm,to obtain a coalition structure. The OCF algorithm guaranteesthat the coalition structure is stable, whereas the SCF algorithmhas a lower computational complexity and less informationexchange among SUs. Moreover, when being assigned a certainchannel, SU implements adaptive transmission power controlscheme to further improve its energy efficiency. Simulation resultsshow that our proposed algorithms achieve a higher aggregatethroughput than the disjoint coalition formation (DCF) algorithmfrom the literature, where each SU can join only one coalition.

Index Terms—Cognitive radio networks, traffic-demand basedcooperation strategy, overlapping coalitional game.

I. INTRODUCTION

THERE is an increasing demand for spectrum resourcesdue to the rapid development of the mobile applications.

However, spectrum channels are under-utilized by licensed

Manuscript received August 12, 2014; revised April 24, 2014 and July 29,2015; accepted November 24, 2015. This work was supported by the NaturalSciences and Engineering Research Council (NSERC) of Canada. Part of thiswork was presented at the IEEE Wireless Communications and NetworkingConference (WCNC), Istanbul, Turkey, Apr. 2014. The review of this paperwas coordinated by Prof. Ozgur Akan.

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 pubs-permissions@ieee.org.

Zhiyu Dai was with the Department of Electrical and Computer En-gineering, The University of British Columbia, Vancouver, BC, V6T 1Z4,Canada. He is now with Dun & Bradstreet Cloud Innovation Center,3rd Floor, 720 Robson Street, Vancouver, BC, V6E 3S7, Canada (e-mail:zhiyud@ece.ubc.ca).

Zehua Wang and Vincent Wong are with the Department of Electrical andComputer Engineering, The University of British Columbia, Vancouver, BC,V6T 1Z4, Canada (e-mail: zwang@ece.ubc.ca; vincentw@ece.ubc.ca).

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

Digital Object Identifier 10.1109/TVT.2015.XXXXXXX

users [1]. The concept of cognitive radio networks (CRNs)provides a promising solution to utilize the spectrum holesand improve spectrum efficiency. In CRNs, secondary users(SUs) are allowed to access the licensed channels to transmitdata as long as the transmission of primary users (PUs) isnot interfered with. In order to protect the transmission ofPUs, SUs have to perform spectrum sensing to detect theavailability of channels before accessing them. To reduce thefalse detection caused by shadowing or path loss, some SUscan sense the spectrum cooperatively [2]. After the cooperativespectrum sensing, the available channels are shared amongthe SUs who participate in the cooperation. In the design ofcooperative spectrum sensing and access strategy, there is atrade-off between throughput and energy efficiency. On theone hand, the SUs who participate in sensing are offered theopportunities to access the channels, so as to satisfy their trafficdemands. On the other hand, cooperative spectrum sensingrequires additional energy consumption. Therefore, how to letSUs obtain high throughput while maintaining high energyefficiency is an important issue.

Many studies have been conducted to address cooperativespectrum sensing and access problem in CRNs. They developdifferent strategies to improve the throughput or energy effi-ciency in CRNs. By optimizing the sensing parameters (e.g.,detection threshold, sensing duration) in cooperative sensingor developing efficient sensing scheduling methods, a bettersensing performance and a higher network throughput can beachieved [3], [4]. When maximizing the network throughput,the protection for the transmission of PUs needs to be guar-anteed. Therefore, the constraints of power consumption andsensing performance are taken into account [5], [6]. In order toimprove the energy efficiency of CRNs, some works focus ondeveloping scheduling algorithms to assign available channelsto different SUs [7], [8]. In this way, the spectrum resourcesare allocated to SUs according to their channel states andtraffic demands. Therefore, the energy efficiency is improved.In [9], the optimal sensing time, energy detection threshold,and the number of SUs are determined to maximize the energyefficiency at system level.

Game theory is widely applied to study the cooperation ofSUs in CRNs. In [10], the problem of cooperative spectrumsensing in CRNs with multiple channels is formulated as anevolutionary game, where each SU makes its own decision onwhether to participate in sensing the spectrum. An entropy-based coalition formation algorithm is developed to help SUsselect which channel to sense. In [11], Jiang et al. study

the channel access problem in CRNs. They first propose anapproach to estimate the channel states based on Bayesianlearning, and then make the channel access decision for eachSU by a Markov chain decision process. In order to obtain abetter cooperative sensing performance or efficiently distributethe spectrum resources among SUs, the concept of coalitionalgame is also applied in cooperation strategy design for CRNs.In [12], Saad et al. investigate the trade-off between spectrumsensing and spectrum access. The problem is formulated asa disjoint coalition formation (DCF) game, where each SUcan join one coalition to maximize its utility. A collection ofcoalitions is referred to as a coalition structure. They propose adistributed algorithm to obtain the Nash-stable coalition struc-ture. In [13], Hao et al. apply the hedonic coalition formationgame theory to the cooperative spectrum sensing and accessproblem. The coalition payoff is related to energy efficiencyand sensing accuracy. They propose a DCF algorithm to find astable coalition structure. In [14], a coalitional game approachis applied to study the spectrum access of SUs, where SUsserve as cooperative relays of PUs and obtain the channelaccess as their payments. It is shown that SUs need to formthe grand coalition structure to maximize the system utility.

Although several algorithms have been proposed to improveeither the energy efficiency or throughput of SUs in CRNs,most of them do not consider the traffic demand of individualSU. In most of the previous works (e.g., [12] and [13]), SUsare assumed to have infinite data to transmit and the channelsare always fully utilized when they are assigned to SUs.This, however, may not be the case in practice. The trafficdemands of SUs may change from time to time and varyfrom one to another. The amount of data that an SU needs totransmit depends on its application (e.g., an SU with a videostreaming application usually requires a higher throughputthan an SU running a best-effort application). In addition, thetraffic demand of an SU may change over time (e.g., an SUwith an environmental monitoring application aims to reportthe changes of temperature). Therefore, when investigating theproblem of spectrum sensing and channel access in CRNs,it is necessary to take into account the traffic demands ofSUs. Although in [7] and [15], the traffic demands of SUsare considered when studying the spectrum resource allocationproblem, they are not considered in the spectrum sensing.Moreover, their objective is to maximize the aggregate systemutility instead of developing a cooperative strategy from theperspective of individual SU.

Most of the existing works assume that all SUs shouldparticipate in cooperative sensing (e.g., [8], [16], and [17]).However, in an energy-constrained CRN, for an SU having nodata to transmit during a certain period of time, it may prefer tostay in idle to conserve energy for future transmission insteadof participating in cooperative sensing. Thus, it is reasonableto let SU make its own decision on whether to join in thecooperative sensing according to its traffic demand. The workin [18] uses an evolutionary game approach to determine thecooperative sensing strategy for SUs. However, it does notconsider the energy efficiency and traffic demands of SUs.

Coalitional game theory is widely used in developing spec-trum sensing and access strategies in CRNs. Most of the

previous works aim at formulating the problem as a DCF gameand finding a non-overlapping coalition structure (e.g., [12]and [13]). They assume that an SU can join only one coalitionand perform cooperative sensing within that coalition. Thisassumption restricts the cooperation of SUs and limits theimprovement of system utility. To relax this assumption,the overlapping coalitional game theory is considered, wherea player is allowed to join multiple coalitions. When theoverlapping coalitional game theory is applied to the CRNs,an SU can contribute to sensing multiple channels at thesame time to increase its probability for channel access. Forexample, the work in [19] studies the cooperative sensing ofSUs. The problem is formulated as an overlapping coalitionalgame, and a distributed algorithm is proposed to find a stablecoalition structure. However, this work focuses on improvingsensing performance and does not consider spectrum resourceallocation. Moreover, it does not consider the additional energyconsumption caused by joining a coalition.

In this paper, we study a traffic demand-based joint coopera-tive spectrum sensing and access strategy in CRNs. To improvethe cooperation opportunities for SUs, we apply overlappingcoalitional game theory to develop a cooperation strategyfrom the perspective of individual SU. When an SU makes acoalition formation decision, it takes into account the expectedthroughput and energy efficiency. In our proposed strategy, anSU can sense multiple channels in a sensing period and joinmultiple coalitions to increase its payoff. When an SU has alow traffic demand, it can choose not to join any coalition andquit sensing. In summary, the main contributions of our workare as follows:

• We study a cooperation strategy in CRNs with multiplechannels, where each SU makes individual decisions onhow many and which channels to sense according to itsown traffic demand.

• We formulate this problem as a non-transferable utility(NTU) overlapping coalition formation game, where eachSU serves as a player.

• We propose an overlapping coalition formation (OCF)algorithm and prove that it converges to a stable coalitionstructure.

• We also propose a sequential coalition formation (SCF)algorithm which requires a lower number of iterationsand less information exchange among SUs than the OCFalgorithm.

• An adaptive transmission power control strategy is pro-posed to minimize the energy consumption spent ontransmission while guaranteeing that the maximum ex-pected throughput is obtained.

• Simulation results show that our proposed OCF and SCFalgorithms provide a higher aggregate throughput thanthe disjoint coalition formation (DCF) algorithm, whereeach SU can join at most one coalition.

The rest of this paper is organized as follows. SectionII describes the system model. In Section III, we formulatethe cooperative spectrum sensing and access problem as anoverlapping coalitional game and propose two algorithms forthe overlapping coalition formation. The properties of these

algorithms are also analyzed. In Section IV, we proposethe adaptive transmission power control scheme. Section Vpresents the performance of our proposed algorithms by com-paring them with an existing coalition formation algorithm.Conclusions are given in Section VI.

II. SYSTEM MODEL

We consider a CRN with M PUs, N SUs, and a base stationfor these SUs. Each PU transmits data via a licensed channel.There are M licensed channels. Let M = {1, . . . ,M} denotethe set of PUs and N = {1, . . . , N} denote the set of SUs.The CRN works in a time slotted manner and the slot durationis T . At the beginning of each time slot, if an SU choosesto participate in cooperative spectrum sensing and access, itwill perform sensing before accessing the available channels.The length of the sensing period is τ < T . Each SU hasmultiple energy detectors to sense multiple channels in asensing period [20]. We consider that each detector is used tosense a specific channel and these detectors can finish sensingwithin the sensing period. There are multiple groups, whereeach group consists of all the SUs that sense the same channel.When a channel is detected as idle, the BS allocates thechannel to a group member. That is, an idle channel can onlybe accessed by one SU at a time. This is to avoid interferencebetween transmission of different SUs. Moreover, we consideran SU can access at most one channel during a time slot.Each SU has different amount of data in its buffer waitingto be transmitted and only those SUs participating in sensingcan obtain access for channel use. SUs selectively participatein cooperative sensing based on the knowledge of the trafficdemand and channel capacity.

Let Sj denote the set of SUs choosing to sense and accesschannel j ∈ M. We use Pd,i,j and Pf,i,j to denote thedetection probability and false alarm probability of SU i ∈ Nafter sensing channel j ∈ M, respectively. Specifically, thedetection probability is the probability that the channel isdetected as busy when it is indeed busy. The false alarmprobability is the probability that the channel is detected asbusy when it is idle. According to the work in [21], thedetection probability of channel j on user i is

Pd,i,j(ε, γi,j) = Q

((ε

σ2n

− γi,j − 1

)√Ns

2γi,j + 1

), (1)

where Q(.) is the tail probability for the standard normaldistribution, ε is the detection threshold, σ2

n denotes noisepower, γi,j is the received SNR at SU i when it senses channelj, and Ns is the number of sensing samples during the sensingperiod in a time slot. Meanwhile, the false alarm probabilityof SU i at channel j can be expressed as

Pf,i,j(Pd,i,j , γi,j) = Q(√

2γi,j + 1Q−1(Pd,i,j) +√Nsγi,j

).

(2)Due to path loss and shadowing, an individual sensing

result may not be accurate. In our system model, SUs performcooperative sensing for channels. The BS collects the sensingresults from SUs. The BS, which also serves as a fusion centre,uses the OR rule to decide the availability for each channel

[21]. The detection probability of the set of SUs Sj who sensechannel j ∈M is given by

Pd,j = 1−∏i∈Sj

(1− Pd,i,j). (3)

A high detection probability leads to a high false alarmprobability. To protect the transmission of PU j, we set adesired target value P̄d,j for SUs in set Sj . When the OR ruleis used to make a sensing decision based on the sensing resultsof SUs in set Sj , the cooperative sensing performance needsto satisfy the target value P̄d,j . We can compute the individualtarget detection probability of SU i ∈ Sj as [22]

P̄d,i,j = 1− (1− P̄d,j)1

|Sj | , (4)

where |Sj | denotes the number of SUs sensing channel j.We substitute P̄d,i,j obtained by (4) to (2), and obtain the

value of Pf,i,j . According to the OR rule, we can calculatethe false alarm probability at channel j as

Pf,j = 1−∏i∈Sj

(1− Pf,i,j). (5)

Let Wt,i,j denote the transmit power of SU i when it per-forms transmission on channel j. Let Bj denote the bandwidthof channel j. SU i can achieve a transmission rate of Ri,j overchannel j as

Ri,j = Bj log2

(1 + |gi|2

Wt,i,j

σ2n

), (6)

where gi denotes the channel gain of transmission link of SU i.One approach to estimate the channel gain is by sending pilotsignals and using the minimum mean square error estimationor the maximum likelihood estimation [23, pp. 105] [24]. Theoptimal value of Wt,i,j can be obtained from our proposedpower control scheme, which will be presented in Section IV.

We use PI,j to denote the probability that channel j is idle.The probability that channel j is correctly detected as idle isPI,j(1−Pf,j). When a channel is detected as idle, we assumethat each member in the same coalition has an equal chance toaccess the channel. The probability that SU i ∈ Sj can accesschannel j given that this channel is detected as idle is 1

|Sj | .Since the channel assignment is coordinated by the BS, an idlechannel is assigned to only one SU who has participated incooperative sensing. That is, if channel j ∈M is detected asidle and is assigned to SU i ∈ Sj , then SU i is the only one thatcan use channel j to transmit data during the time slot. Thus,the probability that SU i is allowed to perform transmissionover channel j without interfering the transmission of PU is

PUi,j = PI,j(1− Pf,j)1

|Sj |. (7)

Since the time slot duration T is short, we assume that thenumber of information bits in the buffer of SU i, which isdenoted by Di, is a constant in a time slot. This assumption isalso made in [7] and [15]. SU i cannot transmit more than thenumber of information bits in its buffer. When SU i obtains theaccess to an available channel j, the time of its transmission,denoted as ti,j , is given by

ti,j = min

{Di

Ri,j, T − τ

}. (8)

Thus, the throughput that SU i can achieve in the time slot is

Ui,j =Ri,jti,jT

. (9)

There are two cases that an SU can perform transmission overchannel j ∈M. The first case is that channel j is detected asidle and it is indeed idle, which has the probability PI,j(1−Pf,j). The second case is that channel j is detected as idle butactually it is busy, which has the probability (1 − PI,j)(1 −Pd,j). Therefore, the probability that SU i is assigned channelj to transmit data is

PEi,j =((1− PI,j)(1− Pd,j) + PI,j(1− Pf,j)

) 1

|Sj |. (10)

SUs can sense multiple channels during the sensing periodby participating in multiple coalitions. We denote the setof channels that SU i chooses to sense as Ai. Now weconsider the expected throughput that SU i can achieve aftersensing the channels in set Ai. Let K denote a subset of Aiwhich contains the channels assigned to SU i. Specifically,set K ⊆ Ai is obtained on SU i with a probability given by∏j∈K(PEi,j)

∏j∈Ai\K(1−PEi,j). Although all the channels in

set K are available for SU i to perform its data transmission,SU i is allowed to transmit data over one channel only. Amongthese |K| channels, we consider that the channel which canmaximize its throughput (i.e., maxj∈K{Ui,j}) is selected bySU i. When SU i chooses channel argmaxj∈K{Ui,j} to accessand transmit data, its transmission may not succeed due tomis-detection of transmission from the PU. The probabilitythat SU i can successfully transmit data over this chan-nel is PUi,argmaxj∈K{Ui,j}/P

Ei,argmaxj∈K{Ui,j}. Thus, the expected

throughput that SU i can obtain is given by the followingfunction of channel set Ai:

Ui(Ai) =∑K⊆Ai

((∏j∈K

PEi,j∏

j∈Ai\K

(1− PEi,j))

×PUi,argmaxj∈K{Ui,j}

PEi,argmaxj∈K{Ui,j}maxj∈K{Ui,j}

).

(11)

As shown in (11), the expected throughput of SU i mayincrease with the size of set Ai. This is because by choosingmore channels to sense, SU i can obtain more opportunity toaccess a channel.

For an energy-constrained SU, it should limit the energyspent on sensing in order to save its energy for data transmis-sion. Now we consider the expected power consumption whenSU i chooses channel set Ai to sense and access. The powerconsumption contains two parts: sensing power consumptionand data transmission power consumption. Let Ws,i,j denotethe sensing power of SU i to channel j. Thus, the energy thatSU i spends to sense channel j is Esi,j = Ws,i,jτ . Note thatWt,i,j is the transmission power of SU i on channel j, theenergy spent by SU i on data transmission over channel j isEti,j = Wt,i,jti,j . The energy spent on sensing the channels inset Ai is inevitable for SU i. The energy spent on transmissionoccurs when SU i transfers data over the selected channelarg maxj∈K{Ui,j}. Thus, the expected power consumption of

SU i is a function of the set of channels Ai that it prefers tosense, which is given by

Ei(Ai) =1

T

( ∑K⊆Ai

((∏j∈K

PEi,j∏

j∈Ai\K

(1− PEi,j))

× Eti,argmaxj∈K{Ui,j}

)+∑j∈Ai

Esi,j

).

(12)

To balance the throughput and power consumption, we useenergy efficiency as the metric to evaluate an SU’s decisionon cooperation. We define the expected energy efficiency ofSU i as [25]

ηi(Ai) =Ui(Ai)

Ei(Ai). (13)

The objective of each SU is to maximize its throughput subjectto an energy efficiency constraint. That is, during a time slot,each SU aims to transmit the data in its buffer as muchas possible under the condition that its energy efficiency isabove a predefined threshold. When the traffic demand ofan energy-constrained SU is low, it may not want to spentenergy to participate in the cooperative sensing to transmitjust several information bits. In this case, the SU can simplychoose not to perform sensing, since the cost of participatingin the cooperation outweighs the payoff. The energy can besaved for data transmission in the future when there are enoughinformation bits in the buffer. Meanwhile, the SUs who chooseto sense the same channel have the equal probability to accessit. Thus, the problem is how each SU should cooperate withother SUs to sense the channels. We will address this problemin the next section.

III. OVERLAPPING COALITIONAL GAME FORCOOPERATION STRATEGY

In this section, we formulate the cooperative spectrumsensing and access problem as an NTU overlapping coalitionalgame. The payoff for each SU captures both the energyefficiency and the expected throughput that can be obtainedby joining multiple coalitions. For SUs to make distributeddecisions on coalition formation, we define three move rulesthat take into account both social welfare and individualpayoff. We first propose the OCF algorithm to enable SUsto form a stable coalition structure. The convergence of thisalgorithm is analyzed. Then, we propose the SCF algorithmwhich requires fewer iterations and less information to beexchanged among SUs than the OCF algorithm.

A. NTU Overlapping Coalitional Game Formation

In this cooperative spectrum sensing and access strategy,SUs choose different channels to maximize their expectedthroughput while satisfying the energy efficiency requirements.For each channel, the SUs who participate in sensing it forma coalition to improve the sensing performance. These SUsshare the spectrum resource of this channel if it is detectedas idle. That is, we have a coalitional game for each channelin set M. Each SU in set N can choose multiple channelsand contribute to multiple coalitions. Thus, we have the

overlapping coalitional game where an SU is allowed to joinmultiple coalitions and each coalition is formed on one channelthat the SU prefers to sense. The payoff of an SU in onecoalition is constrained by its traffic demand, so the utilityobtained by an SU in an overlapping coalition game cannotbe transferred to other SUs. Hence, our overlapping coalitionalgame is an NTU game. We first define the NTU overlappingcoalitional game as follows:

Definition 1 [26]. An NTU overlapping coalitional gameG = (N , v) is given by a set of players N = {1, . . . , N} anda function v : S → R|S|, where S ⊆ N denotes a coalitionformed by players and v(∅) = 0.

Corresponding to our system model, the players of thisgame refer to N SUs. They cooperate with each other and formdifferent coalitions to sense and access different channels. Acoalition refers to a set of SUs choosing to sense the samechannel. The value function v maps each coalition S ⊆ Nto an |S|-dimensional vector. We define the coalition value asthe expected payoff that each SU can obtain in the coalition.Specifically, we define v(Sj) = (xi(Sj),∀i ∈ Sj), wherexi(Sj) = PUi,jUi,j is the payoff of SU i who participates incoalition Sj . This is not the total payoff that SU i can obtain inan overlapping coalitional game since SU i may join additionalcoalitions to sense other channels besides channel j. We willdiscuss the total payoff of SU i in the following context.

In a DCF game, a coalition structure is a partition ofN [27].In an overlapping coalitional game, SUs are allowed to joinmultiple coalitions at the same time. We define the coalitionstructure Π in an overlapping coalitional game as follows:

Definition 2 [28]. An overlapping coalition structure Π overa player set N is defined as a set Π = {S1, . . . , SK}, whereK is the number of coalitions. ∀ 1 ≤ j ≤ K, Sj ⊆ N and∪Kj=1Sj = N . Since coalitions can be overlapped, ∃ Sj , Sk ∈Π, j 6= k such that Sj ∩ Sk 6= ∅.

Recall that there are M channels. Those SUs who chooseto sense and access channel j ∈ M will join coalition Sj .Moreover, some SUs with low traffic demand may prefer tostay in idle to save energy for future transmission. Thus, theymay choose not to perform sensing. We denote the set of SUsthat quit sensing as SM+1. Therefore, in our system model,the number of coalitions K is equal to M + 1. For example,consider an overlapping coalitional game G = (N , v), whereN = {1, 2, 3, 4} and M = {1, 2}. That is, the number ofchannels M = 2. There are K = M + 1 = 3 coalitions inthe overlapping coalition structure. The SUs form coalitions:S1 = {1, 2}1, S2 = {2, 3}2, and S3 = {4}3. Specifically,S1 = {1, 2}1 means that SUs 1 and 2 form a coalition to senseand access channel 1; S2 = {2, 3}2 means that SUs 2 and 3form a coalition to sense and access channel 2; S3 = {4}3means that SU 4 chooses to quit sensing. The collection ofthese coalitions Π = {{1, 2}1, {2, 3}2, {4}3} is an overlappingcoalition structure of N . For SU i ∈ N , let Ai = {j | i ∈Sj , Sj ∈ Π} denote the set of channels that SU i chooses tosense and access. In the above example, we have A1 = {1},A2 = {1, 2}, A3 = {2}, and A4 = ∅.

We now consider the total payoff of SU i in an overlappingcoalitional game. Although more than one channel may beavailable for an SU in a time slot, the SU is allowed to transmit

data over one channel only. Therefore, we cannot calculatethe total payoff of an SU by summing up its payoffs from allcoalitions that it belongs to. We define the total payoff of SUi as follows:

pi(Π) =

{Ui(Ai), if ηi(Ai) ≥ ηmin,−∞, otherwise.

(14)

That is, when the expected energy efficiency of SU i is greaterthan the energy efficiency threshold ηmin, the total payoffof SU i in the overlapping coalitional game is equal to theexpected throughput it obtains by choosing the channel set Aito sense. Otherwise, it is equal to negative infinity. This payoffdefinition guarantees that the expected energy efficiency ofSUs during the coalition formation process will not be lowerthan the threshold. In addition, we define the payoff of eachSU in set SM+1 as zero. Note that an SU cannot belong tosets SM+1 and Sj for channel j ∈M at the same time. Oncean SU decides to join the quit sensing coalition SM+1, it doesnot sense and access any channel. Thus, it does not belong toany other coalitions. Therefore, SM+1 is an isolated coalition,which is not overlapped with any other coalitions.

In some existing works (e.g., [12] and [13]), the SUs areselfish, which means an SU seeks to maximize its own payoffwithout considering the benefits of other SUs. In this case, thechannel may be occupied by SUs with low traffic demandswhile SUs with high traffic demands are still in short ofspectrum resources. This leads to a low utilization of thechannels. We now propose our preference order of SUs, whichtakes into account both social welfare and individual payoff.The proposed preference order can help SUs compare twocoalition structures and select the one which can provide moresocial welfare and individual payoff at the same time. We firstdefine the social welfare of a coalition structure Π as follows:

u(Π) =∑i∈N

pi(Π). (15)

The social welfare u(Π) of a coalition structure Π is referredto as the value of coalition structure Π, which is the sum of thepayoff of each individual player. It also represents the expectedoverall throughput of SUs by meeting their energy efficiencyconstraints. We now define the preference order of coalitionstructures by taking both individual payoff and social welfareinto account as follows:

Definition 3. In an overlapping coalitional game G =(N , v), given two coalition structures Πp and Πq over N ,Πp is i-preferred over Πq , where i ∈ N , is equivalent topi(Πp) > pi(Πq) and u(Πp) > u(Πq). This preference orderis represented as

Πp �i Πq ⇔ pi(Πp) > pi(Πq) and u(Πp) > u(Πq). (16)

According to Definition 3, a coalition structure is preferredby an SU over another if and only if the total payoff of thecoalition structure and the individual payoff of the SU areboth increased from one to the other. This preference ordernot only guarantees the increase of social welfare during thecoalition formation, but also keeps the spectrum efficiencyabove a certain level. In Theorem 1 of Section III-B1, wewill show that by using our proposed preference order, the

SUs can reach a stable coalition structure in a finite numberof iterations during their coalition formation.

B. Coalition Formation AlgorithmsDuring the process of overlapping coalition formation, an

SU makes its own decision on joining or leaving a coalitionaccording to our proposed preference order. We now definethree move rules for the possible moves of an SU. First, theSU joins a new coalition that it does not belong to. Second, theSU leaves one of its current coalitions. Third, the SU switchesfrom one of its current coalitions to a new coalition. To providea mechanism through which SUs can form different coalitionsby performing these moves, we define three move rules asfollows:

Definition 4. Join rule: Consider a coalition structure Πp

over a set of players N , where coalition Sj ∈ Πp and SUi ∈ N \Sj . A new coalition structure is defined as Πq ={Πp\Sj} ∪ {Sj ∪ {i}}. If Πq �i Πp, SU i joins coalition Sjand coalition structure Πp changes into coalition structure Πq .

According to Definition 4, in coalition structure Πp, SUi does not belong to coalition Sj at first. We assume thatSU i joins coalition Sj and the current coalition structureΠp changes into a new coalition structure Πq . If coalitionstructure Πq is preferred by SU i over coalition structure Πp

according to Definition 3, SU i joins coalition Sj and thecurrent coalition structure Πp is replaced by Πq . Althoughthe decision is made by SU i, SU i does not act selfishlywithout considering the effects of its move to other SUs. Thisis because an SU joins a new coalition only if its own payoffand the coalition structure value are both improved by hismove. For example, if SU i can increase its payoff by joiningcoalition Sj but its join move is detrimental to some SUs inthe game and leads to a decrease of the total payoff of SUs,SU i is not allowed to join coalition Sj . Therefore, our joinrule takes both the individual payoff and social welfare intoaccount .

Definition 5. Quit rule: Consider a coalition structure Πp

over a set of players N , where coalition Sj ∈ Πp and SUi ∈ N ∩ Sj . A new coalition structure is defined as Πq ={Πp\Sj}∪{Sj\{i}}. If Πq �i Πp, then SU i leaves coalitionSj and coalition structure Πp changes into coalition structureΠq .

According to quit rule, SU i leaves one of its currentcoalitions Sj and Πp changes into Πq if this newly formedcoalition structure is preferred by SU i over the current one.This occurs in the following two situations. The first situationis that there are too many SUs in coalition Sj to sense channelj, so the channel has a low chance to be assigned to SU i. Inthe other situation, SU i has only several information bits totransmit in a time slot. Therefore, SU i may prefer to leavecoalition Sj to avoid a negative payoff.

Definition 6. Switch rule [13]: Consider a coalition structureΠp over a set of players N , where coalitions Sj , Sk ∈ Πp, SUi ∈ N satisfies i ∈ Sj and i 6∈ Sk. A new coalition structureis defined as Πq = {Πp \{Sj , Sk}}∪{Sj \{i}}∪{Sk ∪{i}}.If Πq�iΠp, then SU i switches from coalition Sj to coalitionSk, and coalition structure Πp changes into coalition structureΠq .

Algorithm 1 The overlapping coalition formation (OCF)algorithm in CRN for SU i ∈ N .

1: Initialization: SM+1 := N ; Sj := ∅ for j ∈ M; Π :={S1, . . . , SM+1}; Π` := Π for each SU ` ∈ N .

2: SU i executes i.broadcast(i,Di).3: SU i executes i.receive(`,D`) for each SU ` ∈ N \ {i}.4: repeat5: SU i randomly selects k ∈ Ai ∪ {M + 1} and j ∈M \Ai.6: ΠQuit := {Π \ Sk} ∪ {Sk \ {i}}.7: SU i calculates u(ΠQuit) and pi(ΠQuit).8: ΠJoin := {Π \ Sj} ∪ {Sj ∪ {i}}.9: SU i calculates u(ΠJoin) and pi(ΠJoin).

10: ΠSwitch := {Π \ {Sj , Sk}} ∪ {Sj ∪ {i}} ∪ {Sk \ {i}}.11: SU i calculates u(ΠSwitch) and pi(ΠSwitch).12: if ΠQuit �i Π then13: Πi := ΠQuit; u(Πi) := u(ΠQuit).14: else if ΠJoin �i Π then15: Πi := ΠJoin; u(Πi) := u(ΠJoin).16: else if ΠSwitch �i Π then17: Πi := ΠSwitch; u(Πi) := u(ΠSwitch).18: end if19: SU i executes i.broadcast(i,Πi, u(Πi)).20: SU i executes i.receive(`,Π`, u(Π`)) for each SU `∈N\{i}.21: Tinfo := {Π1, . . . ,ΠN}.22: Π := arg maxΠl∈Tinfo{u(Πl)}.23: until ∀ i ∈ N , ∀ k ∈ Ai∪{M+1} and j ∈M\Ai, ΠQuit 6�i Π,

ΠJoin 6�i Π, and ΠSwitch 6�i Π.24: SU i executes i.sense(Π).25: if SU i is assigned channel j then26: SU i calculates W opt

t,i,j and transmits data with power W optt,i,j .

27: end if

According to the definition, SU i switches from one ofits coalitions to a new coalition when the new coalitionstructure is preferred over the current one. The switch rulebalances the size of different coalitions and improves thespectrum efficiency. During the coalition structure formation,some channels may be chosen by many SUs while some otherchannels are selected by few ones. When an SU notices thatits payoff can be improved by switching from the coalitionwith many members to another coalition with few members,it performs the switch move. In this way, SUs autonomouslydistribute their contributions to different coalitions and theavailable channels will be equally utilized.

We now define the stability of an overlapping coalitionstructure as follows:

Definition 7. An overlapping coalition structure Π over aset of players N is stable if any SU i ∈ N such that i ∈ Sjand i /∈ Sk for coalitions Sj , Sk ∈ Π, SU i will not deviatefrom coalition Sj or join coalition Sk.

According to Definition 7, for each SU i ∈ N in a stablecoalition structure, it will not leave any of its coalitions or joinany new coalition. Therefore, all the SUs would stay in theircurrent coalitions and do not make any change.

1) Overlapping Coalition Formation (OCF) Algorithm: Toreach a stable coalition structure, we propose an OCF algo-rithm shown in Algorithm 1. This algorithm is a distributedalgorithm, which is executed by each SU i ∈ N . In Algorithm1, we initialize the coalitions by letting all SUs join quitsensing coalition SM+1 (Line 1). Then SU i communicatesthe traffic demand information with other SUs (Lines 2 - 3).This is accomplished by invoking functions i.broadcast() and

i.receive(). We assume that SUs exchange the information viaa control channel, which is orthogonal to the licensed channelsof PUs [29]. After that, SU i makes coalition formation movesaccording to the three move rules. At the beginning of eachiteration, SU i randomly selects a coalition that it currentlybelongs to and a new coalition it does not belong to, whichare denoted by Sk and Sj , respectively (Lines 5 - 10). SUi assumes that it leaves coalition Sk and a new coalitionstructure ΠQuit is formed (Line 6). SU i calculates its resultedpayoff and the value of coalition structure ΠQuit accordingto equations (13) - (15) (Line 7). Similarly, it considers itspayoffs in potential new coalition structures resulting from joinmove and switch move, and calculates the values of coalitionstructures ΠJoin and ΠSwitch by equations (13) - (15) (Lines9 - 11), respectively. If the resulted coalition structure by quitmove is preferred over the current one by SU i, the coalitionstructure is updated (Line 13). If the quit move does notimprove the coalition structure, SU i considers join move (Line14) and switch move (Line 16). SU i updates the informationof coalition structure and its value, and communicates theupdated information with other SUs (Lines 19 - 20). All theupdated coalition structure information form a set Tinfo (Line21). The coalition structure with the greatest value amongTinfo is selected as the new coalition structure (Line 22). SUsrepeat the coalition formation process until for all SUs theywill not deviate from their current coalitions or join other newcoalitions (Line 23). In other words, the process convergesto a stable coalition structure. After the coalition formationprocess, SU i cooperatively senses the channels with other SUsin corresponding coalitions according to the coalition structureΠ. This is accomplished by invoking function i.sense(Π).During the spectrum access stage, if SU i is allocated achannel, it will calculate the optimal transmission power W opt

t,i,j

by solving problems (17) and (21), which will be discussed inthe next section. SU i then sets its transmission power to theoptimal value to transmit data (Line 26).

The convergence of the proposed OCF algorithm is guaran-teed by the following theorem:

Theorem 1: The proposed OCF algorithm converges to astable overlapping coalition structure after a finite number ofiterations.

Proof: Given that the number of channels M and thenumber of players N are finite, the number of possibleoverlapping coalition structures is 2MN . When implementingthe OCF algorithm, the coalition formation process involvesa sequence of moves of SUs, which results in a sequence ofcoalition structures {Π(0)′ ,Π(1)′ , . . . ,Π(r)′}, where r is thetotal number of moves made by SUs. According to Definitions3 - 6, after each move of an SU, a new coalition structure witha higher value is formed. In addition, the number of possiblecoalition structures is finite. Therefore, r is a finite number.We now use the proof by contradiction approach to show thatif Π(r)′ is a final coalition structure after the last move ofSUs, then it must be stable. Assume that Π(r)′ is not stable,according to Definition 7, there exists SU i ∈ N such thatSU i will deviate from one of its current coalitions or join anew coalition. Thus, according to the proposed OCF algorithm,SU i will make a join, quit, or switch move, and coalition

structure Π(r)′ will change into a new one. This contradictsthe fact that Π(r)′ is the final coalition structure. Thus, Π(r)′

is a stable coalition structure. Therefore, after a finite numberof iterations, the proposed OCF algorithm returns SUs a stableoverlapping coalition structure.

During the process of coalition formation in Algorithm 1,each SU seeks to improve its individual utility while increasingthe value of the coalition structure. The moves of SUs lead toa new coalition structure Π after each iteration. Thus, a largertotal payoff for SUs is obtained every time when the coalitionstructure Π changes. Moreover, our proposed algorithm isadaptive to the changes in the CRN (e.g., number of SUs,channel states). When new SUs join in the network, morechannels become available, the traffic demand changes, orthe channel state varies, the SUs can adaptively change theircooperation strategies and form a stable coalition structure.

2) Sequential Coalition Formation (SCF) Algorithm: Al-though the convergence of Algorithm 1 is guaranteed, thenumber of iterations required to reach a stable coalitionstructure may grow exponentially with the number of SUs.Therefore, we propose the SCF algorithm, which has a lowercomputational complexity and requires less information tobe exchanged among SUs to form an overlapping coalitionstructure.

The idea in the SCF algorithm is originated from the conceptintroduced in [30]. In a sequential game of coalition formation,which is defined by the rule of order h, the coalition structureis formed step by step. In each step, only one player canupdate the coalition structure. Players make moves one byone according to the rule of order h. Once a player has joineda coalition, it has to remain in the coalition. In our proposedSCF algorithm, the rule of order h is determined accordingto the traffic demands of SUs. Specifically, the SU that hasthe highest traffic demand is the first one to make a move.The active SU makes a decision to join multiple coalitionsbased on the current coalition structure, and remains in thesecoalition once it has joined them.

The proposed SCF algorithm is shown in Algorithm 2.In this algorithm, SUs make distributed coalition formationdecision. However, their behaviours are coordinated by acentral coordinator. First, each SU reports its traffic demandinformation to the central coordinator (Line 2). This is ac-complished by invoking function i.broadcast(). The trafficdemand information from different SUs forms an informationvector D (Line 4). H(X ) is a sorting function that mapsa vector X to a |X |-dimensional vector. It returns a vectorwith each element representing the sorted index of each X ’selement in descending order. For example, for Y = H(X ),where X = (x1, x2, x3, x4) and x2 ≥ x3 ≥ x1 ≥ x4,Y = (2, 3, 1, 4), which means x2 ranks first, x3 ranks second,x1 ranks third, and x4 ranks fourth in the sequence. Thecoordinator calculates the rule of order ρ through sorting Dwith function H(D). The information of ρ is broadcast to allSUs (Line 5). Then, SUs make coalition formation decisionone by one according to ρ. For example, the SU with thehighest traffic demand (e.g., SU ρ(1)) makes the first choice.It initializes the coalition structure by setting all SUs in quitsensing coalition SM+1 (Line 8). For other SUs, the active SU

Algorithm 2 The sequential coalition formation (SCF) algo-rithm in CRN.

1: for each i ∈ N do2: SU i executes i.broadcast(i,Di).3: end for4: D := (D1, D2, . . . , DN )5: Coordinator calculates ρ := H(D) and broadcasts ρ to all SUs.6: for i = 1 to N do7: if i = 1 then8: SU ρ(i) initializes SM+1 := N ; Sj := ∅, ∀j ∈ M, and

Π := {S1, S2, . . . , SM+1}.9: else

10: SU i executes i.receive(ρ(i− 1),Π).11: end if12: for each j ∈M do13: ΠJoin := {Π \ Sj} ∪ {Sj ∪ {ρ(i)}}.14: SU ρ(i) calculates u(ΠJoin) and pρ(i)(ΠJoin).15: SU ρ(i) randomly selects k ∈ Aρ(i) ∪ {M + 1}.16: ΠSwitch := {Π\{Sj , Sk}}∪{Sj∪{ρ(i)}}∪{Sk\{ρ(i)}}.17: SU ρ(i) calculates u(ΠSwitch) and pρ(i)(ΠSwitch).18: if ΠJoin �ρ(i) Π then19: Π := ΠJoin.20: else if ΠSwitch �ρ(i) Π then21: Π := ΠSwitch.22: end if23: end for24: if i = N then25: SU ρ(i) executes ρ(i).broadcast(ρ(i),Π).26: else27: SU ρ(i) executes ρ(i).send(ρ(i), ρ(i+ 1),Π).28: end if29: end for30: for each i ∈ N \ {ρ(N)} do31: SU i executes i.receive(ρ(N),Π).32: end for33: for each i ∈ N do34: SU i executes i.sense(Π).35: if SU i is assigned channel j then36: SU i calculates W opt

t,i,j and transmits data with power W optt,i,j .

37: end if38: end for

receives the updated information of Π from previously activeSU (Line 10). This is accomplished by invoking functioni.receive(). During the coalition formation process, each SUchecks all the channels one by one to find potential newcoalition when it is active. For a new channel j, if SU prefersto sense channel j, SU can join coalition j or switch tocoalition j, which means quit move is not considered in thisalgorithm. Specifically, the active SU considers the potentialnew coalition structure by assuming that it joins coalition Sj(Line 13). The SU calculates its new payoff and the value ofthe newly resulted coalition structure from equations (13) -(15) (Line 14). Moreover, the active SU randomly selects acoalition it belongs to (Line 15), and assumes that it switchesfrom this selected coalition to coalition Sj (Line 16). Also, thevalue of the potential new coalition structure resulting fromswitch move and SU’s new payoff are calculated (Line 17). Ifthe resulted coalition structure by join move ΠJoin is preferredover the current one Π, the coalition structure is updated (Line19). If the join move cannot improve the coalition structure,the resulted structure by switch move ΠSwitch is considered(Line 20). After checking all the new coalitions, the currently

active SU ρ(i) sends the updated information of Π to the nextactive SU ρ(i+1) (Line 27). This is accomplished by invokingfunction i.send(). For the last active SU, it broadcasts the finalcoalition structure to other SUs (Line 25). After the coalitionformation process, SUs perform sensing cooperatively withineach coalition that they belong to according to the finalcoalition structure. During the transmission stage, SUs performdata transmission with the optimal transmission power (Lines34 - 36).

Although the SCF algorithm does not guarantee the stabilityof the coalition structure obtained, its performance is as goodas the OCF algorithm in terms of the aggregate throughput,which is the sum of throughput of all SUs. This will be shownin the simulation results in Section V. Moreover, the SCFalgorithm has a lower computational complexity than the OCFalgorithm. In the SCF algorithm, each SU makes one movewhen it is active. The active SU checks each potential channelonly once and proposes its coalition structure. Therefore, thereare at most MN iterations when running the SCF algorithm.On the other hand, in the OCF algorithm, the formationof a coalition structure takes place every time when a newcoalition structure is preferred over the current one. Thenumber of the possible overlapping coalition structures in theOCF algorithm is 2MN , which is much more than the numberof iterations required in the SCF algorithm. Furthermore, theSCF algorithm requires less information to be exchanged thanthat in the OCF algorithm. In the SCF algorithm, each SUonly needs to send the updated coalition structure to the nextactive SU. In the OCF algorithm, each SU has to exchangethe information of the updated coalition structures with othersafter each iteration, which consumes much more energy.

IV. ADAPTIVE TRANSMISSION POWER CONTROL

In this section, we present the transmission power controlscheme given that an SU has been assigned a channel. Weprove that this adaptive transmission power control schemeachieves the optimal transmission power for an SU, whichminimizes the energy consumption spent on data transmissionunder the constraint that the maximum throughput of the SUis achieved.

From equations (6), (8), and (9), we notice that the through-put Ri,j that SU i ∈ N can achieve over channel j ∈ Mdepends on the transmission power Wt,i,j and the number ofinformation bits Di in its buffer. When the amount of datain the buffer of SU i is not large and can be completelytransmitted in a time slot with a low data rate, increasingthe transmission power Wt,i,j may consume more energyin vain since SU i cannot increase its throughput further.Therefore, it is reasonable for SU i to adaptively change itstransmission power to balance its transmission data rate andthe energy consumption. There are two steps to determine theoptimal transmission power for an SU. First, we find the setof transmission power that an SU can achieve the maximumthroughput. Second, from this set of values, we determine theone that minimizes the energy consumption for the SU.

Without loss of generality, we consider SU i has beenassigned channel j ∈ M for data transmission. Since the

bandwidth of channel j is fixed, the transmission rate Ri,j isa function of Wt,i,j given by (6). For SU i in a time slot, thevalues of T , τ , and Di are constant. Thus, both the throughputand the energy consumption during data transmission onlydepend on the transmission power Wt,i,j . Let Wmin

t,i andWmaxt,i denote the minimum and maximum transmission power

of SU i, respectively. Thus, the transmission power Wt,i,j canvary from Wmin

t,i to Wmaxt,i . We first consider the throughput

maximization problem as follows:

maximizeWt,i,j

Ui,j(Wt,i,j)

subject to Wmint,i ≤Wt,i,j ≤Wmax

t,i .(17)

The objective function in problem (17) is a piecewise function.We can rewrite Ui,j(Wt,i,j) by substituting (6) in (9) to obtainthe objective function as follows:

Ui,j(Wt,i,j) =Bj log2

(1+|gi|2

Wt,i,j

σ2n

)(T−τ)

T , if Wt,i,j ≤Wht,i,j ,

DiT , otherwise,

(18)

where Wht,i,j > 0 is a threshold of transmission power that

satisfies

Bj log2

(1 + |gi|2

Wht,i,j

σ2n

)(T − τ) = Di. (19)

From the objective function of problem (17) given in (18),we find that the range of transmission power [Wmin

t,i ,Wmaxt,i ],

from which SU i selects its transmission power Wt,i,j totransmit data over channel j, affects the optimal solutions toproblem (17). Let W∗t,i,j denote the set of optimal solutionsto problem (17). We have

W∗t,i,j ={Wmax

t,i }, if Wmaxt,i ≤Wh

t,i,j ,

{Wt,i,j |Wht,i,j≤Wt,i,j≤Wmax

t,i }, ifWmint,i ≤Wh

t,i,j≤Wmaxt,i ,

{Wt,i,j |Wmint,i ≤Wt,i,j≤Wmax

t,i },otherwise.(20)

According to the set in (20), problem (17) has eithera unique solution or multiple solutions depending on thethreshold Wh

t,i,j and the range of the possible transmissionpower of SU i. In other words, an SU can set its transmissionpower to be equal to a certain value or any value within aninterval to obtain its maximum throughput. For example, whenWmint,i ≤ Wh

t,i,j ≤ Wmaxt,i , the optimal solution to problem

(17) can be any value between Wht,i,j and Wmax

t,i . In order tosave energy spent on data transmission while maintaining themaximum throughput, we consider an optimization problemthat minimizes the energy consumption of SU i for data trans-mission under the constraint that the maximum throughput isobtained. This optimization problem is formulated as follows:

minimizeWt,i,j

Eti,j(Wt,i,j)

subject to Ui,j(Wt,i,j) ≥ Umaxi,j ,(21)

where Umaxi,j is the maximum value of Ui,j(Wt,i,j), which canbe achieved only if the transmission power Wt,i,j is an optimal

solution to problem (17) (i.e.,Wt,i,j ∈ W∗t,i,j). In other words,the solution to problem (17) serves as the constraint in problem(21). Thus, the requirement for a maximum throughput on SU ican be guaranteed when we minimize the energy consumptionfor the data transmission. In summary, depending on the valuesof the parameters Wh

t,i,j , Wmint,i , and Wmax

t,i , the solutionsto problem (17) may not be unique. Therefore, the optimalsolution to problem (21), which is denoted by W opt

t,i,j , can beobtained based on the following three cases.

In Case 1, where Wmaxt,i ≤Wh

t,i,j , set W∗t,i,j is a singletonset. The unique solution to problem (17) is also the optimalsolution to problem (21). That is, we have W opt

t,i,j = Wmaxt,i . It

means that SU i has to perform data transmission over channelj with its maximum transmission power in order to achievethe maximum throughput. In this case, the minimum energyconsumption for data transmission during the time slot (i.e.,the optimal value of the objective function in problem (21))is Wmax

t,i (T − τ).In Case 2, we have Wmin

t,i ≤Wht,i,j ≤Wmax

t,i . The optimalsolution to problem (17) can be any element in the setW∗t,i,j ={Wt,i,j | Wh

t,i,j ≤ Wt,i,j ≤ Wmaxt,i }. Note that W∗t,i,j is also

the set of transmission power which is feasible to the constraintin problem (21). By considering equations (6) and (8), andnoting that Eti,j(Wt,i,j) = Wt,i,jti,j , the objective function ofproblem (21) is given by

Eti,j(Wt,i,j) =Wt,i,jDi

Bj log2

(1 + |gi|2Wt,i,j

σ2n

) . (22)

We now have the following proposition:Proposition 1: Function Eti,j(Wt,i,j) in (22) is a monoton-

ically increasing function with Wt,i,j > 0.Proof: We take the first derivative of function Eti,j(Wt,i,j)

with respect to Wt,i,j . We have

Eti,j′(Wt,i,j) =

ln 2Diσ2n

Bj |gi|2(ln(1+|gi|2Wt,i,j

σ2n

))2

×(

ln(1+|gi|2Wt,i,j

σ2n

)−|gi|2Wt,i,j

σ2n

1+|gi|2Wt,i,j

σ2n

).

(23)

Since ln 2Diσ2n

Bj |gi|2/(

ln(1+ |gi|2Wt,i,j

σ2n

))2> 0, to show the mono-

tonicity of function Eti,j(Wt,i,j), we need to prove thatln(1+ |gi|2Wt,i,j

σ2n

)−(1+ |gi|2Wt,i,j

σ2n

)−1(|gi|2Wt,i,j

σ2n

)> 0 for

Wt,i,j > 0. We denote y(Wt,i,j) = ln(1+ |gi|2Wt,i,j

σ2n

)−(1+

|gi|2Wt,i,j

σ2n

)−1(|gi|2Wt,i,j

σ2n

). We determine the first derivative

of y(Wt,i,j) with respect to Wt,i,j , which is given by

y′(Wt,i,j) =|gi|2

σ2n

(1

1 + |gi|2Wt,i,j

σ2n

− 1

(1 + |gi|2Wt,i,j

σ2n

)2

)

=|gi|4Wt,i,j

σ4n

(1 + |gi|2Wt,i,j

σ2n

)2. (24)

That is, we have y′(Wt,i,j) > 0 for Wt,i,j > 0. Thus, y(Wt,i,j)is monotonically increasing with Wt,i,j , and y(Wt,i,j) >y(0) = 0 for Wt,i,j > 0. Therefore, Eti,j

′(Wt,i,j) > 0 and

Eti,j(Wt,i,j) is monotonically increasing with Wt,i,j > 0,which completes the proof.

According to Proposition 1, the value of Eti,j(Wt,i,j) in-creases with Wt,i,j when Wh

t,i,j ≤Wt,i,j ≤Wmaxt,i . Thus, the

optimal solution to problem (17) is W optt,i,j = Wh

t,i,j .In Case 3, we have Wh

t,i,j ≤Wmint,i , the solutions to problem

(17) are given by the setW∗t,i,j = {Wt,i,j | Wmint,i ≤Wt,i,j ≤

Wmaxt,i }. The objective function is the same with that in Case

2. According to Proposition 1, the optimal solution to problem(21) is W opt

t,i,j = Wmint,i .

The optimal solution to problem (21) changes accordingto the relations between Wh

t,i,j and the range of transmissionpower [Wmin

t,i ,Wmaxt,i ]. When the range of the transmission

power of SU i is fixed, the value of Wht,i,j determines the

solution set W∗t,i,j for problem (17), and thus affects thesolution to problem (21). According to the definition of Wh

t,i,j ,the value of Wh

t,i,j depends on the traffic demand of SU i.When the number of information bits in the buffer of SU idecreases (i.e., Di goes smaller), Wh

t,i,j also decreases by theequality in (19). Eventually, Case 3 occurs when Wh

t,i,j issmaller than Wmin

t,i . Therefore, SU i saves energy by settingits transmission power as Wmin

t,i . On the contrary, when Di

is very large, Wht,i,j may be greater than Wmax

t,i . In orderto obtain the highest throughput, SU i needs to perform itsdata transmission with a transmission power Wmax

t,i . Thus,the transmission power of an SU on a channel is adaptivelycontrolled according to its traffic demand. Meanwhile, theenergy spent on data transmission is minimized under theconstraint that the SU achieves its maximum throughput.

V. PERFORMANCE EVALUATION

In this section, we compare the performance of the OCFalgorithm, SCF algorithm, and DCF algorithm from the per-spective of aggregate throughput. In the DCF algorithm, eachSU can only join at most one coalition, which is similar to thealgorithm proposed in [12]. We consider a CRN with N SUsand M PUs (i.e., M licensed channels). SUs are randomlyplaced in a 100 m × 100 m square region. The base stationis placed at the centre of the square region. We model thechannel gain of the link of SU i as |gi|2 = 1/dni , where diis the distance from SU i to the base station, and n is thepath loss exponent. The probability that the channel is idleis randomly chosen between [0.5, 1]. The number of sensingsamples during the sensing period in a time slot is Ns = 5000,which is similar to the parameter setting in [31]. The numberof packets generated by each SU during a time slot followsthe Poisson distribution. A list of simulation parameters issummarized in Table I.

Fig. 1 shows a snapshot of a stable overlapping coalitionstructure obtained by implementing the OCF algorithm. Thereare 3 PUs (i.e., 3 licensed channels), 7 SUs, and a base station(BS) in the CRN. The base station is located at the centreof the square area. The 7 SUs are randomly located in thearea. All the SUs in a same ellipse form a coalition to senseand access a channel. Results in Fig. 1 show that SUs 1, 4and 7 belong to coalition 1, which corresponds to channel1. SUs 3, 4, 5 and 7 form a coalition to sense and access

TABLE ILIST OF SIMULATION PARAMETERS

Parameter ValueNumber of SUs N 10

Number of PUs (licensed channels) M 6

Path loss exponent n 2

Bandwidth of channel j Bj 100 kHzProbability that channel j is being idle PI,j [0.5, 1]

Target detection probability P̄d,i,j 0.99

Noise power σ2n 0.01 mW

Maximum transmission power of SU i Wmaxt,i 150 mW

Minimum transmission power of SU i Wmint,i 50 mW

Received SNR at each SU during the sensingstage γi,j

−15 dB

Sensing power of SU i on channel j Ws,i,j 50 mWSlot duration T 100 msSensing duration τ 5 msNumber of sensing samples during the sensingstage Ns

5000

Average number of packets generated by SUduring a time slot λ

0.5 packet per time slot

Packet size 20 kbitBuffer size of each SU 200 kbitLower bound of the energy efficiency ηmin 500 kbit/Joule

0 20 40 80 1000

20

40

60

80

100

x (m)

y(m

)

SU3

SU5

SU1

SU4

SU7

SU2

SU6

Coalition 1

Coalition 4

Coalition 2

Coalition 3

BS

60

Fig. 1. An example of the stable overlapping coalition structure (M =3, N = 7) by the OCF algorithm.

channel 2. SU 2 forms a singleton coalition to sense andaccess channel 3. SU 6 joins the coalition 4, which is thecoalition of quiting sensing. Thus, the overlapping coalitionstructure is Π = {{1, 4, 7}1, {3, 4, 5, 7}2, {2}3, {6}4}. In thisstable coalition structure, coalitions 1 and 2 are overlappedwith each other. SUs 4 and 7 contribute to both coalitions atthe same time. SU 6 chooses not to cooperate with other SUsdue to the fact that it has no traffic demand in the time slot.Therefore, SU 6 is in the quit sensing coalition.

Fig. 2 shows the aggregate throughput of SUs when thenumber of SUs N increases from 2 to 20. As shown in thisfigure, both OCF and SCF algorithms outperform the DCFalgorithm in terms of aggregate throughput. This is because

2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

Number of SUs N

Aggrega

tethrough

put(kbit/s)

SCF algorithmOCF algorithmDCF algorithm

Fig. 2. Aggregate throughput versus the number of SUs in CRN for M = 6.

1 2 3 4 5 6 7 8 9 10200

300

400

500

600

700

800

900

1000

Number of PUs M

Agg

rega

tethrough

put(kbit/s)

SCF algorithmOCF algorithmDCF algorithm

Fig. 3. Aggregate throughput versus the number of PUs in CRN for N = 10.

in OCF and SCF algorithms, each SU can join multiplecoalitions, which increases its chance to be assigned a channel.However, in the DCF algorithm, an SU can only join onecoalition and share one channel with other SUs. Therefore,its chance of being assigned a channel is limited. This resultshows that the overlapping coalitional game strategy improvesspectrum efficiency. Furthermore, the performance of SCFalgorithm is similar to or slightly better than the OCF algo-rithm. This is because in the SCF algorithm, SUs with highertraffic demands are prioritized to join coalitions. These SUshave a higher chance to transmit data than those SUs withlower traffic demands. Therefore, the SCF algorithm achievesa slightly higher aggregate throughput than the OCF algorithm.

Fig. 3 shows the aggregate throughput of SUs when thenumber of PUs M (i.e., the number of channels) increasesfrom 1 to 10. We find that the SCF algorithm obtains a slightlyhigher aggregate throughput than the OCF algorithm. This isbecause SUs with high traffic demands have priority to choosecoalitions in the SCF algorithm, which enables them to havea higher chance to access the channels. When M is small,the performance gap from the DCF algorithm to our proposedalgorithms is small. The gap of performance becomes largerwhen M increases. When more channels become available,SUs in our OCF or SCF algorithms obtain higher throughputby joining multiple coalitions. However, SUs running the DCFalgorithm obtain limited improvement of throughput due tothe constraint that it can sense and access at most one channel

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

The packet arrival rate λ on each SU (packet per time slot)

Agg

regate

throughput(kbit/s)

SCF algorithmOCF algorithmDCF algorithm

Fig. 4. Aggregate throughput versus the packet arrival rate λ.

only.Fig. 4 shows the aggregate throughput when increasing the

packet arrival rate λ. Both the OCF algorithm and SCF algo-rithm outperform the DCF algorithm in terms of the aggregatethroughput when λ changes from 0.1 to 1. The throughputincreases with the traffic demands for all algorithms whenλ is small. Larger value of λ means more data packets aregenerated at each SU during each time slot, which encouragesSUs to join coalitions to obtain a higher throughput. Whenλ ≥ 0.7, the the aggregate throughput does not increase sig-nificantly with the traffic demand. This is because the increaseof throughput is constrained by the spectrum resources. In theOCF algorithm, each SU in the same coalition has an equalchance to access an available channel. Some available channelsmay be used by SUs with low traffic demands, which can leadto a relatively low utilization of spectrum resources. SUs withhigher traffic demands in the SCF algorithm are prioritizedto be assigned an available channel. Thus, the SCF algorithmachieves a slightly higher aggregate throughput than the OCFalgorithm.

We also perform simulations to determine the channelaccess delay of SUs who have different traffic demands ina CRN. The channel access delay is defined as the averageduration from the time that a packet is generated by an SUto the time until the packet is being transmitted. In this set ofsimulations, the packet arrival rate λ of SU i = 1, 2, . . . , 10is 0.03, 0.06, . . . , 0.3, respectively. These SUs have the samedistance to the base station, which is 60 m. We set thetransmission power of SU i over channel j as a constant byletting Wmin

t,i,j = Wmaxt,i,j = 100 mW. The lower bound of

the energy efficiency ηmin is 1500 kbit/Joule. We have 1000time slots in each simulation run, where the slot duration T isequal to 0.5 sec. The average simulation results of the DCF,SCF, and OCF algorithms are given in Fig. 5. An SU does nothave any incentive to join coalitions until a certain amount ofinformation bits have been accumulated. An SU with a lowertraffic demand needs to wait a longer time to have enough datain its buffer. Thus in Fig. 5, the SU who has a lower trafficdemand has a longer channel access delay than the SU whohas a higher traffic demand. In general, our proposed SCF andOCF algorithms obtain smaller channel access delay than theDCF algorithm because an SU can join multiple coalitions in

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

Index of SU i

Chan

nel

access

delay

(tim

eslot)

DCF algorithmSCF algorithmOCF algorithm

Fig. 5. Channel access delay versus indexes of SUs. SUs 1, 2, . . . , 10 havepacket arrival rate λ = 0.03, 0.06, . . . , 0.3, respectively.

the SCF and OCF algorithms to increase its chance of beingassigned a channel. We find the channel access delay of SUi = 1 in the OCF algorithm is greater than its channel accessdelay in the SCF and DCF algorithms. This can be explainedas follows. An SU in the OCF algorithm can form a newcoalition structure in each round by increasing its payoff andthe social welfare. The SU with a higher traffic demand hasmore potential to form such a coalition structure which canbenefit itself and improve the social welfare at the same time.That is, the converged coalitions in the OCF algorithm canbetter benefit the SUs with higher traffic demands. Thus, theOCF algorithm results in the longest channel access delay forSU i = 1, whose packet arrival rate is λ = 0.03. This alsoexplains why the SU i = 10 with λ = 0.3 obtains the shortestchannel access delay in the OCF algorithm compared with theSCF and DCF algorithms.

In Fig. 6, we present the traffic demand satisfaction ratio ofeach SU with the same simulation settings as those used forFig. 5. Let γi denote the traffic demand satisfaction ratio ofSU i, which is defined as the number of packets transmittedby SU i and successfully received by the base station overthe number of packets generated by SU i. From Fig. 6, wefind that the DCF, SCF, and OCF algorithms maintain a goodfairness for SUs with different traffic demands, since theseSUs can obtain almost the same traffic demand satisfactionratio. Specifically, each SU i = 1, . . . , 10 can obtain the trafficdemand satisfaction ratio γi > 0.991 in these algorithms,which means each SU can successfully transmit almost all ofits generated packets to the base station. The traffic demandsatisfaction ratio is not equal to 1 because some packets aredropped when we terminate each simulation run after 1000time slots. The high traffic demand satisfaction ratio is becausethe channel capacity available for SUs is greater than theirtraffic demand. In our simulations, we also determine thetraffic demand to available channel capacity ratio for theSUs, which is defined as the ratio between the total trafficdemand of SUs and the total channel capacity available fortheir data transmission. With the simulation settings used inFigs. 5-6, this ratio is equal to 0.24. Note that when thetraffic demand to available channel capacity ratio is greaterthan one, there will be competition between SUs. In that case,admission control can be used at the BS to keep this ratio to

1 2 3 4 5 6 7 8 9 100.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Index of SU i

Trafficdemandsatisfactionratioγi

DCF algorithm

SCF algorithm

OCF algorithm

Fig. 6. Traffic demand satisfaction ratio versus indexes of SUs. SUs1, 2, ..., 10 have packet arrival rate λ = 0.03, 0.06, . . . , 0.3, respectively.

0 500 1000 1500 2000 2500 3000 3500100

150

200

250

300

350

400

Energy efficiency threshold ηmin (kbit/Joule)

Agg

rega

tethrough

put(kbit/s)

OCF algorithmSCF algorithmDCF algorithm

Fig. 7. Aggregate throughput versus the energy efficiency threshold ηmin.

be less than one and to maintain fairness among the SUs.Fig. 7 shows the aggregate throughput for various energy

efficiency threshold ηmin. As the same transmission powercontrol scheme is used in both OCF and SCF algorithms, theyhave similar aggregate throughput and outperform the DCFalgorithm when the energy efficiency threshold varies from0 kbit/J to 3500 kbit/J. When ηmin is smaller than 1500 kbit/J,the energy efficiency threshold has little effect on the numberof coalitions that an SU can join. Therefore, SUs are able tojoin enough coalitions to satisfy their traffic demand. In thiscase, an increase of ηmin has little effect on SUs’ throughput.However, when ηmin ≥ 1500 kbit/J, the number of coalitionsthat an SU is allowed to join is limited. SUs are refrainedfrom transmitting data in their buffer until the expected energyefficiency becomes greater than the threshold. This leads to thedecrease of throughput.

Fig. 8 shows the number of iterations when running the OCFand SCF algorithms as the number of SUs increases. When Nis small, the number of iterations for these two algorithms aresimilar. This is because when there are only a few SUs, thecooperation possibilities are limited. The number of iterationsthat the OCF algorithm needs to converge is not very large.However, when N > 10, the performance gap between thesetwo algorithms becomes larger. In the SCF algorithm, eachSU checks each new coalition only once when it is active.However, in the OCF algorithm, all SUs try to form newcoalitions whenever there is a change in coalition structure

2 6 10 14 18 22 26 300

20

40

60

80

100

120

Number of SUs N

NumberofIterations

OCF algorithm

SCF algorithm

Fig. 8. Number of iterations versus the number of SUs N .

until the process converges. The number of possible coalitionstructures increases exponentially with N . Thus, the OCFalgorithm requires more iterations than the SCF algorithm toform a coalition structure.

VI. CONCLUSION

In this paper, we studied a traffic-demand based cooperationstrategy in CRNs with multiple channels. We proposed ajoint cooperative spectrum sensing and access scheme toenable energy-constrained SUs to obtain a high throughputwhile maintaining a high energy efficiency. An overlappingcoalitional game is formulated to solve this problem, in whicheach SU makes its own decision to form overlapping coalitionswith other SUs to sense and access multiple channels cooper-atively. To reach a stable coalition structure, we proposed anOCF algorithm based on three move rules, which capturesboth individual payoff and social welfare. We proved thatour proposed OCF algorithm converges to a stable coalitionstructure. We also proposed an SCF algorithm which has alower computational complexity and requires less informationexchange. Moreover, an adaptive transmission power controlscheme is proposed. Simulation results show that the OCFand SCF algorithms have similar performance in terms ofaggregate throughput. Both algorithms outperform the DCFalgorithm proposed in [12]. For future work, we will extendour system model to a general setting, where an SU canadapt its sensing duration and sensing power according to thenetwork settings.

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Zhiyu Dai received the B.Eng. degree in Controland System Engineering from Zhejiang University,Hangzhou, China, in 2012, and the M.A.Sc. degreein Electrical and Computer Engineering from theUniversity of British Columbia (UBC), Vancouver,BC, Canada, in 2014. He is currently working as asoftware engineer in D&B Cloud Innovation Cen-ter, Vancouver, BC, Canada. His research interestsinclude cognitive radio networks, data analysis, ma-chine learning, and data mining.

Zehua Wang (S’11) received the B.Eng. degreein Software Engineering from Wuhan University,Wuhan, China, in 2009, and the M.Eng. degree inElectrical and Computer Engineering from Memo-rial University of Newfoundland, St Johns, NL,Canada, in 2011. He is currently a Ph.D. candidateat the University of British Columbia (UBC),Vancouver, BC, Canada. His research interestsinclude machine-type communications, device-to-device communications, social networks, and routingand forwarding in mobile ad hoc networks. He has

been the recipient of the Four Year Doctoral Fellowship (4YF) at UBC since2012. He was also awarded the Graduate Support Initiative (GSI) Award fromUBC. Mr. Wang served as technical program committee (TPC) members forseveral conferences including the IEEE International Conference on Com-munications (ICC) 2012, 2014–2016 and the IEEE Global CommunicationsConference (GLOBECOM) 2014–2015.

Vincent W.S. Wong (S’94, M’00, SM’07, F’16)received the B.Sc. degree from the University ofManitoba, Winnipeg, MB, Canada, in 1994, theM.A.Sc. degree from the University of Waterloo,Waterloo, ON, Canada, in 1996, and the Ph.D. de-gree from the University of British Columbia (UBC),Vancouver, BC, Canada, in 2000. From 2000 to2001, he worked as a systems engineer at PMC-Sierra Inc. He joined the Department of Electricaland Computer Engineering at UBC in 2002 andis currently a Professor. His research areas include

protocol design, optimization, and resource management of communicationnetworks, with applications to wireless networks, smart grid, and the Internet.Dr. Wong is an Editor of the IEEE Transactions on Communications. Heis a Guest Editor of IEEE Journal on Selected Areas in Communications,special issue on “Emerging Technologies” in 2016. He has served on theeditorial boards of IEEE Transactions on Vehicular Technology and Journalof Communications and Networks. He has served as a Technical Program Co-chair of IEEE SmartGridComm’14, as well as a Symposium Co-chair of IEEESmartGridComm’13 and IEEE Globecom’13. He is the Chair of the IEEECommunications Society Emerging Technical Sub-Committee on Smart GridCommunications and the IEEE Vancouver Joint Communications Chapter. Dr.Wong received the 2014 UBC Killam Faculty Research Fellowship. He is aFellow of the IEEE.