channel assignment with access contention resolution for cognitive radio networks

2808 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 6, JULY 2012

Channel Assignment With Access ContentionResolution for Cognitive Radio Networks

Le Thanh Tan, Student Member, IEEE, and Long Bao Le, Member, IEEE

Abstract—In this paper, we consider the channel assignmentproblem for cognitive radio networks with hardware-constrainedsecondary users (SUs). In particular, we assume that SUs exploitspectrum holes on a set of channels where each SU can use at mostone available channel for communication. We present the optimalbrute-force search algorithm to solve the corresponding nonlinearinteger optimization problem and analyze its complexity. Becausethe optimal solution has exponential complexity with the numbersof channels and SUs, we develop two low-complexity channel as-signment algorithms that can efficiently utilize the spectrum holes.In the first algorithm, SUs are assigned distinct sets of channels.We show that this algorithm achieves the maximum throughputlimit if the number of channels is sufficiently large. In addition,we propose an overlapping channel assignment algorithm that canimprove the throughput performance compared with its nonover-lapping channel assignment counterpart. Moreover, we designa distributed medium access control (MAC) protocol for accesscontention resolution and integrate it into the overlapping channelassignment algorithm. We then analyze the saturation throughputand the complexity of the proposed channel assignment algo-rithms. We also present several potential extensions, including thedevelopment of greedy channel assignment algorithms under themax–min fairness criterion and throughput analysis, consideringsensing errors. Finally, numerical results are presented to validatethe developed theoretical results and illustrate the performancegains due to the proposed channel assignment algorithms.

Index Terms—Channel assignment, cognitive radio, mediumaccess control (MAC) protocol, spectrum sensing, throughputmaximization.

I. INTRODUCTION

EMERGING broadband wireless applications have de-manded unprecedented increase in radio spectrum re-

sources. As a result, we have been facing a serious spectrumshortage problem. However, several recent measurements re-veal very low spectrum utilization in most useful frequencybands [1]. Cognitive radio technology is a promising technol-ogy that can fundamentally improve the spectrum utilization oflicensed frequency bands through secondary spectrum access.However, transmissions from primary users (PUs) should sat-isfactorily be protected from secondary spectrum access due totheir strictly higher access priority.

Protection of primary communications can be achievedthrough interference avoidance or interference control approach

Manuscript received October 12, 2011; accepted April 17, 2012. Date ofpublication April 26, 2012; date of current version July 10, 2012. The reviewof this paper was coordinated by Dr. Y.-C. Liang.

The authors are with the Institut National de la Recherche Scientifique–Énergie, Matériaux et Télécommunications, Université du Québec, Montréal,QC J3X 1S2, Canada (e-mail: [email protected]; [email protected]).

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

Digital Object Identifier 10.1109/TVT.2012.2196532

(i.e., spectrum overlay or spectrum underlay) [1]. For the in-terference control approach, transmission powers of secondaryusers (SUs) should carefully be controlled so that the aggre-gated interference that they create at primary receivers does notseverely affect ongoing primary communications [2]. In mostpractical scenarios where direct coordination between PUs andSUs is not possible and/or if distributed communications strate-gies are desired, it would be very difficult to maintain theseinterference constraints. The interference avoidance approachinstead protects primary transmissions by requiring SUs to per-form spectrum sensing to discover spectrum holes over whichthey can transmit data [3], [4]. This paper focuses on developingefficient channel assignment algorithms for a cognitive radionetwork with hardware-constrained secondary nodes using theinterference avoidance spectrum-sharing approach.

In particular, we consider the scenario where each SU canexploit at most one available channel for communication. Thiscan be the case if SUs are equipped with only one radio thatemploys a narrowband radio frequency front end [5]. In addi-tion, it is assumed that white spaces are very dynamic such thatit is not affordable for each SU to sense all channels to discoveravailable channels and/or to exchange sensing results with oneanother. Under this setting, we are interested in determining aset of channels that were allocated to each SU in advance sothat the maximum network throughput can be achieved in adistributed manner. To the best of our knowledge, this importantproblem has not been considered. The contributions of thispaper can be summarized as follows.

• We formulate the channel assignment problem forthroughput maximization as an integer optimization prob-lem. We then derive the user and total network throughputfor the case that SUs are assigned distinct sets of channels.We present the optimal brute-force search algorithm andanalyze its complexity.

• We develop two greedy nonoverlapping and overlappingchannel assignment algorithms to solve the underlyingNP-hard problem. We prove that the proposed nonoverlap-ping channel assignment algorithm achieves the maximumthroughput, because the number of channels is sufficientlylarge. For the overlapping channel assignment algorithm,we design a medium access control (MAC) protocol foraccess contention resolution, and we integrate the MACprotocol overhead analysis into the channel assignmentalgorithm.

• We analyze the saturation throughput and complexity ofthe proposed channel assignment algorithms. Moreover,we investigate the impact of contention collisions on thedeveloped throughput analytical framework.

0018-9545/$31.00 © 2012 IEEE

TAN AND LE: CHANNEL ASSIGNMENT WITH ACCESS CONTENTION RESOLUTION FOR COGNITIVE RADIO NETWORKS 2809

• We show how we can extend the proposed channel as-signment algorithms when the max–min fairness is con-sidered. We also extend the throughput analytical modelto consider sensing errors and propose an alternative MACprotocol that can relieve congestion on the control channel.

• We demonstrate through numerical studies the interactionsamong various MAC protocol parameters and suggestits configuration. We show that the overlapping channelassignment algorithm can achieve noticeable networkthroughput improvement compared to its nonoverlappingcounterpart. In addition, we present the throughput gainsdue to both proposed channel assignment algorithmscompared with the round-robin algorithms, which donot exploit the heterogeneity in the channel availabilityprobabilities.

The remainder of this paper is organized as follows. InSection II, we discuss important related works on spectrum-sharing algorithms and MAC protocols. Section III describesthe system model and problem formulation. We present thenonoverlapping channel assignment algorithm and describe itsperformance in Section IV. The overlapping channel assign-ment and the corresponding MAC protocol design are presentedin Section V. The performance analysis of the overlappingchannel assignment algorithm and the MAC protocol is givenin Section VI. Several potential extensions are discussed inSection VII. Section VIII demonstrates the numerical results,followed by the concluding remarks in Section IX. For brevity,we sometimes express different quantities in vector form(e.g., x describes the channel assignment variables whose el-ements are xij).

II. RELATED WORK

Developing efficient spectrum-sensing and access mecha-nisms for cognitive radio networks has been a very activeresearch topic over the past several years [3], [6]–[19]. A greatsurvey of recent works on MAC protocol design and analysisis given in [6]. In [3], it was shown that, by optimizing thesensing time, a significant throughput gain can be achievedfor an SU. In [7], we extended the result in [3] to the multi-user setting, where we design, analyze, and optimize a MACprotocol to achieve optimal tradeoff between sensing time andcontention overhead. In fact, in [7], we assumed that each SUcan simultaneously use all available channels. Therefore, thechannel assignment problem and the exploitation of multiuserdiversity do not exist in this setting, which is the topic of thispaper. Another related effort along this line was conducted in[8], where sensing-period optimization and optimal channel-sequencing algorithms were proposed to efficiently discoverspectrum holes and to minimize the exploration delay.

In [9], a control-channel-based MAC protocol was proposedfor SUs to exploit white spaces in the cognitive ad hoc net-work setting. In particular, the authors of this paper developedboth random- and negotiation-based spectrum-sensing schemesand performed throughput analysis for both saturation andnonsaturation scenarios. There exist several other synchronouscognitive MAC protocols that rely on a control channel forspectrum negotiation and access, including the approaches in

[10]–[14]. A synchronous MAC protocol without using a con-trol channel was proposed and studied in [15]. In [16], a MAC-layer framework was developed to dynamically reconfigureMAC- and physical-layer protocols. Here, by monitoring thecurrent network metrics, the proposed framework can achievegreat performance by selecting the best MAC protocol and itscorresponding configuration.

In [17], a power-controlled MAC protocol was developedto efficiently exploit spectrum access opportunities while satis-factorily protecting PUs by respecting interference constraints.Another power control framework was described in [18], whichaims at meeting the rate requirements of SUs and interferenceconstraints of PUs. A novel clustering algorithm was devised in[19] for network formation, topology control, and exploitationof spectrum holes in a cognitive mesh network. It was shownthat the proposed clustering mechanism can efficiently adapt tothe changes in the network and radio transmission environment.

Optimal sensing and access design for cognitive radio net-works were designed by using the optimal stopping theory in[21]. In [22], a multichannel medium access control (McMAC)protocol was proposed, taking into account the distance amongusers so that white spaces can efficiently be exploited whilesatisfactorily protecting PUs. Different power and spectrumallocation algorithms were devised to maximize the secondarynetwork throughput in [23]–[25]. Optimization of spectrumsensing and access, in which either cellular or TV bands canbe employed, was performed in [26].

In [27], cooperative sequential spectrum sensing and packetscheduling were designed for cognitive radios that are equippedwith multiple spectrum sensors. An energy-efficient MACprotocol was proposed for cognitive radio networks in [28].Spectrum-sensing, access, and power control algorithms weredeveloped, considering quality-of-service (QoS) protection forPUs and QoS provisioning for SUs, in [29] and [30]. Finally,a channel-hopping-based MAC protocol was proposed in [31]for cognitive radio networks to alleviate the congestion problemin the fixed control channel design. All these existing works,however, did not consider the scenario where cognitive radioshave hardware constraints, which allows them to access at mostone channel at any time. Moreover, exploiting the multichanneldiversity through efficient channel assignment is very critical tooptimize the throughput performance of the secondary networkfor this problem. We will investigate this problem, consideringits unique design issues, in this paper.

III. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

We consider a collocated cognitive radio network in whichM SUs exploit spectrum opportunities in N channels. Weassume that any SU can hear the transmissions of other SUs.In addition, each SU can use at most one channel for itsdata transmission. In addition, time is divided into a fixed-sizecycle, where SUs perform sensing on assigned channels at thebeginning of each cycle to explore available channels for com-munication. We assume that perfect sensing can be achievedwith no sensing error. Extension to the imperfect spectrum


sensing will be discussed in Section VII-B. It is assumed thatSUs transmit at a constant rate with a normalized value of one.

B. Problem Formulation

We are interested in performing channel assignment to max-imize the system throughput. Let Ti denote the throughputachieved by SU i. Let xij describe the channel assignmentdecision where xij = 1 if channel j is assigned to SU i;otherwise, xij = 0. The throughput maximization problem canformally be written as follows:

maxx

M∑i=1

Ti. (1)

For nonoverlapping channel assignments, we have the follow-ing constraints:

M∑i=1

xij = 1 for all j. (2)

We can derive the throughput that was achieved by SU i fornonoverlapping channel assignment as follows. Let Si be the setof channels solely assigned to SU i. Let pij be the probabilitythat channel j is available at SU i. For simplicity, we assumethat pij are independent of one another. This assumption holdswhen each SU impacts different set of PUs on each channel.This can, indeed, be the case, because spectrum holes dependon space. Note, however, that this assumption can be relaxedif the dependence structure of these probabilities is available.Under this assumption, Ti can be calculated as

Ti = 1 −∏j∈Si

pij = 1 −N∏j=1

(p̄ij)xij (3)

where pij = 1 − pij is the probability that channel j is notavailable for SU i. In fact, 1 −

∏j∈Si

pij is the probability thatat least one channel is available for SU i. Because each SUcan use at most one available channel, its maximum through-put is 1. In the overlapping channel assignment scheme, theconstraints in (2) are not needed. Based on this calculation,it can be observed that the optimization problem (1) and (2)is a nonlinear integer program, which is an NP-hard problem(interested readers can refer to [35, pt. VIII] for a detailedtreatment of this hardness result).

C. Optimal Algorithm and Its Complexity

Due to the nonlinear and combinatorial structure of the for-mulated channel assignment problem, it would be impossibleto obtain its closed-form optimal solution. However, we canemploy the brute-force search (i.e., exhaustive search) to deter-mine the best channel assignment that results in the maximumtotal throughput. In particular, we can enumerate all possiblechannel assignment solutions and then determine the best so-lution by comparing their achieved throughput. This solutionmethod requires a throughput analytical model to calculate thethroughput for any particular channel assignment solution. Wewill develop such a model in Section VI-A.

We now quantify the complexity of the optimal brute-forcesearch algorithm. Let us consider SU i (i.e., i ∈ {1, . . . ,M}).Suppose that we assign k channels to this SU i, where k ∈{1, . . . , N}. Then, there are Ck

N ways of doing so. Becausek can take any values in k ∈ {1, . . . , N}, the total number of

ways of assigning channels to SU i isN∑

k=1

CkN ≈ 2N . Hence,

the total number of ways of assigning channels to all SUs isasymptotically equal to (2N )M = 2NM . Recall that we needto calculate the throughput that was achieved by M SUs foreach potential assignment to determine the best one. Therefore,the complexity of the optimal brute-force search algorithm isO(2NM ). Given the exponentially large complexity requiredto find the optimal channel assignment solution, we willdevelop suboptimal and low-complexity channel assignmentalgorithms in the following sections. In particular, we con-sider the following two different channel assignment schemes:1) nonoverlapping channel assignment and 2) overlappingchannel assignment.

IV. NONOVERLAPPING CHANNEL

ASSIGNMENT ALGORITHM

We develop a low-complexity algorithm for nonoverlappingchannel assignment in this section. Recall that Si is the set ofchannels solely assigned for SU i (i.e., Si ∩ Sj = ∅, i �= j).The greedy channel assignment algorithm iteratively allocateschannels to SUs that achieve the maximum increase in thethroughput. A detailed description of the proposed algorithm ispresented in Algorithm 1. In each channel allocation iteration,each SU i calculates its increase in throughput if the best avail-able channel (i.e., channel j∗i = argmaxj∈Sa

pij) is allocated.This increase in throughput can be calculated as follows:

ΔTi =T ai − T b

i

=

⎡⎣1 − (1 − pij∗

i)∏j∈Si

(1 − pij)

⎤⎦−

⎡⎣1 −

∏j∈Si

(1 − pij)

⎤⎦

= pij∗i

∏j∈Si

(1 − pij). (4)

Based on (4), it can be observed that ΔTi will quickly decreaseover allocation iterations, because

∏j∈Si

(1 − pij) tends to zeroas the set Si is expanded. We have the following property forthe resulting channel assignment due to Algorithm 1.

Algorithm 1: Nonoverlapping channel assignment.1: Initialize the set of available channels Sa := {1, 2, . . . , N}

and Si := ∅ for i = 1, 2, . . . ,M .2: for i = 1 to M do3: j∗i = argmaxj∈Sa

pij .4: if Si �= 0 then5: Find ΔTi = T a

i − T bi , where T a

i and T bi are the

throughputs after and before assigning channel j∗i .6: else7: Find ΔTi = pij∗

i.


8: end if9: end for10: i∗ = argmaxi ΔTi.11: Assign channel j∗i∗ to user i∗.12: Update Sa = Sa \ j∗i∗ .13: If Sa is empty, terminate the algorithm. Otherwise, return

to step 2.

Proposition 1: If we have N � M , then the throughputachieved by any SU i due to Algorithm 1 is very close to themaximum value of 1.

Proof: This proposition can be proved by showing that, ifthe number of channels is much larger than the number of SUs(i.e., N � M ), then each SU will be assigned a large numberof channels. Recall that Algorithm 1 assigns channels to aparticular SU i based on the increase-in-throughput metric ΔTi.This property can be proved by observing that, if a particular SUi has been assigned a large number of channels, its ΔTi is veryclose to zero. Therefore, other SUs who have been assigned asmall number of channels will have a good chance of receivingmore channels. As a result, all SUs are assigned a large numberof channels if N � M . According to (3), the throughput thatis achieved by SU i will reach its maximum value of 1 if itsnumber of assigned channels is sufficiently large. Hence, wehave proved the proposition. �

In practice, we do not need a very large number of channelsto achieve the close-to-maximum throughput. In particular, ifeach channel is available for secondary spectrum access with aprobability of at least 0.8, then the throughput that is achievedby an SU that was assigned three channels is not smallerthan 1 − (1 − 0.8)3 = 0.992, which is less than 1% below themaximum throughput. Note that, after running Algorithm 1, wecan establish the set of channels allocated to each SU, fromwhich we calculate its throughput by using (3).

V. OVERLAPPING CHANNEL ASSIGNMENT

Overlapping channel assignments can improve the networkthroughput by exploiting the multiuser diversity gain. However,a MAC protocol is needed to resolve the access contentionunder the overlapping channel assignments. The MAC protocolincurs overhead that offsets the throughput gain due to themultiuser diversity. Hence, a sophisticated channel assignmentalgorithm is needed to balance the protocol overhead andthroughput gain.

A. MAC Protocol

Let Si be the set of channels solely assigned for SU iand Scom

i be the set of channels assigned for SU i and someother SUs. Let denote Stot

i = Si ∪ Scomi , which is the set of all

channels assigned to SU i. Assume that one control channelis always available and used for access contention resolution.We consider the following MAC protocol run by any particularSU i, which belongs to the class of synchronized MAC pro-tocols [20]. The MAC protocol is illustrated in Fig. 1, wherethe synchronization and sensing phases are employed before

Fig. 1. Timing diagram for the proposed McMAC protocol.

the channel contention and transmission phase in each cycle.A synchronization message is exchanged among SUs during thesynchronization phase to establish the same starting epoch ofeach cycle. After sensing the assigned channels in the sensingphase, each SU i proceeds as follows. If there is at least onechannel in Si available, then SU i randomly chooses one ofthese available channels for communication. If this is not thecase, SU i will randomly choose one available channel inScomi if there is any channel in this set available. (For brevity,

we will simply use users instead of SUs when there is noconfusion.) Then, it chooses a random backoff value that isuniformly distributed in the interval [0,W − 1] (i.e., W is thecontention window) and starts decreasing its backoff counterwhile listening on the control channel.

If it overhears transmissions of request to send/clear to send(RTS/CTS) from any other users, it will freeze from decreasingits backoff counter until the control channel is again free. Assoon as a user’s backoff counter reaches zero, its transmittersends an RTS message that contains a chosen channel to itsreceiver. If the receiver successfully receives the RTS, it willreply with a CTS, and user i starts its communication on thechosen channel for the remainder of the cycle. If the RTS/CTSmessage exchange fails due to collisions, the correspondinguser will quit the contention and wait until the next cycle. Inaddition, by overhearing the RTS/CTS messages of neighboringusers, which convey information about the channels that werechosen for communication, other users will compare thesechannels with their chosen ones.

Any user whose chosen channel coincides with the overheardchannels quits the contention and waits until the next cycle.Otherwise, it will continue to decrease its backoff counter be-fore exchanging RTS/CTS messages. The standard short inter-frame space (SIFS) is assumed to be used between the backoffinterval, RTS/CTS, and DATA messages. Note that the funda-mental aspect that makes this MAC protocol different from theapproach proposed in [7] is that, in [7], we assumed that eachwinning user can use all available channels for communication,whereas at most one available channel can be exploited byhardware-constrained SUs in this paper. Therefore, the channelassignment problem does not exist for the setting considered in[7]. One example of overlapping channel assignment for threeusers and six channels is illustrated in Table I, where channelassignments are indicated by an “x.”


TABLE ICHANNEL ASSIGNMENT EXAMPLE (M = 3, N = 6)

Remark 1: We focus on the saturation buffer scenario inthis paper. In practice, cognitive radios may have low backlogor, sometimes, even empty buffers. In addition, because thedata transmission phase is quite large compared to a typicalpacket size, we should allow users to transmit several packetsto completely fill the transmission phase in our MAC design.This condition can be realized by allowing only sufficientlybacklogged users to participate in the sensing and accesscontention processes.

B. Overlapping Channel Assignment Algorithm

We develop an overlapping channel assignment algorithmthat possesses two phases as follows. We run Algorithm 1 toobtain the nonoverlapping channel assignment solution in thefirst phase. Then, we perform overlapping channel assignmentby allocating channels that have been assigned to some users toother users in the second phase. We devise a greedy overlappingchannel assignment algorithm using the increase-of-throughputmetric similar to the metric employed in Algorithm 1.However, the calculation of this metric exactly turns out tobe a complicated task. Hence, we employ an estimate of theincrease-of-throughput, which is derived as follows to performchannel assignment, assuming that the MAC protocol overheadis δ < 1. In fact, δ depends on the outcome of the channelassignment algorithm (i.e., sets of channels assigned to differentusers). We will show how we can calculate δ and integrate it intothis channel assignment algorithm later.

Consider the case where channel j is the common channelof users i1, i2, . . . , iMS . Here, MS is the number of users whoshare this channel. We are interested in estimating the increasein throughput for a particular user i if channel j is assigned tothis user. Indeed, this increase of throughput can be achieved,because user i may exploit channel j if this channel is notavailable or not used by other users i1, i2, . . . , iMS . To estimatethe increase of throughput, in the remainder of this paper, weare interested only in a practical scenario where all pij are closeto 1 (e.g., at least 0.8). This assumption would be reasonable,given several recent measurements that reveal that the spectrumutilization of useful frequency bands is very low (e.g., less that15%). Under this assumption, we will show that the increase ofthroughput for user i and channel j can be estimated as

ΔTMS,esti (j)

=(1 − 1/MS)(1−δ)pij

(∏h∈Si

pih

)

×

⎛⎝1 −

∏h∈Scom

〉

pih

⎞⎠ MS∑

k=1

⎡⎣pikj

⎛⎝ MS∏

q=1,q �=k

piqj

⎞⎠⎤⎦ (5)

+ (1 − δ)pij∏h∈Si

pih∏

h∈Scomi

pih

MS∏q=1

piqj

×MS∏q=1

⎛⎝1 −

∏h∈Siq

piqh

⎞⎠ (6)

+ (1 − 1/MS)(1 − δ)pij∏h∈Si

pih

×

⎛⎝1−

∏h∈Scom

〉

pih

⎞⎠MS∏

q=1

piqj

MS∏q=1

⎛⎝1−

∏h∈Siq

piqh

⎞⎠ . (7)

Algorithm 2: Overlapping channel assignment.1: Initialize the sets of allocated channels for all users Si :=

∅ for i = 1, 2, . . . ,M and δ02: Run Algorithm 1 to obtain a nonoverlapping channel as-

signment solution.3: Let the group of channels that are shared by l users be Gl

and Uj be the set of users who share channel j and setU tempj := Uj ∀j = 1, 2, . . . , N .

4: continue := 1; h = 1; updoverhead := 0.5: while continue = 1 do6: Find the group of channels shared by h users, Gh.7: for j = 1 to |Gh| do8: for l = 1 to M do9: if l ∈ Uj then10: ΔTh,est

l (j) = 0.11: else12: User l calculates ΔTh,est

l (j), assuming thatchannel j is allocated to user l.

13: end if14: end for15: l∗j = argmaxl ΔTh,est

l (j).16: end for17: j∗l∗ = argmaxj ΔTh,est

l∗j

(j).

18: if ΔTh,estl∗ (j∗l∗) ≤ ε and updoverhead = 1 then

19: Set: continue := 0.20: Go to step 35.21: end if22: if ΔTh,est

l∗ (j∗l∗) > ε then23: Temporarily assign channel j∗l∗ to user l∗, i.e., update

U tempj∗l∗

= Uj∗l∗∪ {l∗}.

24: Calculate W and δ with U tempj∗l∗

by using the methodsin Sections V-C and V-D, respectively.

25: if |δ − δ0| > εδ then26: Set: updoverhead := 1;27: Return to step 7 using the updated δ0 = δ.28: else29: Update Uj∗

l∗:= U temp

j∗l∗

(i.e., assign channel j∗l∗ touserl∗), calculateW and δ0 with Uj∗

l∗, and update Gh.

30: Update: updoverhead := 0.31: end if32: end if33: Return to step 7.34: h = h+ 1.35: end while


This estimation is obtained by listing all possible scenarios/events in which user i can exploit channel j to increase itsthroughput. Because the user throughput is bounded by 1, weonly count events that occur with nonnegligible probabilities.In particular, assuming that pij are high (or pij are small), weonly count events whose probabilities have at most two suchelements pij in the product. In addition, we can determine theincrease of throughput for user i by comparing its achievablethroughput before and after channel j is assigned to it. It can beverified that we have the following events for which the averageincreases of throughput are significant.

• Channel j is available for all users i and iq , q =1, 2, . . . ,MS , except for ik, where k = 1, 2, . . . ,MS .In addition, all channels in Si are not available, and atleast one channel in Scom

i is available for user i. Useri can achieve a maximum average throughput of 1 − δby exploiting channel j, whereas its minimum averagethroughput before being assigned channel i is at least(1 − δ)/MS (when user i needs to share the availablechannel in Scom

i with MS other users). The increase ofthroughput for this case is at most (1 − 1/MS)(1 − δ),and the upper bound for the increase of throughput of useri is written in (5).

• Channel j is available for user i and all users iq , q =1, 2, . . . ,MS , but each user iq uses other available chan-nels in Siq for transmission. Moreover, no channel in Stot

i

is available. In this case, the increase of throughput for useri is 1 − δ, and the average increase of throughput of user iis written in (6).

• Channel j is available for user i and all users iq , q =1, 2, . . . ,MS , but each user iq uses other available chan-nels in Siq for transmission. Moreover, at least one channelin Scom

i is available. In this case, the increase of through-put for user i is upper bounded by (1 − 1/MS)(1 − δ),and the average increase of throughput of user i is writtenin (7).

The detailed description of the algorithm is given inAlgorithm 2. This algorithm has outer and inner loops, wherethe outer loop increases the parameter h, which represents themaximum number of users allowed to share any particularchannel (i.e., MS in the aforementioned estimation of theincrease of throughput), and the inner loop performs channelallocation for one particular value of h = MS . In eachassignment iteration of the inner loop, we assign one “best”channel j to user i that achieves maximum ΔTh,est

i (j). Thisassignment continues until the maximum ΔTh,est

i (j) is lessthan a predetermined number ε > 0. As will be clear in thethroughput analysis, it is beneficial to maintain at least onechannel in each set Si. This case is because the throughputcontributed by channels in Si constitutes a significant fractionof the total throughput. Therefore, we will maintain thisconstraint when running Algorithm 2.

C. Calculation of Contention Window

We show how we can calculate the contention windowW so that collision probabilities among contending SUs aresufficiently small. In fact, there is a tradeoff between collision

probabilities and the average overhead of the MAC protocol,which depends on W . In particular, larger values of W reducecollision probabilities, at the cost of higher protocol overhead,and vice versa, because there can be several collisions duringthe contention phase, each of which occurs if two or more usersrandomly choose the same value of backoff time. In addition,the probability of the first collision is largest, because thenumber of contending users decreases for successive potentialcollisions.

Let Pc be the probability of the first collision. In the fol-lowing discussion, we determine contention window W byimposing a constraint Pc ≤ εP , where εP controls the collisionprobability and overhead tradeoff. Let us calculate Pc as afunction of W , assuming that there are m SUs in the contentionphase. Without loss of generality, assume that the randombackoff times of m users are ordered as r1 ≤ r2 ≤ . . . ≤ rm.The conditional probability of the first collision if there are musers in the contention stage can be written as

P(m)c =

m∑j=2

Pr(j users collide)

=

m∑j=2

W−2∑i=0

Cjm

(1W

)j (W − i− 1

W

)m−j

(8)

where each term in the double sum represents the probabilitythat j users collide when they choose the same backoff valueequal to i. Hence, the probability of the first collision can becalculated as

Pc =

M∑m=2

P(m)c × Pr {m users contend} (9)

where P(m)c is given in (8), and Pr{m users contend} is the

probability that m users join the contention phase. To computePc, we now derive Pr{m users contend}. It can be verified thatuser i joins the contention if all channels in Si are busy and atleast one channel in Scom

i is available. The probability of thisevent can be written as

P(i)con = Pr {all channels in Si are busy and

∃! some channels in Scomi are available}

=

⎛⎝∏

j∈Si

pij

⎞⎠

⎛⎝1 −

∏j∈Scom

i

pij

⎞⎠ . (10)

The probability of the event that m users join the contentionphase is

Pr {m users contend} =

CmM∑

n=1

∏i∈Λn

P(i)con

∏j∈ΛM\Λn

P(j)con (11)

where Λn is one particular set of m users, and ΛM is the set ofall M users ({1, 2, . . . ,M}). Substituting the result in (11) into(9), we can calculate Pc. Finally, we can determine W as

W = min {W such that Pc(W ) ≤ εP } (12)


where, for clarity, we denote Pc(W ), which is given in (9) as afunction of W .

D. Calculation of the MAC Protocol Overhead

Using the contention window calculated in (12), we canquantify the average overhead of the proposed MAC protocol.Toward this end, let r be the average value of the backoff valuechosen by any SU. Then, we have r = (W − 1)/2, becausethe backoff counter value is uniformly chosen in the interval[0,W − 1]. As a result, the average overhead can be calculatedas follows:

δ(W )=[W−1]θ/2+tRTS+tCTS+3tSIFS+tSEN+tSYN

Tcycle(13)

where θ is the time that corresponds to one backoff unit; tRTS,tCTS, and tSIFS are the corresponding time of RTS, CTS, andSIFS messages; tSEN is the sensing time; tSYN is the length ofthe synchronization message; and Tcycle is the cycle time.

E. Update δ Inside Algorithm 2

The overhead δ that was calculated in (13) depends on thechannel assignment outcome, which is not known when we runAlgorithm 2. Therefore, in each allocation step in Algorithm 2,we update δ based on the current channel assignment outcome.Because δ does not change much in two consecutive allocationdecisions, Algorithm 2 smoothly runs in practice.

F. Practical Implementation Issues

To perform channel assignments, we need to know pij forall users and channels. Fortunately, we only need to per-form the estimation of pij once these values change, whichwould be infrequent in practice. These estimation and channelassignment tasks can be performed by one secondary node orcollaboratively be performed by several of them. For example,for the secondary network that support communication betweenM secondary nodes and a single secondary base station (BS),the BS can take the responsibility of estimating pij and per-forming channel assignments. Once the channel assignmentsolution has been determined and forwarded to all SUs, eachSU will perform spectrum sensing and run the underlying MACprotocol to access the spectrum in each cycle.

It is again emphasized that, although sensing and the MACprotocol are performed and run in every cycle, estimating pijand performing channel assignment (given the values of pij)are performed only if the values of pij change, which shouldbe infrequent. Therefore, it would be affordable to accuratelyestimate pij by employing sufficiently long sensing time. Thisis because, for most spectrum-sensing schemes, including anenergy detection scheme, misdetection and false-alarm proba-bilities tend to zero when the sensing time increases for a givensampling frequency [3], [4].

VI. PERFORMANCE ANALYSIS

Suppose that we have run Algorithm 2 and obtained the setof users Uj associated with each allocated channel j. Based onthis condition, we have the corresponding sets Si and Scom

i

for each user i. Given this channel assignment outcome, wederive the throughput as follows, assuming that there is nocollision due to MAC protocol access contention. We will showthat, by appropriately choosing contention parameters for theMAC protocol, the throughput analysis under this assumptionachieves accurate results.

A. Throughput Analysis

Because the total throughput is the sum of throughput of allusers, it is sufficient to analyze the throughput of one particularuser i. We will perform the throughput analysis by consideringall possible sensing outcomes performed by the considered useri for its assigned channels. We will have the following cases,which correspond to nonzero throughput for the considereduser.

• Case 1: If at least one channel in Si is available, thenuser i will exploit this available channel and achieve athroughput of one. Here, we have

Ti {Case 1} = Pr {Case 1} = 1 −∏j∈Si

p̄ij . (14)

• Case 2: In this case, we consider scenarios where all chan-nels in Si are not available, at least one channel in Scom

i

is available, and user i chooses the available channel j fortransmission. Suppose that channel j is shared by MSj

SUs, including user i (i.e., MSj = |Uj |). The followingfour possible groups of users ik, k = 1, . . . ,MSj , sharechannel j.

– Group 1: Channel j is available for user ik, and userik has at least 1 channel in Sik available.

– Group 2: Channel j is not available for user ik.– Group 3: Channel j is available for user ik, all chan-

nels in Sik are not available, and another channel j′

in Scomik

is available for user ik. In addition, user ikchooses channel j ′ for transmission in the contentionstage.

– Group 4: Channel j is available for user ik, and allchannels in Sik are not available. In addition, userik chooses channel j for transmission in the con-tention stage. Hence, user ik competes with user i forchannel j.

The throughput that was achieved by user i in this casecan be written as

Ti {Case 2} = (1 − δ)Θi

MSj∑A1=0

MSj−A1∑A2=0

MSj−A1−A2∑A3=0

Φ1(A1)Φ2(A2)Φ3(A3)Φ4(A4) (15)

where the following conditions hold.


– Θi is the probability that all channels in Si are notavailable and user i chooses some available channelj in Scom

i for transmission.– Φ1(A1) denotes the probability that A1 users belong

to group 1 among MSj users who share channel j.– Φ2(A2) represents the probability that A2 users belong

to group 2 among MSj users who share channel j.– Φ3(A3) describes the probability that A3 users belong

to group 3 among MSj users who share channel j.– Φ4(A4) denotes the probability that A4 = MSj −

A1 −A2 −A3 remaining users belong to group 4scaled by 1/(1 +A4), where A4 is the number ofusers, excluding user i, who compete with user i forchannel j.

We now proceed to calculate these quantities. We have

Θi =∏k∈Si

pik.

Hi∑Bi=1

CBiHi∑

h=1

∑j∈Ψh

i

1Bi

∏j1∈Ψh

i

pij1∏

j2∈Scomi

\Ψhi

pij2

(16)

where Hi denotes the number of channels in Scomi . The

first product term in (16) represents the probability thatall channels in Si are not available for user i. The secondterm in (16) describes the probability that user i chooses anavailable channel j among Bi available channels in Scom

i

for transmission. Here, we consider all possible subsetsof Bi available channels, and for one such particularcase, Ψh

i describes the corresponding set of Bi availablechannels, i.e.,

Φ1(A1) =

CA1MSj∑c1=1

∏m1∈Ω(1)

c1

pm1j

⎛⎝1 −

∏l∈Sm1

pm1l

⎞⎠ . (17)

In (17), we consider all possible subsets of size A1 thatbelong to group 1 (there are CA1

MSjsuch subsets). Each

term inside the sum represents the probability for thecorresponding event whose set of A1 users is denoted byΩ

(1)c1 , i.e.,

Φ2(A2) =

CA2MSj−A1∑c2=1

∏m2∈Ω(2)

c2

pm2j . (18)

In (18), we capture the probability that channel j is notavailable for A2 users in group 2 whose possible sets aredenoted by Ω

(2)c2 , i.e.,

Φ3(A3) =

CA2MSj−A1−A2∑

c3=1

∏m3∈Ω(3)

c3

pm3j

∏l3∈Sm3

pm3l3 (19)

β∑n=0

Cnβ∑

q=1

∏h1∈Scom,q

j,m3

pm3h1

∏h2∈S

com,qj,m3

pm3h2

(1− 1

n+1

). (20)

For each term in (19), we consider different possiblesubsets of A3 users, which are denoted by Ω

(3)c3 . Then,

each term in (19) represents the probability that channelj is available for each user m3 ∈ Ω

(3)c3 whereas all chan-

nels in Sm3for the user m3 are not available. In (20),

we consider all possible sensing outcomes for channelsin Scom

m3performed by user m3 ∈ Ω

(3)c3 . In addition, let

Scomj,m3

= Scomm3

\ {j} and β = |Scomj,m3

|. Then, in (20), weconsider all possible scenarios in which n channels inScomj,m3

are available and user m3 chooses a channel dif-ferent from channel j for transmission [with probability(1 − (1/n+ 1))], where Scom

j,m3= Scom,q

j,m3∪ Scom,q

j,m3, and

Scom,qj,m3

∩ Scom,qj,m3

= ∅. We have

Φ4(A4) =1

1 +A4

∏m4∈Ω(4)

⎛⎝pm4j

∏l4∈Sm4

pm4l4

⎞⎠ (21)

γ∑m=0

Cmγ∑

q=1

∏h1∈Scom,q

j,m4

pm4h1

∏h2∈Scom,q

j,m4

pm4h2

(1

m+ 1

). (22)

The sensing outcomes captured in (21) and (22) are similarto the outcomes in (19) and (20). However, given threesets of A1, A2, and A3 users, the set Ω(4) whose size is|Ω(4)| = A4 can be determined. Here, γ denotes the cardi-nality of the set Scom

j,m4= Scom

m4\ {j}. Other sets are similar

to the sets in (19) and (20). However, all users in Ω(4)

choose channel j for transmission in this case. Therefore,user i wins the contention with probability 1/(1 +A4),and its achievable throughput is (1 − δ)/(1 +A4).

Summarizing all the cases considered, the throughputachieved by user i is given as

Ti = Ti{Case 1}+ Ti{Case 2}. (23)

In addition, the total throughput of the secondary networkT is the sum of throughput achieved by all SUs.

B. Impacts of Contention Collision

We have presented the saturation throughput analysis, as-suming that there is no contention collision. Intuitively, if theMAC protocol is designed such that the collision probability issufficiently small, then the impact of collision on the throughputperformance would be negligible. For our MAC protocol, usersperform contention resolution in case 2 considered in the previ-ous throughput analysis, which occurs with a small probability.Therefore, if the contention window in (12) is chosen for asufficiently small εP , then contention collisions would havenegligible impacts on the network throughput. We formallystate this intuitive result in the following proposition.

Proposition 2: The throughput T that was derived in theprevious section has an error, which can be upper bounded as

Et ≤ εP

M∑i=1

∏j∈Si

p̄ij

⎛⎝1 −

∏j∈Scom

i

p̄ij

⎞⎠ (24)


where εP is the target collision probability that is used todetermine the contention window in (12).

Proof: As aforementioned, contention collision can occuronly in case 3 of the aforementioned throughput analysis. Theprobability that covers all possible events for user i in thiscase is

∏j∈Si

p̄ij(1 −∏

j∈Scomi

p̄ij). In addition, the maximumaverage throughput that a particular user i can achieve is1 − δ < 1 (because no other users contend with user i to exploita chosen channel). In addition, if contention collision happens,then user i will quit the contention and may experience amaximum average throughput loss of 1 − δ compared to theideal case with no contention collision. In addition, the collisionprobabilities of all potential collisions is bounded above by εP .Therefore, the average error due to the proposed throughputanalysis can be upper bounded as in (24). �

To illustrate the throughput error bound presented in thisproposition, let us consider an example where p̄ij ≤ 0.2 andεP ≤ 0.03. Because the sets Si returned by Algorithm 2contain at least one channel, the throughput error can bebounded by M × 0.2 × 0.03 = 0.006M . In addition, the totalthroughput will be at least

∑Mi=1(1 −

∏j∈Si

p̄ij) ≥ 0.8M ifwe consider only the throughput contribution from case 1.Therefore, the relative throughput error can be upper boundedby 0.006M/0.8M ≈ 0.75%, which is quite negligible. Thisexample shows that the proposed throughput analytical modelis very accurate in most practical settings.

C. Complexity Analysis

We analyze the complexity of Algorithms 1 and 2 in thissection. Let us proceed by analyzing the steps taken in eachiteration of Algorithm 1. To determine the best assignmentfor the first channel, we have to search over M SUs and Nchannels, which involves MN cases. Similarly, to assign thesecond channel, we need to perform searching over all SUs andN − 1 channels (one channel is already assigned in the firstiteration). Hence, the second assignment involves M(N − 1)cases. Similar analysis can be applied for other assignments inlater iterations. In summary, the total number of cases involvedin assigning all channels to M SUs is M(N + · · ·+ 2 + 1) =MN(N + 1)/2, which is O(MN2). In Algorithm 1, the in-crease of throughput used in the search is calculated using (4).

In Algorithm 2, we run Algorithm 1 in the first phase andthen perform further overlapping channel assignments usingAlgorithm 2 in the second phase. Hence, we need to analyzethe complexity of the second phase (i.e., Algorithm 2). InAlgorithm 2, we increase the value of h from 1 to M − 1over iterations of the while loop (to increase the number ofusers who share one channel). For a particular value of h, wesearch over the channels that have been shared by h users andover all M users. Therefore, we have NM cases to considerin the worst case for each value of h, each of which requirescalculating the corresponding increase of throughput by using(5). Therefore, the worst case complexity of the second phase isNM(M − 1), which is O(NM2). Considering the complexityof both phases, the complexity of Algorithm 2 is O(MN2 +NM2) = O(MN(M +N)), which is much lower than theoptimal brute-force search algorithm (O(2NM )).

VII. FURTHER EXTENSIONS AND DESIGN ISSUES

A. Fair Channel Assignment

We extend the channel assignment problem to consider themax–min fairness objective, which maximizes the minimumthroughput achieved by all SUs [36]. In particular, the max–minchannel assignment problem can be stated as follows:

maxx

mini

Ti. (25)

Intuitively, the max–min fairness criterion tends to allocatemore radio resources for “weak” users to balance the through-put performance among all users. Due to the exact through-put analytical model developed in Section VI-A, the optimalsolution of the optimization problem (1) can be found bythe exhaustive search, which, however, has an extremely highcomputational complexity.

To resolve this complexity issue, we devise greedy fairnonoverlapping and overlapping channel assignment algo-rithms, which are described in Algorithms 3 and 4, respec-tively. In Section VIII, we compare the performance of thesealgorithms with the optimal exhaustive search algorithm. Thesealgorithms are different from Algorithms 1 and 2 mainly inthe way we choose the user to allocate one “best” channel ineach iteration. In Algorithm 3, we find the set of users whoachieve a minimum throughput in each iteration. For each userin this set, we find one available channel that results in thehighest increase of throughput. Then, we assign the “best”channel that achieves the maximum increase of throughput,considering all minimum-throughput users. Therefore, this as-signment attempts to increase the throughput of a weak userwhile exploiting the multiuser diversity.

Algorithm 3: Fair nonoverlapping channel assignment.1: Initialize SU i’s set of available channels, Sa

i :={1, 2, . . . , N} and Si := ∅, for i = 1, 2, . . . ,M , whereSi denotes the set of channels assigned for SU i.

2: continue := 1.3: while continue = 1 do4: Find the set of users who currently have minimum

throughput, Smin = argmini Tbi

where Smin = {i1, . . . , im} ⊂ {1, . . . ,M} is the setof minimum-throughput SUs.

5: if ORil∈Smin(Sail�= ∅) then

6: For each SU il ∈ Smin and channel jil ∈ Sail

, findΔTil(jil) = T a

il− T b

il

where T ail

and T bil

are the throughput after andbefore assigning channel jil ; set ΔTil=0 if Sa

il=∅.

7: {i∗l , j∗i∗l} = argmaxil∈Smin,jil∈S

ail

ΔTil(jil).

8: Assign channel j∗i∗l

to SU i∗l .9: Update Si∗

l= Si∗

l∪ j∗i∗

land Sa

k = Sak \ j∗i∗

lfor all

k ∈ {1, . . . ,M}.10: else11: Set continue := 0.12: end if13: end while


Algorithm 4: Fair overlapping channel assignment.1: Run Algorithm 3 and obtain the sets Si for all SU i.

Initialize Scomi = ∅ for i.

2: continue := 1.3: while continue = 1 do4: Find i∗ = argmini∈{1,...,M} T

bi and Tmin = T b

i∗ , whereties are randomly broken.

5: SSepi∗ = ∪i,i�=i∗Si, SUni

i∗ = ∪iScomi \ Scom

i∗ .6: Run Algorithm 5.7: if ORiScom,temp

i �= ∅ then8: Assign Scom

i = Scom,tempi and Si = Stemp

i .9: else10: Set continue := 0.11: end if12: end while

Algorithm 5: Searching for potential channel assignment.1: (∗ Search channel assignment from separate sets ∗)2: for j ∈ SSep

i∗ do3: Find SU i′, where j ∈ Si′ . Let nc = M − 2.4: for l = 0 to nc do5: for k = 1 to Cl

ncdo

6: Find T ai∗ , T a

i′ , and T am|m∈Ul

j, where U l

j is the setof l new SUs who share channel j.

7: if min(T ai∗ , T

am|m∈Ul

j, T a

i′ ) > Tmin then

8: Temporarily assign channel j to SUs i∗, i′ andall SUs m: Scom,temp

i∗ = Scomi∗ ∪ j, Scom,temp

i′ =

Scomi′ ∪j, Stemp

i′ =Si′\j and Scom,tempm =Scom

m ∪j.9: Update Tmin = min(T a

i∗ , Tam|m∈Ul

j, T a

i′ ).10: Reset all temporary sets of other SUs to be empty.11: end if12: end for13: end for14: end for15: (∗ Search channel assignment from common sets ∗)16: for j ∈ SUni

i∗ do17: Find the subset of SUs, except for SU i∗, SUse who use

channel j as an overlapping channel.18: for l = 0 to M − 1− |SUse| do19: for k = 1 to Cl

M−1−|SUse| do

20: Find T ai∗ , T a

i′ |i′∈SUse , T am|m∈Ul

j, where U l

j is theset of l new SUs who share channel j.

21: if min(T ai∗ , T

ai′ |i′∈SUse , T a

m|m∈Ulj) > Tmin then

22: Temporarily assign channel j to SU i∗, allSUs i′, and all SUs m :Scom,temp

i∗ =Scomi∗ ∪ j,

Scom,tempm = Scom

m ∪ j.23: Update Tmin=min(T a

i∗ , Tai′ |i′∈SUse , T a

m|m∈Ulj).

24: Reset all temporary sets of other SUs to be empty.25: end if26: end for27: end for28: end for

In Algorithm 4, we first run Algorithm 3 to obtain nonover-lapping sets of channels for all users. Then, we seek to improvethe minimum throughput by performing overlapping channelassignments. In particular, we find the minimum-throughputuser and an overlapping channel assignment that results in thelargest increase in its throughput. The algorithm terminateswhen there is no such overlapping channel assignment. Thesearch of an overlapping channel assignment in each iterationof Algorithm 4 is performed in Algorithm 5. In particular, wesequentially search over channels that have been allocated for asingle user or shared by several users (i.e., channels in separateand common sets, respectively). Then, we update the currenttemporary assignment with a better one (if any) during thesearch. This search requires throughput calculations for whichwe use the analytical model developed in Section VI-A withthe MAC protocol overhead δ < 1 derived in Section V-D. Itcan be observed that the proposed throughput analysis is veryuseful, because it can be used to evaluate the performanceof any channel assignment solution and to perform channelassignments in greedy algorithms.

B. Throughput Analysis Under Imperfect Sensing

We extend the throughput analysis by considering imperfectsensing. The following two important performance measuresare used to quantify the sensing performance: 1) detectionprobabilities and 2) false-alarm probabilities. Let Pij

d and Pijf

be the detection and false-alarm probabilities, respectively, ofSU i on channel j. In particular, a detection event occurs whena secondary link successfully senses a busy channel, and a falsealarm represents the situation when a spectrum sensor returns abusy state for an idle channel (i.e., a transmission opportunityis overlooked). In addition, let us define Pij

d = 1 − Pijd and

Pijf = 1 − Pij

f . Under imperfect sensing, the following fourscenarios are possible for channel j and SU i.

• Scenario 1: A spectrum sensor indicates that channel j isavailable and the nearby PU is not using channel j (i.e.,correct sensing). This scenario occurs with the probabilityPij

f pij .• Scenario 2: A spectrum sensor indicates that channel j is

available and the nearby PU uses channel j (i.e., misde-tection). This scenario occurs with the probability Pij

d pij .In this case, the potential transmission of SU i will collidewith the nearby PU. We assume that both transmissionsfrom SU i and the nearby PU fail.

• Scenario 3: A spectrum sensor indicates that channel j isnot available and the nearby PU uses channel j (i.e., cor-rect detection). This scenario occurs with the probabilityPijd pij .

• Scenario 4: A spectrum sensor indicates that channel j isnot available and the nearby PU is not using channel j(i.e., false alarm). This scenario occurs with the probabilityPijf pij , and the channel opportunity is overlooked.

Because SUs make channel access decisions based on theirsensing outcomes, the first two scenarios can result in spectrumaccess on channel j by SU i. Moreover, spectrum access inscenario 1 leads to successful data transmission. Let us define


Pijidle = Pij

f pij + Pijd pij and Pij

busy = 1 − Pijidle as the proba-

bilities under which SU i may and may not access channel j,respectively. The same synchronized MAC protocol describedin Section V-A is assumed here. In addition, the MAC proto-col overhead can be calculated as presented in Section V-D,where the contention window W is determined as described inSection V-C. However, pij and pij are substituted by Pij

idle and

Pijbusy, respectively, in the calculation of contention window

in Section V-C for this case, because Pijidle and Pij

busy capturethe probabilities that channel j is available and busy for useri as indicated by sensing, respectively, considering potentialsensing errors. Because the total throughput is the sum ofthroughput of all users, it is sufficient to analyze the throughputof one particular user i. To analyze the throughput of user i, weconsider the following cases.

Case 1: At least one channel in Si is available, and user ichooses one of these available channels for its transmis-sion. User i can achieve a throughput of one in sucha successful access, which occurs with the followingprobability:

Pr {Case 1} =

|Si|∑k1=1

Ck1|SI |∑

l1=1

∏j1∈Sl1

i

pij1∏

j2∈Si\Sl1i

pij2 (26)

k1∑k2=1

Ck2k1∑

l2=1

∏j3∈Sl2

i

Pij3f

∏j4∈Si\Sl2

i

Pij4f (27)

|Si|−k1∑k3=0

Ck3|Si|−k1∑l3=1

k2k2 + k3

∏j5∈Sl

i3

Pij5d

∏j6∈Si\Sl1

i\Sl3

i

Pij6d (28)

where we have Ti{Case 1} = Pr{Case 1}. The quantity(26) represents the probability that there are k1 availablechannels in Si (which may or may not correctly be sensedby SU i). Here, Sl1

i denotes a particular set of k1 avail-able channels whose index is l1. In addition, the quantity(27) describes the probability that there are k2 availablechannels as indicated by sensing (the remaining availablechannels are overlooked due to sensing errors), where Sl2

i

denotes the l2th set with k2 available channels. For thequantity in (28), k3 denotes the number of channels thatare not available but the sensing outcomes indicate thatthey are available (i.e., due to misdetection). Moreover,k2/(k2 + k3) represents the probability that SU i choosesthe available channel for transmission, given that its sens-ing outcomes indicate k2 + k3 available channels. Theremaining quantity in (28) describes the probability thatthe sensing outcomes due to SU i incorrectly indicates k3available channels.

Case 2: All channels in Si are indicated as not available by sens-ing; at least one channel in Scom

i is indicated as availableby sensing, and user i chooses an available channel j fortransmission. Suppose that channel j is shared by MSj

SUs, including user i (i.e., MSj = |Uj |). The following

four possible groups of users ik, k = 1, . . . ,MSj sharechannel j.

– Group 1: Channel j is available for user ik, anduser ik has at least one channel in Sik available asindicated by sensing.

– Group 2: Channel j is indicated as not available foruser ik by sensing.

– Group 3: Channel j is available for user ik, all chan-nels in Sik are not available, and another channelj ′ in Scom

ikis available for user ik as indicated by

sensing. In addition, user ik chooses channel j ′ fortransmission in the contention stage.

– Group 4: Channel j is available for user ik, andall channels in Sik are not available as indicated bysensing. In addition, user ik chooses channel j fortransmission in the contention stage. Hence, user ikcompetes with user i for channel j.

The throughput that was achieved by user i in this case canbe written as

Ti{Case 2} = (1 − δ)Θi

MSj∑A1=0

MSj−A1∑A2=0

MSj−A1−A2∑A3=0

Φ1(A1)Φ2(A2)Φ3(A3)Φ4(A4). (29)

Here, we use the same notations as in the perfect sensingscenario investigated in Section VI-A, where the followingconditions hold.

– Θi is the probability that all channels in Si are indicatedas not available by sensing and user i chooses someavailable channel j in Scom

i as indicated by sensing fortransmission.

– Φ1(A1) denotes the probability that A1 users belong togroup 1 among MSj users who share channel j.

– Φ2(A2) represents the probability that A2 users belongto group 2 among MSj users who share channel j.

– Φ3(A3) describes the probability that A3 users belong togroup 3 among MSj users who share channel j.

– Φ4(A4) denotes the probability that A4 = MSj −A1 −A2 −A3 remaining users belong to group 4 scaled by1/(1 +A4), where A4 is the number of users, excludinguser i, who compete with user i for channel j.

We now proceed to calculate these quantities. We have

Θi =

|Si|∑k1=0

Ck1|Si|∑

l1=1

∏j1∈Sl1

i

Pij1d pij1

∏j2∈Si\Sl1

i

pij2 (30)

H1∑k2=1

Ck2H1∑

l2=1

∏j3Ψ

l2i

pij3∏

j4∈Scomi

\Ψl2i

pij4 (31)

k2∑k3=1

Ck3k2∑l3

∑j5∈Γl3

1

∏j5∈Γl3

1

Pij5f

∏j6∈Ψl2

i\Γl3

1

Pij6f (32)

Hi−k2∑k4=0

Ck4Hi−k2∑l4=1

1k3 + k4

∏j7∈Γl3

2

Pij7d

∏j8∈Scom

i\Ψl2

i\Γl4

2

pij8d (33)


where Hi denotes the number of channels in Scomi . The quantity

in (30) is the probability that all available channels in Si (ifany) are overlooked by user i due to false alarms. Therefore,user i does not access any channels in Si. The quantity in (31)describes the probability that there are k2 available channels inScomi , and Ψl2

i denotes such a typical set with k2 available chan-nels. The quantity in (32) describes the probability that user icorrectly detects k3 channels out of k2 available channels. Thelast quantity in (33), excluding the factor 1/(k3 + k4), denotesthe probability that user i misdetects k4 channels among theremaining Hi − k2 busy channels in Scom

i . Finally, the factor1/(k3 + k4) is the probability that user i correctly choosesone available channel in Scom

i for transmission out of k3 + k4channels, which are indicated as available by sensing, i.e.,

Φ1(A1) =

CA1MSj∑c1=1

∏m1∈Ω(1)

c1

Pm1jidle

⎛⎝1 −

∏l∈Smi

pm1lbusy

⎞⎠ . (34)

In (34), we consider all possible subsets of users of size A1 thatbelong to group 1 (there are CA1

MSjsuch subsets). Each term

inside the sum represents the probability of the correspondingevent whose set of A1 users is denoted by Ω

(1)c1 , i.e.,

Φ2(A1) =

CA2MSj−A1∑c2=1

∏m2∈Ω(2)

c2

Pm2jbusy. (35)

In (35), we capture the probability that channel j is indicatedas not available by sensing for A2 users in group 2. Possiblesets of these users are denoted by Ω

(2)c2 , i.e.,

Φ3(A3)=

CA3MSj−A1−A2∑

c3=1

∏m3∈Ω(3)

c3

Pm3jidle

∏l3∈Sm3

Pm3l3busy (36)

β∑n=0

Cnβ∑

q=1

∏h1∈Scom,q

j,m3

Pm3h1

idle

∏h2∈S

com,qj,m3

Pm3h2

busy

[1− 1

n+1

]. (37)

For each term in (36), we consider different possible subsetsof A3 users, which are denoted by Ω

(3)c3 . Then, each term

in (36) represents the probability that channel j is indicatedas available by sensing for each user m3 ∈ Ω

(3)c3 whereas all

channels in Sm3are indicated as not available by sensing.

In (37), we consider all possible sensing outcomes forchannels in Scom

m3performed by user m3 ∈ Ω

(3)c3 . In addition,

let Scomj,m3

= Scomm3

\ {j} and β = |Scomj,m3

|. Then, in (37), weconsider all possible scenarios in which n channels in Scom

j,m3

are indicated as available by sensing and user m3 chooses achannel that is different from channel j for transmission [withprobability (1 − (1/n+ 1))], where Scom

j,m3= Scom,q

j,m3∪ Scom,q

j,m3,

and Scom,qj,m3

∩ Scom,qj,m3

= ∅. We have

Φ4(A4) =1

1 +A4

∏m4∈Ω(4)

Pm4jidle

∏l4∈Sm4

Pm4l4busy (38)

γ∑m=0

Cmγ∑

q=1

∏h1∈Scom,q

j,m4

Pm4h1

idle

∏h2∈S

com,qj,m4

Pm4h2

busy

(1

m+1

). (39)

The sensing outcomes captured in (38) and (39) are similar tothe outcomes in (36) and (37). However, given three sets of A1,A2, and A3 users, the set Ω(4) whose size is |Ω(4)| = A4 can bedetermined. Here, γ denotes the cardinality of the set Scom

j,m4=

Scomm4

\ {j}. Other sets are similar to the sets in (36) and (37).However, all users in Ω(4) choose channel j for transmission inthis case. Therefore, user i wins the contention with probability1/(1 +A4), and its achievable throughput is (1 − δ)/(1 +A4).

Summarizing all the cases considered, the throughput thatwas achieved by user i is written as

Ti = Ti {Case 1}+ Ti {Case 2} . (40)

In addition, the total throughput T can be calculated by sum-ming the throughput of all SUs.

C. Congestion of the Control Channel

Under our design, contention on the control channel is mildif the number of channels N is relatively large compared to thenumber of SUs M . In particular, there is no need to employa MAC protocol if we have N � M , because distinct sets ofchannels can be allocated for SUs by using Algorithm 1. Incontrast, if the number of channels N is small compared to thenumber of SUs M , then the control channel may experiencecongestion due to excessive control message exchanges. Thecongestion of the control channel in such scenarios can bealleviated if we allow RTS/CTS messages to be exchanged inparallel on several channels (i.e., multiple rendezvous [32]).

We describe a potential design of a multiple-rendezvousMAC protocol as follows using similar ideas of the McMACprotocol in [31] and [32]. We assume that each SU hops throughall channels by following a particular hopping pattern, whichcorresponds to a unique seed [32]. In addition, each SU putsits seed in every packet so that neighboring SUs can learnits hopping pattern. The same cycle structure as described inSection V-A is employed here. Suppose that SU A wishes totransmit data to SU B in a particular cycle. Then, SU A turns tothe current channel of B and senses this channel and its assignedchannels in Stot

AB, which is the set of allocated channels for linkAB. If SU A’s sensing outcomes indicate that the current chan-nel of SU B is available, then SU A sends RTS/CTS messages,with SU B containing a chosen available communication chan-nel. Otherwise, SU A waits until the next cycle to again performsensing and contention. If the handshake is successful, SU Atransmits data to SU B on the chosen channel in the data phase.Upon completing data transmission, both SUs A and B return totheir home hopping patterns. In general, collisions among SUsare less frequent under a multiple-rendezvous MAC protocol,because contentions can occur in parallel on different channels.

VIII. NUMERICAL RESULTS

We present numerical results to illustrate the throughput per-formance of the proposed channel assignment algorithms. Toobtain the results, the probabilities pi,j are randomly realizedin the interval [0.7, 0.9], unless stated otherwise. We choosethe length of control packets as follows. An RTS, includingthe PHY header, is 288 b, whereas a CTS, including the PHY


Fig. 2. Collision probability versus the contention window (for M = 15).

header, is 240 b, which correspond to tRTS = 48 μs and tCTS =40 μs for a transmission rate of 6 Mb/s, which is the basicrate of 802.11a/g standards [37]. Other parameters are chosenas follows. The cycle time is Tcycle = 3 ms, θ = 20 μs, andtSIFS = 28 μs. The target collision probability is εP = 0.03, andtSEN and tSYN are assumed to be negligible, and therefore, theyare ignored. Note that these values of θ and tSIFS are typical(interested readers can refer to [33] for related information).The value of cycle time Tcycle is relatively small, given thatpractical cognitive systems such as systems that operate on thetelevision bands standardized in the 802.22 standard requiresthe spectrum evacuation time of a few seconds [34]. We willpresent the total throughput under the throughput maximizationdesign and the minimum throughput under max–min fairnessdesign in all numerical results. All throughput curves are ob-tained by averaging over 30 random realizations of pi,j .

A. MAC Protocol Configuration

We first investigate interactions between MAC protocolparameters and the achievable throughput performance. Inparticular, we plot the average probability of the first collision,which is derived in Section V-C, versus the contention windowin Fig. 2 when Algorithm 2 is used for channel assignment. Thisfigure shows that the collision probability first increases andthen decreases with N . This case can be interpreted as follows.When N is relatively small, Algorithm 2 tends to allow moreoverlapping channel assignments for increasing the numberof channels. However, more overlapping channel assignmentsincrease the contention level, because more users may wantto exploit same channels, which results in a larger collisionprobability. Because N is sufficiently large, a few overlappingchannel assignments are needed to achieve the maximumthroughput.Therefore, thecollisionprobabilitydecreaseswithN.

We now consider the impact of target collision probabil-ity εP on the total network throughput, which is derived inSection VI-A. Recall that, in this analysis, the collision prob-ability is not taken into account, which is shown to havenegligible errors in Proposition 2. In particular, we plot thetotal network throughput versus εP for M = 10 and differentvalues of N in Fig. 3. This figure shows that the total through-put slightly increases with εP . However, the increase is quite

Fig. 3. Total throughput versus target collision probability under the through-put maximization design for M = 10.

Fig. 4. Total throughput versus the number of channels under the throughputmaximization design for M = 2 (Theo: Theory; Sim: Simulation; Over: Over-lapping; Non: Nonoverlapping; Opt: Optimal assignment).

marginal as εP ≥ 0.03. In fact, the required contention windowW given in (12) decreases with increasing εP (as observed inFig. 2), which leads to decreasing the MAC protocol overheadδ(W ), as confirmed by (13), and, therefore, the increase in thetotal network throughput. Moreover, the total throughput maydegrade with increasing εP because of the increasing numberof collisions. Therefore, we will choose εP = 0.03 to presentthe following results, which would be reasonable to balance be-tween throughput gain due to moderate MAC protocol overheadand throughput loss due to contention collision.

B. Proposed Algorithms Versus Optimal Algorithms

We demonstrate the efficacy of the proposed algorithms bycomparing their throughput performances with those obtainedby the optimal brute-force search algorithms for small values ofM and N . Numerical results are presented for both throughput-maximization and max–min fair objectives. In Figs. 4 and 5, wecompare the throughput of the proposed and optimal algorithmsfor M = 2 and M = 3 under the throughput-maximizationobjective. These figures confirm that Algorithm 2 achievesthroughput very close to the throughput attained by the optimalsolution for both values of M .


Fig. 5. Total throughput versus the number of channels under the throughputmaximization design for M = 3 (Theo: Theory; Sim: Simulation; Over: Over-lapping; Non: Nonoverlapping; Opt: Optimal assignment).

Fig. 6. Minimum throughput versus the number of channels under max–minfairness for M = 2 (Theo: Theory; Sim: Simulation; Over: Overlapping; Non:Nonoverlapping; Opt: Optimal assignment).

In Figs. 6 and 7, we plot the throughput achieved by ourproposed algorithms and the optimal algorithm for M = 2 andM = 3 under the max–min fair design. Again, Algorithm 4achieves throughput very close to the optimal throughput underthis design. In addition, analytical results match the simula-tion results very well, and nonoverlapping channel assignmentalgorithms achieve noticeably lower throughput than thethroughput attained by their overlapping counterparts if thenumber of channels is small. It can also be observed thatthe average throughput per user under the throughput maxi-mization design is higher than the minimum throughput thatwas attained under the max–min fair design. This result is quiteexpected, because the max–min fairness trades throughput forfairness.

C. Throughput Performance of the Proposed Algorithms

We illustrate the total throughput T versus the number ofchannels obtained by both Algorithms 1 and 2, where eachpoint is obtained by averaging the throughput over 30 differ-ent realizations of pi,j in Fig. 8. Throughput curves due toAlgorithms 1 and 2 are indicated by “P-ware” in this figure. Inaddition, for comparison, we show the throughput performance

Fig. 7. Minimum throughput versus the number of channels under max–minfairness for M = 3 (Theo: Theory; Sim: Simulation; Over: Overlapping; Non:Nonoverlapping; Opt: Optimal assignment).

Fig. 8. Total throughput versus the number of channels under the throughputmaximization design for M = 15 (Theo: Theory; Sim: Simulation; Over:Overlapping; Non: Nonoverlapping; 5-Over: Five-user sharing overlapping).

achieved by the “P-blind” algorithms, which simply allocatechannels to users in a round-robin manner without exploitingthe heterogeneity of pi,j (i.e., multiuser diversity gain). For theP-blind algorithms, we show the performance of both nonover-lapping and overlapping channel assignment algorithms. Here,the overlapping P-blind algorithm allows at most five usersto share one particular channel. We have observed throughnumerical studies that allowing more users to share one channelcannot achieve better throughput performance because of theexcessive MAC protocol overhead.

As shown in Fig. 8, the analytical and simulation resultsachieved by both proposed algorithms match each other verywell. This result validates the accuracy of our throughput ana-lytical model developed in Section VI-A. It also indicates thatthe total throughput reaches the maximum value, which is equalto M = 15 as the number of channels becomes sufficientlylarge for both Algorithms 1 and 2. This case confirms theresult stated in Proposition 1. In addition, Algorithm 2 achievessignificantly larger throughput than Algorithm 1 for low ormoderate values of N . This performance gain comes fromthe multiuser diversity gain, which arises due to the spatialdependence of white spaces. For large N (i.e., more thantwice the number of users M ), the negative impact of the


Fig. 9. Throughput gain between Algorithms 2 and 1 versus the number ofchannels.

Fig. 10. Throughput gain between Algorithm 2 and P-blind five-user sharingversus the number of channels.

MAC protocol overhead prevents Algorithm 2 from performingoverlapped channel assignments. Therefore, both Algorithms 1and 2 achieve similar throughput performance.

Fig. 8 also indicates that both proposed algorithms outper-form their round-robin channel assignment counterparts. In par-ticular, Algorithm 1 significantly improves the total throughputcompared to the round-robin algorithm under nonoverlappingchannel assignments. For the overlapping channel assignmentschemes, we show the throughput performance of the round-robin assignment algorithms when five users are allowed toshare one channel (denoted as five-user sharing in the fig-ure). Although this approach achieves larger throughput forthe round-robin algorithm, it still performs worse comparedto the proposed algorithms. Moreover, we demonstrate thethroughput gain due to Algorithm 2 compared with Algorithm 1for different values of N and M in Fig. 9. This figure showsthat performance gains up to 5% can be achieved when thenumber of channels is small or moderate. In addition, Fig. 10presents the throughput gain due to Algorithm 2 versus theP-blind algorithm with five-user sharing. It can be observed thata significant throughput gain of up to 10% can be achieved forthese investigated scenarios.

Fig. 11 illustrates the throughput of Algorithms 3 and 4,where pij are chosen in the range of [0.5, 0.9]. It can beobserved that the overlapping channel algorithm also signifi-

Fig. 11. Minimum throughput versus the number of channels under max–minfairness for M = 5 (Theo: Theory; Sim: Simulation; Over: Overlapping; Non:Nonoverlapping).

Fig. 12. Total throughput versus the number of channels under the throughputmaximization design for M = 5, Pij

f∈ [0.1, 0.15], and Pij

d= 0.9 (Theo:

Theory; Sim: Simulation; Over: Overlapping; Non: Nonoverlapping; Per: Per-fect sensing; Imp: Imperfect sensing).

cantly improves the minimum throughput performance com-pared to its nonoverlapping counterpart. Finally, we plot thethroughput achieved by Algorithms 1 and 2 under perfect andimperfect spectrum sensing for M = 5 in Fig. 12, where thedetection probabilities are set as Pij

d = 0.9, whereas false-alarm probabilities are randomly realized as Pij

f ∈ [0.1, 0.15].This figure shows that sensing errors can significantly degradethe throughput performance of SUs. In addition, the presentedresults validate the throughput analytical model described inSection VII-B.

IX. CONCLUSION

We have investigated the channel assignment problem forcognitive radio networks with hardware-constrained SUs. Wehave first presented the optimal brute-force search algorithmand analyzed its complexity. Then, we have developed thefollowing two channel assignment algorithms for throughputmaximization: 1) the nonoverlapping overlapping channel as-signment algorithm and 2) the overlapping channel assignmentalgorithm. In addition, we have developed an analytical modelto analyze the saturation throughput that was achieved bythe overlapping channel assignment algorithm. We have also


presented several potential extensions, including the design ofmax–min fair channel assignment algorithms and throughputanalysis, considering imperfect spectrum sensing. We have val-idated our results through numerical studies and demonstratedsignificant throughput gains of the overlapping channel assign-ment algorithm compared with its nonoverlapping and round-robin channel assignment counterparts in different networksettings.

REFERENCES

[1] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access,” IEEESignal Process. Mag., vol. 24, no. 3, pp. 79–89, May 2007.

[2] L. B. Le and E. Hossain, “Resource allocation for spectrum underlay incognitive radio networks,” IEEE Trans. Wireless Commun., vol. 7, no. 12,pp. 5306–5315, Dec. 2008.

[3] Y.-C. Liang, Y. Zeng, E. C. Y. Peh, and A. T. Hoang, “Sensing-throughputtradeoff for cognitive radio networks,” IEEE Trans. Wireless Commun.,vol. 7, no. 4, pp. 1326–1337, Apr. 2008.

[4] T. Yucek and H. Arslan, “A survey of spectrum sensing algorithms forcognitive radio applications,” IEEE Commun. Surveys Tuts., vol. 11, no. 1,pp. 116–130, 1st Quarter, 2009.

[5] J. So and N. H. Vaidya, “Multichannel MAC for ad hoc networks:Handling multichannel hidden terminals using a single transceiver,” inProc. ACM MobiHoc, 2004, pp. 222–233.

[6] C. Cormio and K. R. Chowdhury, “A survey on MAC protocols for cogni-tive radio networks,” Elsevier Ad Hoc Netw., vol. 7, no. 7, pp. 1315–1329,Sep. 2009.

[7] L. T. Tan and L. B. Le, “Distributed MAC protocol for cognitive radionetworks: Design, analysis, and optimization,” IEEE Trans. Veh. Technol.,vol. 60, no. 8, pp. 3990–4003, Oct. 2011.

[8] H. Kim and K. G. Shin, “Efficient discovery of spectrum opportunitieswith MAC-layer sensing in cognitive radio networks,” IEEE Trans. Mo-bile Comput., vol. 7, no. 5, pp. 533–545, May 2008.

[9] H. Su and X. Zhang, “Cross-layer-based opportunistic MAC protocols forQoS provisionings over cognitive radio wireless networks,” IEEE J. Sel.Areas Commun., vol. 26, no. 1, pp. 118–129, Jan. 2008.

[10] H. Su and X. Zhang, “Opportunistic MAC protocols for cognitive-radio-based wireless networks,” in Proc. CISS, 2007, pp. 363–368.

[11] H. Nan, T.-I. Hyon, and S.-J. Yoo, “Distributed coordinated spectrumsharing MAC protocol for cognitive radio,” in Proc. IEEE DySPAN, 2007,pp. 240–249.

[12] L. Le and E. Hossain, “OSA-MAC: A MAC protocol for opportunisticspectrum access in cognitive radio networks,” in Proc. IEEE WCNC,2008, pp. 1426–1430.

[13] C. Cordeiro and K. Challapali, “C-MAC: A cognitive MAC protocolfor multichannel wireless networks,” in Proc. IEEE DySPAN, 2007,pp. 147–157.

[14] A. C.-C. Hsu, D. S. L. Weit, and C.-C. J. Kuo, “A cognitive MAC protocolusing statistical channel allocation for wireless ad hoc networks,” in Proc.IEEE WCNC, 2007, pp. 105–110.

[15] Y. R. Kondareddy and P. Agrawal, “Synchronized MAC protocolfor multihop cognitive radio networks,” in Proc. IEEE ICC, 2008,pp. 3198–3202.

[16] C. Doerr, M. Neufeld, J. Fifield, T. Weingart, D. C. Sicker, andD. Grunwald, “MultiMAC: An adaptive MAC framework for dynamicradio networking,” in Proc. IEEE DySPAN, 2005, pp. 548–555.

[17] H. Salameh, M. M. Krunz, and O. Younis, “MAC protocol for opportunis-tic cognitive radio networks with soft guarantees,” IEEE Trans. MobileComput., vol. 8, no. 10, pp. 1339–1352, Oct. 2009.

[18] T. Shu, S. Cui, and M. Krunz, “Medium access control for multichannelparallel transmission in cognitive radio networks,” in Proc. IEEE GLOBE-COM, 2006, pp. 1–5.

[19] T. Chen, H. Zhang, G. M. Maggio, and I. Chlamtac, “CogMesh: Acluster-based cognitive radio network,” in Proc. IEEE DySPAN, 2007,pp. 168–178.

[20] J. Shi, E. Aryafar, T. Salonidis, and E. W. Knightly, “Synchronized CSMAcontention: Model, implementation and evaluation,” in Proc. IEEE INFO-COM, 2009, pp. 2052–2060.

[21] J. Jia, Q. Zhang, and X. Shen, “HC-MAC: A hardware-constrained cog-nitive MAC for efficient spectrum management,” IEEE J. Sel. AreasCommun., vol. 26, no. 1, pp. 106–117, Jan. 2008.

[22] H. B. Salameh, M. Krunz, and O. Younis, “Cooperative adaptive spectrumsharing in cognitive radio networks,” IEEE/ACM Trans. Netw., vol. 18,no. 4, pp. 1181–1194, Aug. 2010.

[23] T. Shu and M. Krunz, “Exploiting microscopic spectrum opportunities incognitive radio networks via coordinated channel access,” IEEE Trans.Mobile Comput., vol. 9, no. 11, pp. 1522–1534, Nov. 2010.

[24] W. Wang, K. G. Shin, and W. Wang, “Joint spectrum allocation and powercontrol for multihop cognitive radio networks,” IEEE Trans. Mobile Com-put., vol. 10, no. 7, pp. 1042–1055, Jul. 2011.

[25] X. Zhang and H. Su, “Opportunistic spectrum sharing schemes forCDMA-based uplink MAC in cognitive radio networks,” IEEE J. Sel.Areas Commun., vol. 29, no. 4, pp. 716–730, Apr. 2011.

[26] Y.-J. Choi and K. G. Shin, “Opportunistic access of TV spectrum usingcognitive-radio-enabled cellular networks,” IEEE Trans. Veh. Technol.,vol. 60, no. 8, pp. 3853–3864, Oct. 2011.

[27] X. Zhang and H. Su, “CREAM-MAC: Cognitive radio-enabled multi-channel MAC protocol over dynamic spectrum access networks,” IEEEJ. Sel. Topics Signal Process., vol. 5, no. 1, pp. 110–123, Feb. 2011.

[28] M. Timmers, S. Pollin, A. Dejonghe, L. Van der Perre, and F. Catthoor,“A distributed multichannel MAC protocol for multihop cognitiveradio networks,” IEEE Trans. Veh. Technol., vol. 59, no. 1, pp. 446–459,Jan. 2010.

[29] W. Jeon, J. Han, and D. Jeong, “A novel MAC scheme for multichannelcognitive radio ad hoc networks,” IEEE Trans. Mobile Comput., vol. 11,no. 6, pp. 922–934, Jun. 2012.

[30] S. C. Jha, U. Phuyal, M. M. Rashid, and V. K. Bhargava, “Design ofOMC-MAC: An opportunistic multichannel MAC with QoS provisioningfor distributed cognitive radio networks,” IEEE Trans. Wireless Commun.,vol. 10, no. 10, pp. 3414–3425, Oct. 2011.

[31] H. Su and X. Zhang, “Channel-hopping-based single-transceiver MAC forcognitive radio networks,” in Proc. CISS, 2008, pp. 197–202.

[32] J. Mo, H. So, and J. Walrand, “Comparison of multichannel MAC proto-cols,” IEEE Trans. Mobile Comput., vol. 7, no. 1, pp. 50–65, Jan. 2008.

[33] G. Bianchi, “Performance analysis of the IEEE 802.11 distributed coordi-nation function,” IEEE J. Sel. Areas Commun., vol. 18, no. 3, pp. 535–547,Mar. 2000.

[34] G. Ko, A. Franklin, S.-J. You, J.-S. Pak, M.-S. Song, and C.-J. Kim,“Channel management in IEEE 802.22 WRAN systems,” IEEE Commun.Mag., vol. 48, no. 9, pp. 88–94, Sep. 2010.

[35] J. Lee and S. Leyffer, Eds., “Mixed Integer Nonlinear Programming,”in The IMA Volumes in Mathematics and Its Applications, vol. 154.New York: Springer, 2012.

[36] L. Tassiulas and S. Sarkar, “Max–min fair scheduling in wireless ad hocnetworks,” IEEE J. Sel. Areas Commun., vol. 23, no. 1, pp. 163–173,Jan. 2005.

[37] Wireless Local Area Networks, IEEE Std. 802.11.

Le Thanh Tan (S’11) received the B.Eng. andM.Eng. degrees from Ho Chi Minh City Univer-sity of Technology, Ho Chi Minh City, Vietnam, in2002 and 2004, respectively. He is currently workingtoward the Ph.D. degree with the Institut Nationalde la Recherche Scientifique–Énergie, Matériauxet Télécommunications, Université du Québec,Montréal, QC, Canada.

From 2002 to 2010, he was a Lecturer with HoChi Minh City University of Technical Education.His research activities include wireless communi-

cations and networking, cognitive radios (protocol design, spectrum sensing,detection, and estimation), statistical signal processing, random matrix theory,compressed sensing, and compressed sampling.

Long Bao Le (S’04–M’07) received the B.Eng. de-gree from Ho Chi Minh City University of Tech-nology, Ho Chi Minh City, Vietnam, in 1999, theM.Eng. degree from the Asian Institute of Tech-nology, Pathumthani, Thailand, in 2002, and thePh.D. degree from the University of Manitoba,Winnipeg, MB, Canada, in 2007.

He was a Postdoctoral Researcher withMassachusetts Institute of Technology, Cambridge.He is currently an Assistant Professor with the InstitutNational de la Recherche Scientifique–Énergie,

Matériaux et Télécommunications, Université du Québec, Montréal, QC,Canada. His research interests include cognitive radio, radio resourcemanagement for heterogeneous wireless networks, cooperative diversity,stochastic control, and cross-layer design for communication networks.

channel assignment with access contention resolution for cognitive radio networks

Technology

radio spectrum

channel assignment problem

spectrum sensing

secondary spectrum access

form spectrum

sus exploit spectrum

lowcomplexity channel

low spectrum utilization