1372 ieee transactions on mobile computing, vol....

Efficient and Fair Bandwidth Allocation inMultichannel Cognitive Radio Networks

Dan Xu, Student Member, IEEE, Eric Jung, Student Member, IEEE, and Xin Liu, Member, IEEE

Abstract—Cognitive radio (CR) improves spectrum efficiency by allowing secondary users (SUs) to dynamically exploit the idle

spectrum owned by primary users (PUs). This paper studies optimal bandwidth allocation of SUs for throughput efficiency. Consider

the following tradeoff: an SU increases its instantaneous throughput by accessing more spectrum, but channel access/switching

overhead, contention among multiple SUs, and dynamic PU activity create higher liability for larger bandwidths. So how much is too

much? In this paper, we study the optimal bandwidth allocation for multiple SUs. Our approach is twofold. We first study the optimal

bandwidth an SU should use to maximize the per-SU throughput in the long term. The optimal bandwidth is derived in the context of

dynamic PU activity, where we consider both independent and correlated PU channel scenarios while accounting for the effects of

channel switching overhead. We further consider the case of suboptimal spectrum use by SUs in the short term due to PU activity

dynamics. We propose an efficient channel reconfiguration (CREC) scheme to improve SUs’ performance. We use real PU channel

activity traces in the simulations to validate our results. The work sheds light on the design of spectrum sharing protocols in cognitive

radio networks.

Index Terms—Cognitive radio, opportunistic spectrum access, bandwidth allocation, channel correlation.

Ç

1 INTRODUCTION

COGNITIVE radio (CR) can capture or “sense” temporaland spatial variations in the radio environment,

allowing it to find unoccupied portions of spectrum in realtime [1]. However, the spectrum efficiency of cognitiveradio networks is limited by the need to protect primaryusers (PUs’) transmission, especially when PU activity isdynamic. Moreover, secondary users (SUs’) access over-head and competition also challenge efficient dynamicspectrum access.

Consider a set of licensed channels that are madeavailable to SUs. This is often referred to as spectrumpooling [2], [3]. There can be a secondary service provider(SSP) coordinating a set of SUs to access the PU bands.Using a set of spectrum sensors, the SSP needs to sense PUbands to find channels that are not occupied by PUs. At agiven time, SUs can use a subset of these pooled channels.How SUs should share these channels is not a trivialproblem. A popular paradigm is to let the SSP to allocatechannels based on the current channel availability informa-tion only, e.g., in [4], [5], [6]. However, this paradigm maynot be efficient when the channel availability changes fastdue to dynamic PU activity. First, performing sophisticatedspectrum allocation to achieve good performance may havetime overhead, which decreases channel availability time.Second, frequent channel reallocation may not be desirableif we consider the channel evacuation and access overhead

of SUs. Third, since the available channel pool changes overtime, it is difficult to guarantee the long-term throughputfairness for SUs only with the current channel availabilityinformation.

To address these issues, in this paper, we propose anefficient and fair bandwidth allocation scheme for SUs indynamic PU channel environments, i.e., where PU channelavailability changes over time. Our scheme exploits thestatistical PU channel information. Intuitively, bandwidthselection of an SU is affected by the PU activity. If PUs usethe channels more intensively, e.g., longer PU transmittingtime, each SU should probably access fewer channels.Therefore, it is important for the SSP to obtain informationon the intensity of PU activity.

In this paper, we assume the SSP senses the PU bands inthe long term to obtain statistical information of each PUchannel. Based on this information, the SSP can calculatehow much bandwidth is optimal for SUs to maximize per-SU throughput in the long term. Our scheme considers thechannel access overhead and SUs competition. Note that toguarantee throughput-fairness in the long term, we let eachSU use the same number of channels. In the short term, theSSP simply lets the SUs which have evacuated due to PUs’return access the current idle channels, based on thepredetermined optimal bandwidth. Note that in certaincases, some existing SUs in the spectrum need to berearranged, which is defined as channel reconfiguration(CREC) in this paper. Thus, our scheme is easy toimplement and has a low overhead.

Based on the statistical channel information, we can alsostudy the impact of PU behaviors on the SUs’ performanceand bands selection. Another important contribution of thispaper is the study on PU channel correlation. PU channelcorrelation due to adjacent channel interference (ACI), mayhave important effects on the SUs’ performance. Mean-while, it is also interesting to learn how the SSP should

1372 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 8, AUGUST 2012

. D. Xu and X. Liu are with the Department of Computer Science,University of California, Kemper Hall, One shields Avenue, Davis, CA95617. E-mail: {danxu, xinliu}@ucdavis.edu.

. E. Jung is with the Department of Electrical and Computer Engineering,University of California, Kemper Hall, One Shields Avenue, Davis, CA95617. E-mail: [email protected].

Manuscript received 25 Jan. 2011; revised 29 May 2011; accepted 15 July2011; published online 4 Aug. 2011.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2011-01-0042.Digital Object Identifier no. 10.1109/TMC.2011.168.

1536-1233/12/$31.00 � 2012 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS

observe and select bands for SUs, and how the SSP candetermine bandwidth for each SU if channels are correlated.

In summary, we study the following central issue in thispaper: How much bandwidth should SUs seek to optimizeper-SU throughput, accounting for dynamic PU channelavailability, channel access overhead and SUs’ competition?Our main contributions are as follows:

. We first derive the closed-form approximation ofper-SU throughput in an independent PU channelscenario, as a function of the number of channels anSU uses, by which we can find optimal bandwidthnumerically. We consider two cases where an SU canaccess nonadjacent channels and adjacent channelsonly.

. We study PU channel correlation caused by adjacentchannel interference. We propose an ACI model anduse a performance metric for SUs, N-channel holdingtime, to show how PU channel correlation affects theSUs’ performance and band selection. Motivated bythe study, we derive a closed-form approximation ofper-SU throughput in a correlated PU channelscenario.

. In the context of using adjacent channels, we find thesuboptimality of SUs’ spectrum access due todynamic PU activity. We then define the channelreconfiguration problem and present an efficientdynamic programming scheme for channel reconfi-guration.

. In addition to extensive simulations, we also test ourresults using a set of real channel trace data, whichexhibits both heterogeneous PU behavior and somedependence among PU activities in adjacent chan-nels. Our scheme is shown to be robust.

The rest of the paper is organized as follows: we describethe PU and SU models in Section 2. In Section 3, we derivethe closed-form per-SU throughput approximation in anindependent PU channel scenario. In Section 4, we studyhow PU channel correlation affects the SUs’ performance,followed by the derivation on per-SU throughput approx-imation in a correlated PU channel scenario. We thenpropose channel reconfiguration in Section 5. We evaluateour schemes in Section 6. After reviewing related work inSection 7, we conclude the paper in Section 8.

2 MODELS AND ASSUMPTIONS

2.1 Primary User Model

We consider PUs as legacy devices that access a block ofspectrum through static channelization. We assume thatthere is a set of M consecutive PU channels, each with thesame bandwidth denoted by BP . For example, in the US, aTV channel has a bandwidth of 6 MHz and there are morethan 100 TV channels.

A PU channel is modeled as being in either a busy or idlestate, which refers, respectively, to times when a PUoccupies the channel or not. We also refer to these as theON/OFF states. We use SðiÞ to denote the state of channel i,where SðiÞ ¼ 0 if channel i is idle, and SðiÞ ¼ 1 if it is busy.Let P denote the probability that a channel is idle, andassume that each channel has the same idle probability. In

practice, PU channels are likely to be heterogenous. We cantake P as the average idle probability of all PU channelsconsidered. We can write P ¼ EðTIÞ=ðEðTIÞ þ EðTBÞÞ,where TI and TB denote the idle and busy time,respectively. We consider the scenarios of both independentPU channels and correlated PU channels. In this paper,there is no assumption on the arrival patterns of PUs unlessspecifically stated. Main notations used in this paper arelisted in Table 1.

2.2 Secondary User Model

An SU is an opportunistic user equipped with a cognitiveradio which can dynamically change its operating fre-quency and bandwidth. For simplicity, we assume that acognitive radio transmits on a positive integral number oflicensed PU channels. This assumption is common in theliterature: for example, in the Microsoft KNOWS prototype,the minimum bandwidth of a cognitive radio is 5 MHz andthe prototype operates on a bandwidth that is a multiple of5 MHz [7].

In our model, we consider a network of K SU transmitterand receiver pairs (or links) operating on M licensedchannels. Later, we abbreviate an SU transmitter andreceiver pair as an SU. The SUs can belong to one SSPwhich allocates channels to each SU. All SUs are withininterference range of each other and must be scheduled ondisjoint channels when they communicate concurrently. LetN denote the number of channels an SU operates on, whereN ¼ 1; . . . ; Nmax, with Nmax being the upper limit ofchannels that an SU can operate on. The objective is to findthe optimal value of N for each SU such that the (average)per-SU throughput is optimized. Note that the optimalvalue of N for each SU is the same. In this paper, we use“bandwidth” and “number of channels” interchangeably.Depending on its physical layer capability (such as OFDM),an SU may only transmit on a set of adjacent (orconsecutive) channels or may be able to use nonadjacent(or noncontinuous) channels. We refer to the former case asUAC and the latter as UNC. We note that in principle an SUwith cognitive radio may be capable of transmitting onnonadjacent channels. But in practice transmitting onadjacent channels is easier to implement. The differencebetween the adjacent and nonadjacent channel requirementis very important, as will be shown in Section 6.

We use Shannon capacity to model the achievable rate ofan SU. To elaborate, when an SU uses N channels, itsachievable rate �ðNÞ is NBp logð1þ Pt

n0NBpÞ, where Pt denotes

the transmit power of the SU. We assume that each SU has

XU ET AL.: EFFICIENT AND FAIR BANDWIDTH ALLOCATION IN MULTICHANNEL COGNITIVE RADIO NETWORKS 1373

TABLE 1Main Notations

the same fixed transmit power throughout the paper. Inaddition, n0 denotes noise variance, which we assume isuniform throughout the spectrum in which the SU networkoperates. Our analytical model can be directly extended toother rate models.

The time overhead that an SU incurs before it can accessthe spectrum is denoted by C. The time overhead C can bedifferent for different network configurations and protocols.For example, when an SU needs to sense the channels byitself, C includes the channel sensing time. In the model ofIEEE 802.22 WRAN [8], there is a central infrastructureusing a number of spectrum sensors to perform channelsensing. An SU obtains idle channels from the infrastruc-ture and does not need to sense the channel individually.Therefore, the overhead C mainly consists of notificationoverhead (e.g., the infrastructure notifies SUs the informa-tion on idle channels), channel switching or access over-head, and probable link setup time. In a distributednetwork where SUs contend for access, C includes thechannel contention and backoff time. In this paper, wefollow the model of IEEE 802.22 WRAN [8] and treat C as aconstant (or the mean value of the overhead when it is arandom variable). We assume that C includes notificationoverhead, switching/access, and link setup time, and C isindependent of N , number of channels an SU accesses. Allresults in this paper can be easily extended to otheroverhead models. For example, in the earlier version [35],we also consider different channel sensing schemes and thecorresponding overhead. In practice, the notification over-head, switching/access, and link setup times are based onthe system’s specific hardware and operating environment.For example, in XG field tests [9], the switching and linksetup time is 0.165 s.

To protect PU communications, an SU can only operateon idle channels, and must evacuate a channel immediatelywhen PUs return. When an SU accesses N channels and oneof them is reoccupied by PUs, the SU can keep transmittingon the remaining N � 1 channels. The SU can also switch toa new set of N channels which may include the remainingN � 1 channels or part of them. Note that in the switchingprocess, the SU does not need to stop transmitting on thoseremaining N � 1 channels. In this paper, our objective is tofind the optimal number of channels N� to maximize per-SU throughput. An SU transmits on N� channels if they areavailable. In the absence of N � channels, the SU can usethose available.

2.3 Real Channel Trace Collection

We also collected real trace data (RT) to test the robustness ofour results. Spectral measurements were taken in the publicsafety band (850-870 MHz) in Howard County, Maryland.The spectral data was collected in 0.01 s snapshots, with aDFT frequency resolution of 8.333 kHz. The measurementswere taken over a duration of 100 minutes. Please refer to[10] for more details of the spectrum measurement.

Within the measured band, 60 channels were selected.The channels exhibit high-power PU transmissions andlow-noise levels when idle, resulting in easily detectableON/OFF PU activity traces. PU activity is determined bythe application of a simple energy threshold method, withlow-level processing to eliminate false alarms caused by

noise. We note that each of the channels only has abandwidth of 25 kHz; these channels are of interest becausethey exhibit a large number of on/off cycles, and statisticalheterogeneity. Some of the selected channels are adjacent inthe spectrum, and experience adjacent channel interferencewhich leads to some correlation among the channels.

Fig. 1 visualizes the real trace data used in the paper. Weuse an energy threshold to determine whether each channelis busy, denoted by a black tick, or idle, denoted by a whitetick, for any 0.01 s time interval, and each channel in ourdata trace exhibits a number of busy/idle cycles. ObservingFig. 1, we see PUs transmission is dynamic and idle channelsset varies fast. Thus, performing spectrum allocation basedon instantaneous channel availability may not be efficient. Inthis case, PU channel statistical information is needed topredetermine the proper bandwidth each SU should use.

3 OPTIMAL BANDWIDTH ALLOCATION IN

INDEPENDENT PU CHANNEL CASE

In this section, we investigate the optimal bandwidthallocation problem when there are multiple SUs competingfor a limited number of PU channels. We assume each channelis independent. We consider both UNC and UAC cases.

Assuming SUs have the same channel access opportu-nity, we have the following per-SU throughput equation, asa function of the number of channels N that each SU uses,

RsðNÞ ¼KavgðNÞ�ðNÞ

K: ð1Þ

In (1), KavgðNÞ denotes the average number of SUs that canaccess the spectrum simultaneously, which clearly dependson N , the number of channels an SU uses. Intuitively,KavgðNÞ is constrained by both the number of SUs K, andthe number of SUs that M channels can support. To find N�

that maximizes average SU throughput, we must firstderive the expression for KavgðNÞ, which is different in theUAC and UNC cases. We first consider the UNC case,which leads to a much simpler derivation for KavgðNÞ.

3.1 UNC Case

In this case, the derivation of KavgðNÞ is simple. We firstconsider the case C ¼ 0. First, when each SU uses Nchannels, the maximum number of SUs that the M channelscan accommodate is bMNc. Thus, the maximum number of


Fig. 1. Channel activity map from trace data of 60 real channels.

SUs that can access the M channels simultaneously isKmax ¼ minðK; bMNcÞ. In a given time, there are 0 � i < Kmax

SUs accessing simultaneously only when the number of idlechannels, j, is at least iN , but no more than ðiþ 1ÞN � 1(otherwise there will be iþ 1 SUs using the channels). Notethat there are Kmax SUs using the channels when j is at leastKmaxN (and less than M). Thus, we can write KavgðNÞ as

KavgðNÞ ¼XKmax�1

i¼0

i �Xðiþ1ÞN�1

j¼iNPMðjÞ

" #þ

XMj¼KmaxN

KmaxPMðjÞ;

ð2Þ

where PMðjÞ ¼ ðMj ÞPjð1� P ÞM�j, which is the probabilitythat there are j idle channels.

To expand this model to include nonzero overhead C, wecan consider the overhead as a part of the channel busyperiod. In other words, we modify the channel idleprobability P as

P̂ ¼ EðTIÞ � CEðTIÞ þ EðTBÞ : ð3Þ

Note that C should be smaller than EðTIÞ in practice for thechannels to be useful for dynamic access. If other overheadmodels are considered where C may be a function of N ,e.g., in the case that each SU needs sensing the channels tofind N idle channels, we see that P̂ is also a function of N .We will also apply P̂ to the UAC case in the independentPU channel scenario.

3.2 UAC Case

To calculate KavgðNÞ in this case, we first examine theproperties of the PU activity in the M independent PUchannels. From Fig. 2, we see that at any time, the Mchannels appear as a series of busy and idle channel blocks.Let WI

i and WBi refer, respectively, to the width (i.e., the

number of channels) of the ith idle channel block and ithbusy channel block. Obviously, SUs can only transmit overthe idle channel blocks, and if we know the size of an idlechannel block i, we can calculate the number of SUs that thisblock can support. If we also find the number of idle channelblocks that are occupied by SUs, we can obtain the averagenumber of SUs that M channels can support. Note that bothWI

i and the number of idle channel blocks are randomvariables. Note that the number of SUs in the spectrum isconstrained by either the number of channels M or thenumber of SUs K. First, we consider the channel constraint.

Let us define a random variable TM as

TM ¼ arg mini

�WI

1 þWB1 þ � � � þWI

i þWBi �M

�: ð4Þ

TM is the minimum number of idle and busy channelblocks which taken together have a larger width than M.

We note that TM is a stopping time for the random process

fWIi þWB

i ; i � 1g, since the event fTM ¼ ig only depends

on fWIj þWB

j ; 1 � j � ig, and has no relation with fWIj þ

WBj ; j > ig.Now consider an idle channel block i with width WI

i .

The block can support bWIi

N c SUs. Since WIi is a random

variable, it follows that the number of SUs that can be

supported by each idle channel block forms a random

process, defined as fbWIi

N c; i � 1g. TM is also a stopping time

for fbWIi

N c; i � 1g. Let KM denote the number of SUs that can

be accommodated by the channels. We have

XTM�1

i¼1

WIi

N

� �� KM �

XTMi¼1

WIi

N

� �: ð5Þ

When there are K SUs, the actual number of SUs

occupying the channels is minðKM;KÞ. Now we can write

KavgðNÞ ¼ EðminðKM;KÞÞ: ð6Þ

The intuition is as follows: when M is large, or P̂ is large,

the channel accommodating capacity is large, and the

number of SUs in the spectrum is close to K. On the other

hand, when K is large, the channels are saturated with SUs

due to the limited number of channels M. The distribution

for KM is difficult to derive in practice, so we make the

following approximation:

KavgðNÞ � minðEðKMÞ; KÞ: ð7Þ

Since min function is concave, we have minðEðKMÞ; KÞ �EðminðKM;KÞÞ by Jensen’s inequality. Therefore, minðEðKMÞ;KÞ is an upper bound of KavgðNÞ. Note that we observe in

simulations that minðEðKMÞ; KÞ is close to KavgðNÞ.Now let us calculate EðKMÞ. According to (5), we have

EXTM�1

i¼1

WIi

N

� � !� EðKMÞ � E

XTMi¼1

WIi

N

� � !; ð8Þ

which is equivalent to

EðTMÞEWI

i

N

� �� E

WITM

N

$ % !� EðKMÞ

� EðTMÞEWI

i

N

� �� ;

ð9Þ

where EðPTM

i¼1 bWI

i

N cÞ ¼ EðTMÞEðbWI

i

N cÞ by Wald’s equation

[12]. We note that bWI

TM

N c has a different distribution from

bWIi

N c, i < TM , since TM is the stopping time. Therefore, we

cannot write

EXTM�1

i¼1

WIi

N

� � !¼ ðEðTMÞ � 1ÞE WI

i

N

� �� :

We can use EðTMÞEðbWIi

N cÞ to approximate EðKMÞ. We

calculate EðTMÞ and EðbWIi

N cÞ as follows.From (4) and the fact that TM is the stopping time for

fWIi þWB

i ; i � 1g, we can write


Fig. 2. The alternative idle and busy channel blocks and the stoppingtime TM due to channel boundary M.

XTM�1

i¼1

�WI

i þWBi

�< M �

XTMi¼1

�WI

i þWBi

�; ð10Þ

which is equivalent to

EðTMÞE�WI

i þWBi

�� E

�WI

TMþWB

TM

�< M

� EðTMÞE�WI

i þWBi

�:

ð11Þ

We can use M=EðWIi þWB

i Þ to approximate EðTMÞ. Note

that WIi is a geometrically distributed random variable.

Since the event fWIi ¼ kg is conditioned on the event that

the first channel of the idle channel block i is idle, we have

PrfWIi ¼ kg ¼ P̂

k�1ð1� P̂ Þ; k � 1, and we can ignore the

boundary effect. Similarly, we also have PrfWBi ¼ kg ¼

P̂ ð1� P̂ Þk�1; k � 1. Then, we can write EðWIi Þ ¼ 1=ð1� P̂ Þ

and EðWBi Þ ¼ 1=P̂ . Therefore, we have

EðTMÞ �M

11�P̂ þ

1P̂

¼MP̂ ð1� P̂ Þ: ð12Þ

Now, we calculate EðbWIi

N cÞ. In the event that fbWIi

N c ¼ kg,meaning that the idle channel block i can support no morethan k users, we can then write that fkN �WI

i < ðkþ 1ÞNg.Based on this relation, we can calculate the expected valueof users that idle channel block i can support:

EWI

i

N

� �� ¼XbMNc�1

i¼1

Xðiþ1ÞN�1

j¼i�NiP̂

j�1ð1� P̂ Þ

þ M

N

� � XMj¼bMNcN

P̂j�1ð1� P̂ Þ:

ð13Þ

We again ignore the boundary effect of M. Then, (13)becomes

EWI

i

N

� �� ¼ P̂

N�1

1� P̂N: ð14Þ

Combining (12) and (14), we have

EðKMÞ �MP̂

Nð1� P̂ Þ1� P̂N

: ð15Þ

Finally, according to (7), we have

KavgðNÞ � minMP̂

Nð1� P̂ Þ1� P̂N

;K

!: ð16Þ

Combining this result with ð1Þ, we derive the closed-formapproximation of per-SU throughput in the UAC case. Per-SU throughput is a function of N , i.e., the number ofchannels each SU uses. The optimal number of channels N�

can be determined numerically.

4 OPTIMAL BANDWIDTH IN THE PRESENCE OF PUCHANNEL CORRELATION

In the last section, we assume each PU channel isindependent. In practice, PU channels are often correlateddue to adjacent channel interference. That is, severaladjacent channels may be effectively occupied by a high-power PU in a channel. Jones et al. [10] have observed the

PU channel correlation phenomena in the public safety PUbands. In this section, we are motivated to study theoptimal bandwidth allocation problem in the correlated PUchannel case. We first theoretically study the essentialproperty of channel correlation, and its importance to theSUs’ band selection and performance.

4.1 PU Channel Correlation: The Impact on SUs’Access Opportunity

We model the adjacent channel interference caused by PUtransmission. We first study the so-called 1-ACI case wherea PU interferes only with its nearest adjacent channels.Then, we extend the results to a more general �-ACI case.

Consider a channel i. A PU may use a low transmitpower which causes no notable interference to its neighbor-ing channels. On the other hand, a PU with high transmitpower can cause interference to its adjacent channels i� 1and iþ 1 for simplicity, we assume the arrival processes ofthe high-power PUs (packets or transmission) and low-power PUs in a channel i are two independent Poissonprocesses. The interarrival times of low-power PUs andhigh-power PUs are denoted by �0

i and �1i , respectively.

They are exponentially distributed with parameters �0i and

�1i , respectively.

To evaluate the SU spectrum access opportunity, weintroduce a metric called N-channel holding time, definedas the time an SU can transmit on N channels simulta-neously. Therefore, it is the time between when an SUswitches to N channels and PUs return one of the N

channels. Therefore, the N-channel holding time, ThðNÞ, isthe minimum residual idle time of all N channels. That isThðNÞ ¼ minðTr1 ; T r2 ; . . . ; T rNÞ, where Tri , ði ¼ 1; . . . ; NÞ is theresidual idle time of channel i. Let FTIi ðxÞ denote the CDFof the idle time of channel i, denoted by TIi . Then, theunconditional PDF of Tri is fTri ðxÞ ¼ ð1� FTIi ðxÞÞ=EðT

Ii Þ.

If each channel’s residual idle time is i.i.d, the PDF ofThðNÞ is fThðxÞ ¼ Nð1� FTri ðxÞÞ

N�1fTri ðxÞ. For exponen-tially distributed idle time of each channel, i.e., PUs’transmission in each channel follows a Possion process, wehave EðThðNÞÞ ¼ EðTIÞ=N . We show how PU channelcorrelation impacts the N-channel holding time of an SU.

In [11], the fatal shock model is introduced to modelmultivariant exponential distribution. The model has thefollowing physical meaning in our system. When a PUaccesses a channel, this event acts as a “fatal shock” thatterminates the idle period of the channel. Therefore, boththe occurrence of low-power PUs and high-power PUs inchannel i are “fatal shocks” to channel i. Furthermore, theoccurrence of a high-power user at channel i is also a “fatalshock” to both channels i� 1 and iþ 1. The “fatal shocks”of a channel force the SU to stop using that channel.

We model the channel idle time correlation using thefatal shock model. Consider an SU that operates on N

adjacent channels, indexed by i ¼ j; . . . ; jþN � 1. Foreach channel i, the idle period follows an exponentialdistribution since for each channel the PUs’ arrival followsa Poisson process. The residual idle time of a channel i,Tri , also follows an exponential distribution. We calculateN-(correlated) channel holding time, denoted by Thc ðNÞ.We have


Pr�Thc ðNÞ > t

¼ Pr

�Trj > t; T rjþ1 > t; . . . ; T rjþN�1 > t

¼ Pr

��1j�1 > t; �0

j > t; �1j > t; . . . ; �0

jþN�1 > t;

�1jþN�1 > t; �1

jþN > t

¼ exp � �1j�1 þ

XjþN�1

i¼j

��0i þ �1

i

�þ �1

jþN

!t

" #;

i ¼ j; . . . ; jþN � 1:

ð17Þ

Note j� 1 and jþN are the channels whose high-power

PUs interfere with channel j and jþN � 1, respectively.

Therefore, Thc ðNÞ is exponentially distributed with para-

meter �1j�1 þ

PjþN�1i¼j ð�0

i þ �1i Þ þ �1

jþN .

To highlight the impact of channel correlation, let us

compare it with the independent channel case. First, for

channel i, both the idle time and the residual idle time are

still exponentially distributed with parameter �1i�1 þ �0

i þ�1i þ �1

iþ1. If an SU observes each channel independently, it

will infer that the minimum residual idle time is exponen-

tially distributed with parameterPjþN�1

i¼j ð�1i�1 þ �0

i þ �1i þ

�1iþ1Þ. Or alternatively, we can consider N independent

channels, each of which has an exponentially distributed

residual idle time with parameter �hi�1 þ �li þ �hi þ �hiþ1.

Clearly, the average N-channel holding time of correlated

PU channels, EðThc ðNÞÞ, is larger than that of N indepen-

dent channels, i.e., EðThðNÞÞ.To better understand channel correlation due to ACI, we

next extend the study to the �-ACI case, i.e., PU transmissionin one channel interferes with up to � adjacent channels. Fora channel i, we may have a set of PUs that have differenttransmit power levels, and therefore interfere with differentranges of adjacent channels. Let 1

��idenote the rate of PUs that

interfere with � adjacent channels from i, 0 � � � �. If achannel does not have such a PU, the rate is 0. It can bederived that the N-channel holding time (indexed byi ¼ j; . . . ; jþN � 1) is exponentially distributed as

Pr Thc�ðNÞ > t� ¼ exp �

XjþN�1

i¼j

X��¼0

��i þX�n¼1

X��¼nð��j�n þ ��jþNþn�1Þ

!t

" #;

i ¼ j; . . . ; jþN � 1:

ð18Þ

On the other hand, under the assumption of channelindependence, the inferred N-channel holding time isexponentially distributed with parameter

XjþN�1

i¼j

X�n¼0

X��¼n

��i�n:

For simplicity, consider homogenous channels, where��i ¼ ��, 8i, 0 � 8� � �, the parameters for the correlationand independence are N

P��¼0 �

� þ 2ðP�

�¼1 ��Þ and

NðP�

�¼0 �� þ 2

P��¼1 ��

�Þ, respectively. Therefore, in the�-ACI case, the average N-channel holding time is

EðThc�ðNÞÞ ¼1

NP�

�¼0 �� þ 2ð

P��¼1 ��

�Þ : ð19Þ

Under the independence assumption, the inferred mean ofN-channel holding time is

EðTh� ðNÞÞ ¼1

NðP�

�¼0 �� þ 2

P��¼1 ��

�Þ : ð20Þ

We see EðTh� ðNÞÞ in (20) is larger than EðThc�ðNÞÞ. Wefurther use a simple example to show the impact of � onN-channel holding time. For the cases of different �,we always fix the mean idle time of a channel as 1 s. Wefurther assume arrival rates for different types of PUs arethe same. From (19), EðThc�ðNÞÞ ¼ �þ1

Nþ� ðsÞ. Then, we see thatgiven a fixed channel idle time, the larger � leads to thelarger N-channel holding time.

Note that this does not mean that PUs with highertransmit powers are better for SUs. As stated before, in thecomparison, channel idle time distribution of an individualchannel of the independent channels case is the same as thatof the correlated channels case. Our results on channelcorrelation provide meaningful guideline for spectrummeasurements. It shows that independently observing thePU activity on each individual channel is not optimal.Instead, it would be preferable for the SSP to identify thechannel correlation, which provides additional informationon the intensity of PU activity. With such information, SUscan find more desirable spectrum bands. It also shows thatan SU has more optimistic performance when to accessmultiple channels in practice than the expected perfor-mance if observing each channel independently. Thissimple study motivates us to investigate optimal bandwidthallocation with PU channel correlation.

In this paper, we focus on the impact PU activity on SUs’performance. On the other hand, SUs’ behaviors also affecttheir spectrum access. For example, there can also be high-power SUs and low-power SUs. To protect PUs’ transmis-sion, high-power SUs can use a channel only if both itsadjacent channels are empty, whereas the low-power SUscan use any idle channels. In future work, we will considerthe heterogenous SUs’ transmit power, as well as hetero-genous bandwidth demand and channel availability.

4.2 Optimal Bandwidth Allocation with PU ChannelCorrelation

We have already given the closed-form approximation ofper-SU throughput for UNC and UAC cases in theindependent PU channel scenario. We extend the study tothe correlated PU channel scenario. Our objective is also toderive the representation of per-SU throughput. We firstconsider the UAC case. To derive the throughput, we alsoneed to derive the average number of SUs in the Mchannels, denoted by Kc

avgðNÞ. We first derive the averagenumber of SUs that can be supported by M channels, whichis more challenging in the correlated PU channel scenario.First, we need to derive the distribution of the width of theith idle channel block, denoted by ~WI

i , and the mean widthof the ith idle and busy channel block, Eð ~WI

i þ ~WBi Þ.

Second, we need to derive the optimal stopping time ~TM .We follow the same PUs arrival model as in the last

section. We further assume the arrival processes of bothhigh- and low-power PUs are homogenous for differentchannels. For simplicity, we study the 1-ACI case where


high-power PUs cause interference to two adjacent chan-nels. Here, we replace the notations of �0 and �1 by �l and�h, respectively. We assume PU activity is independent fordifferent channels. That is, a PU can transmit on its ownchannel even if high-power PUs are transmitting in adjacentchannels. In each channel, at most one PU can transmit at agiven time. Therefore, the transmission of high-power PUsand that of the low-power PUs of a channel are exclusive.We further assume average transmission time of each PU isthe same, denoted by tp. Therefore, the idle probability of achannel is

P ¼ ð1� �ltp � �htpÞð1� �htpÞ2; ð21Þ

where ð1� �ltp � �htpÞ is the probability that the channel is

not occupied by either the lower power PUs or high-power

PUs of the channel, and ð1� �htpÞ2 is the probability that

the channel is not interfered by adjacent high-power PUs. In

the rest of the paper, we use �l (�h) to denote �ltp (�htp). We

have P ¼ ð1� �l � �hÞð1� �hÞ2. Note that in this section, we

use P rather than P̂ to denote the channel idle probability.

We later incorporate the overhead C.

First, we derive the distribution of ~WIi . The same as the

independent PU channel scenario, an idle channel block

exists implies the first channel of it is idle. The event f ~WIi ¼

kg is conditioned on the event that its first channel is idle.

Let j denote the first idle channel of the ith idle channel

block. Then, the following equation holds:

Prf ~WIi ¼ kg ¼ PrfSðjþ kÞ ¼ 1; Sðjþ k� 1Þ ¼ 0;

. . . ; Sðjþ 1Þ ¼ 0jSðj� 1Þ ¼ 1; SðjÞ ¼ 0g¼ ðPrfSðjþ kÞ ¼ 1; Sðjþ k� 1Þ ¼ 0; . . . ; SðjÞ ¼ 0;

Sðj� 1Þ ¼ 1gÞ=ðPrfSðj� 1Þ ¼ 1; SðjÞ ¼ 0gÞ:

ð22Þ

To calculate Prf ~WIi ¼ kg, we define two types of events

for channel i. Let Ai and Bi denote the events when thehigh power PUs and low power PUs transmit on channeli, respectively. We have PrfAig ¼ �h, and PrfBig ¼ �l.Note that Ai and Bi are two exclusive events, and theyare independent from the events of any other channel.First, we calculate the denominator in (22), i.e.,PrfSðj� 1Þ ¼ 1; SðjÞ ¼ 0g. We have (23) as follows:

PrfSðj� 1Þ ¼ 1; SðjÞ ¼ 0g ¼ PrfðAj�2 [Aj�1 [Bj�1 [AjÞ\ ðAj�1 \Aj \Bj \Ajþ1Þg¼ PrfððAj�2 [Bj�1Þ \Aj�1Þ \ ðAj \BjÞ \Ajþ1g¼ PrfðAj�2 [Bj�1Þ \Aj�1gPrfAj \BjgPrfAjþ1g¼ ð�l þ �hð1� �hÞ � �l�hÞð1� �l � �hÞð1� �hÞ;

ð23Þ

Similar to (23), we have (24) hold for the numerator in (22).

PrfSðjþ kÞ ¼ 1; Sðjþ k� 1Þ ¼ 0; . . . ; SðjÞ ¼ 0;

Sðj� 1Þ ¼ 1g¼ PrfðAj�2 [Bj�1Þ \Aj�1gPrfAj \Bjg . . .

PrfAjþk�1 \Bjþk�1gPrfðAjþkþ1 [BjþkÞ \Ajþkg¼ ð�l þ �hÞð1� �hÞð1� �l � �hÞkð�l þ �hÞð1� �hÞ;

ð24Þ

Therefore, we have

Pr� ~WI

i ¼ k¼ ð1� �l � �hÞk�1ð�l þ �hÞ; ð25Þ

and further

E� ~WI

i

�¼ 1

�l þ �h: ð26Þ

We note that unlike the independent PU channel scenario,

where we can write PrfWIi ¼kg¼Pk�1ð1�P Þ directly, here

Prf ~WIi ¼ kg just “happens” to be ð1� �l � �hÞk�1ð�l þ �hÞ,

because �l þ �h does not represent the probability that

channel jþ k is busy. In this case,

PrfðAjþkþ1 [BjþkÞ \AjþkgPrfAjþ1g

happens to be �l þ �h due to the channel homogeneity

assumption.

Given the same channel idle probability, if channels are

independent, we have EðWIi Þ ¼ 1

1�P ¼ 11�ð1��l��hÞð1��hÞ2

. We

note thatEð ~WIi Þ � EðWI

i Þ. That is, if channels are correlated,

an idle channel block is longer on average. When SUs use

adjacent channels, more SUs can be supported by an idle

channel block on average.We next calculate the mean width of the ith idle and

busy channel block, i.e., Eð ~WIi þ ~WB

i Þ, which is needed to

find the number of idle channel blocks. Consider infinite

number of channels. At a given time, the ratio of idle

channels is equal to the probability of a channel being idle,

i.e., P . Since the number of idle channel blocks is the same

as the busy channel blocks, we have

E� ~WI

i þ ~WBi

�¼ Eð

~WIi Þ

P¼ 1

ð�l þ �hÞð1� �l � �hÞð1� �hÞ2:

ð27Þ

Therefore, in the correlated PU channel scenario, given the

same channel idle probability, both the mean width of an idle

channel block and a busy channel block are, respectively,

larger than their counterparts in the independent PU channel

scenario. However, the total number of idle channels is the

same as that of the independent PU channel scenario.We next calculate the number of idle channel blocks.

Similar to the independent PU channel scenario, we define~TM as

~TM ¼ arg mini

� ~WI1 þ ~WB

1 þ � � � þ ~WIi þ ~WB

i �M�: ð28Þ

If ~WIi þ ~WB

i are independent for different i, ~TM is the

stopping time for f ~WIi þ ~WB

i ; i � 1g and we can apply

Wald’s equation to calculate ~TM . To show ~WIi þ ~WB

i are

independent for different i, let us first consider two adjacent

idle and busy channel blocks with width of ~WIi þ ~WB

i and~WIiþ1 þ ~WB

iþ1, respectively. We need to show that

Prf ~WIiþ1 þ ~WB

iþ1 ¼ mj ~WIi þ ~WB

i ¼ ng

¼ Prf ~WIiþ1 þ ~WB

iþ1 ¼ mg;ð29Þ


holds for arbitrary m > 1 and n > 1. Since for two adjacentidle and busy channel blocks, the last channel for the firstblock, denoted by j, is busy, and the first channel of thesecond block, denoted by jþ 1, must be idle, we have

Pr� ~WI

iþ1 þ ~WBiþ1 ¼ mj ~WI

i þ ~WBi ¼ n

¼ Pr

� ~WIiþ1 þ ~WB

iþ1 ¼ mjSðjþ 1Þ ¼ 0; SðjÞ ¼ 1;

~WIi þ ~WB

i ¼ n

¼ Pr� ~WI

iþ1 þ ~WBiþ1 ¼ mjSðjþ 1Þ ¼ 0; SðjÞ ¼ 1

¼ Pr

� ~WIiþ1 þ ~WB

iþ1 ¼ m:

ð30Þ

Note that the last equality holds since the event f ~WIiþ1 þ

~WBiþ1 ¼ mg is conditioned on the event that fSðjÞ ¼ 1g andfSðjþ 1Þ ¼ 0g. Following similar arguments, for any twoidle and busy channel blocks, their widths are independent.

Therefore, the event f ~TM ¼ ig is independent of f ~WIk þ

~WBk ; k > ig, and ~TM is a stopping time for f ~WI

i þ ~WBi ; i � 1g.

Similar to (11) in the independent PU channel scenario, weapply Wald’s equation to calculate the expectation of ~TM .That is

Eð ~TMÞ �M

Eð ~WIi Þ þEð ~WB

i Þ¼Mð�l þ �hÞð1� �l � �hÞð1� �hÞ2:

ð31Þ

Now, consider the number of SUs that the ith idle

channel block can support when each SU uses N channels,

i.e., b~WIi

N c. Similar to (13), we have

E~WIi

N

$ % !¼XbMNc�1

i¼1

Xðiþ1ÞN�1

j¼i�Nið1� �l � �hÞj�1ð�l þ �hÞ

þ M

N

� � XMj¼bMNcN

ð1� �l � �hÞj�1ð�l þ �hÞ:ð32Þ

Then, we have

E~WIi

N

$ % !¼ ð1� �l � �hÞ

N�1

1� ð1� �l � �hÞN: ð33Þ

Note that ~TM is also the stopping time for fb~WIi

N c; i � 1g.Then the number of SUs that can be supported by M

correlated channels is Eð ~TMÞEðb~WIi

N cÞ on average. Therefore,

we further have

KcavgðNÞ � min Eð ~TMÞE

~WIi

N

$ % !; K

!

¼ minMð�l þ �hÞð1� �hÞ2ð1� �l � �hÞN

1� ð1� �l � �hÞN;K

!:

ð34Þ

Applying (34) to (1), we obtain the per-SU throughput in theUAC case for the correlated PU channel scenario.

Now let us consider the UNC case in the correlated PUchannel scenario. At a given time, for the correlatedchannels, the ratio of the number of idle channels is thesame as that of the set of independent channels with the

same idle probability. Therefore, given the same channelidle probability, PU channel correlation has no impact onthe SUs’ performance if they use discrete channels. Then tocalculate Kc

avgðNÞ for the UNC case, we can apply P ¼ð1� �l � �hÞð1� �hÞ2 to (2).

We now incorporate nonzero channel switch overheadC. We can substitute �l (�h) by �̂l (�̂h). Similarly, by treatingchannel access overhead as the busy period of the channel,we have

1

3�̂h þ �̂l¼ EðTIÞ � C;

ð1� �̂htpÞ2ð1� �̂ltp � �̂htpÞ ¼EðTIÞ � C

EðTIÞ þ EðTBÞ :

8>><>>: ð35Þ

In (35), EðTIÞ ¼ 13�hþ�l is the mean idle time of a channel,

according to (17). EðTIÞ � C is the new channel idle

time, while EðTI Þ�CEðTI ÞþEðTBÞ is the new channel idle probability.

Although we obtain the closed-form per-SU throughputequation in the correlated PU channel scenario, it is notstraightforward to see how channel correlation affects per-SU throughput and N� by (34). We will numerically studythe impacts of PU channel correlation in Section 6.Intuitively, given the same channel idle probability, channelcorrelation implies less PU activity, which in this case resultsin a longer idle channel block. When SUs use adjacentchannels, more aggregated idle channels can accommodatemore SUs. A higher per-SU throughput is expected.Although (34) discloses per-SU throughput in the correlatedPU channel case, it may be difficult for SUs to determine �land �h to precisely determine N�. We will also show howSUs should calculate N� in the correlated PU channel case.

5 CHANNEL RECONFIGURATION

We have derived the optimal bandwidth each SU shoulduse, to maximize (average) per SU-throughput in the longterm. The optimal bandwidth is derived under theassumption that idle PU channels can be best utilized bySUs given a value of N . That is, the number of SUs that canaccess the bands is only constrained by the number of idlePU channels, and the total number of SUs K. In the UNCcase, the SSP can easily coordinate SUs to best utilize theidle bands, because an SU can access the bands with theoptimal bandwidth whenever there are enough idlechannels. However, in the UAC case, SUs cannot alwaysfully utilize the idle channels, which is caused by bothdynamic PU activity and existing SUs’ transmissions (asshown in next section). To improve the SU performancetoward the optimal per-SU throughput derived, we let theSSP coordinate the SUs by an efficient channel reconfigura-tion scheme.

5.1 Channel Reconfiguration in the UAC Case

In the UAC case, due to the requirement of accessingadjacent channels and PU dynamics, SUs may not fullyutilize the idle channels. Fig. 3 shows a simple example ofthis situation. In the figure, channels 1, 2, and 6 are initiallyoccupied by PUs, which forms an idle channel block fromchannels 3 to 5. An SU occupies channels 3 and 4, which isoptimal in this case since we assume N� ¼ 2 and there are


only three channels available. If channel 2 becomes idle,however, this channel configuration becomes suboptimal.In the resulting configuration, despite having 2 channelswhich remain available, channels 2 and 5 will remainunused because of the UAC requirement. This is clearly notoptimal. Note that these two channels can be allocated totwo SUs. But in consequence each SU may only transmit onone channel, which is also suboptimal. If the current SUreconfigures to use channels 2 and 3 (or 4 and 5), anotherSU could fit into the idle block. This is the basic idea ofchannel reconfiguration in this case: rearrange the exitingSUs in an idle block to make room for more SUs.

There are several issues to consider when developing aCREC scheme that maximizes the SUs’ overall throughput.First, there is an inherent tradeoff to CREC. Althoughreconfiguration allows more SUs to be accommodated, italso harms the SUs already transmitting by forcing them toreconfigure to new channels because channel access over-head is incurred.1 The unpredictable nature of PU activitycan affect the scheme as well. In Fig. 3, if channels 1 or 6were to become idle, no reconfiguration would be requiredto add another SU to the resulting idle block.

Due to the difficulty in predicting PU activity, wesimplify the model by only considering current channelstates. We make some key observations on the nature of theCREC scheme. The main benefit of CREC is that new SUscan be accommodated. When a new SU is accommodatedby CREC, it can then transmit on N� channels over aduration with mean EðThðN�ÞÞ. The cost of CREC is theoverhead added for relocated SUs, which includes the timeto evacuate the channels and set up a new link, C. Thesebenefits and costs can be directly mapped to a throughputgain expression, and the objective of the CREC scheme is tomaximize this expression. We define an optimal CRECscheme as follows.

Optimal channel reconfiguration. A channel reconfigura-tion scheme Q� is optimal when it satisfies

Q� ¼ argmaxQ

�ðQÞEðThðN�ÞÞ � ðQÞC� �

; ð36Þ

where �ðQÞ and ðQÞ refer, respectively, to the number ofadded and reconfigured SUs under scheme Q. The optimalCREC scheme yields the maximum throughput gain in (36).

To find the optimal CREC scheme, let us first examinethe properties of CREC. First, CREC is conducted on a set ofblocks, each of which is bounded by channels being used byPUs, i.e., idle channel blocks as defined in Section 3.2. Werefer to these as CREC blocks here. We note an SU does not

need to reconfigure to another CREC block since this willnot reduce any cost or bring any benefit. Therefore, a CRECscheme Q is a set of CREC schemes w.r.t. each CREC block,i.e., Q ¼ fQ1; Q2; . . . ; Qrg, where r denotes the number ofCREC blocks.

To find Q�, we first focus on a specific CREC block i. It isefficient to find all the possible CREC schemes on i, due tolimited potential idle channel combinations. Based on theseCREC schemes, we can set up a table of gains correspond-ing to the potential number of added SUs. We note thatdifferent CREC schemes may result in the same number ofadded SUs, but they may also result in different throughputgains. We can find the Q� that results in maximum benefitfor the entire system using Algorithm 1. We note that theproblem is equivalent to the classical knapsack problem.2

Algorithm 1. Channel reconfiguration1: For every block i, 1 � i � r, find all the potential

numbers of added SUs in block i, and the

corresponding gains.

2: Find the number of added SUs in block i, �iopt, which

yields the maximum gain for block i.

3: ifPr

i¼1 �iopt is no more than KL, i.e., the number of SUs

that are waiting to obtain channels then

4: Let each block i add extra SUs with an amount of�iopt, respectively.

5: Keep the remaining SUs, i.e., with an amount of

KL �Pr

i¼1 �iopt, waiting for accessing the spectrum.

6: else

7: Use dynamic programming to calculate the number

of SUs added in each CREC block to maximize the

total gain.

8: Add the determined number of SUs to each CRECblock.

9: end if

In Fig. 4, we present a simple example of the optimalCREC scheme algorithm. In the example, KL is 4. For eachblock, the number of SUs that can be added and theassociated gains are shown in the �ðQiÞ=Gain. In Step 2 ofAlgorithm 1, �iopt for the four blocks are 0, 2, 1, 2 from left toright. Then, the optimal scheme would be to accommodatetwo new SUs in CREC blocks 2 and 4, which results in amaximum gain of 6. If KL � 5, the optimal scheme wouldbe to let each CREC block accommodate new SUs with anamount of 0, 2, 1, and 2, respectively.

6 PERFORMANCE EVALUATION

In this section, we evaluate our proposed solutions onbandwidth configuration in the multiple SUs scenariothrough simulations on both simulated channel activityand real trace data.

First, let us examine the correctness of our derivation onper-SU throughput representations. In Fig. 5, we comparethe analytical and numerical results of per-SU throughputfor the UNC and UAC in the independent PU channelscenario. Here, overhead C is set to 0. We will study the


Fig. 3. An example of the opportunity for channel reconfiguration. In thiscase, N� for an SU is 2.

1. For an existing SU using the bands, its channel access overhead isalready considered in deriving per-SU throughput equation. The channelaccess overhead in CREC is not incorporated in the per-SU throughputequation, so we need explicitly consider it in designing the CREC scheme.

2. Knapsack problem can be solved in pseudopolynomial time usingdynamic programming. Moreover, our problem can be solved efficiently byAlgorithm 1 due to limited number of CREC blocks in practice.

case where C > 0 later. We set M ¼ 100 and study the per-

SU throughput as a function of the number of SUs K, where

1 � K � 40. For analytical results, we simply follow (2) and

(16) to find N� and calculate per-SU throughput for the

UAC case and the UNC case, respectively. For numerical

results, we create 100 channels, each of which is randomly

set as idle or busy following a probability. We then let each

SU access N channels, which is chosen from 1 to Nmax. We

repeat the simulation for 500 iterations and obtain average

throughput for each N chosen. The one with maximum

throughput is taken as N�.We observe that for both UNC and UAC cases, under

different channel idle probabilities, numerical results match

the analytical results very well. For UNC, the numerical

results match theory better than in the UAC case, since we

are able to derive an exact closed-form expression for per-

SU throughput when C ¼ 0. As stated previously, our

closed-form expression for the per-SU throughput in UAC

is actually an upper bound. We observe that the analytical

throughput is slightly larger than the numerical result.Fig. 5 shows that although UNC generally leads to

higher per-SU throughput than UAC, this performance

difference decreases as K increases, or when P is very

small. The key factor that mitigates the difference is the

optimal number of channels. When K is large, UNC and

UAC exhibit little performance difference because the

optimal number of channels approaches 1 in both cases.

When P is very small, as in the P ¼ 0:1 curves, the optimal

number of channels is 1 for both UNC and UAC over a

wide range of K. When P is very large, however, the

number of optimal channels in both cases tends to be

rather large because of the high channel idle probability.

Therefore, the limits imposed by the consecutive channelrequirement tend to have less effect on per-SU throughput.

In Fig. 6, we show how N� is affected by K and P . We setM ¼ 60. In the left side figure, we fix P ¼ 0:5. We can see inboth UNC and UAC cases, N� becomes smaller as K

increases. In the right side, we fix K ¼ 15. We see N� isincreasing as P increases. We also observe that UNC leadsto a larger N� than UAC, and our analytical results lead toalmost the same N� as simulation results.

In Fig. 7, we study how channel correlation impacts theSU performance in the multi-SU scenario. We set M ¼ 60,C ¼ 0 and let K vary between 1 and 25. We simulate aslotted system where for each channel, there are a low-power PU arrival process and a high-power PU arrivalprocess. For a channel, if there is a PU transmitting or ahigh-power PU transmitting from a neighbor channel, weset it as busy. We set a homogenous channel scenario. Foreach channel, we consider �l ¼ 0:34, �h ¼ 0:085 and�l ¼ 0:05, �h ¼ 0:2, respectively. In both scenarios, thechannel idle probability is equal to 0.48. In the latterscenario, channels are more correlated due to a larger ratiobetween the arrival rates of high-power PUs and low-powerPUs. We also compare the independent PU channel case


Fig. 5. Optimal performance of UNC and UAC in the independent PUchannel scenario.

Fig. 6. Optimal number of channels an SU accesses (N�) under differentnumber of SUs, and different channel idle probabilities.

Fig. 7. Optimal performance under correlated PU channels.

Fig. 4. An example of CREC scheme. EðThðN�ÞÞ ¼ 3 s and C ¼ 1 s. �ðQiÞ is the potential number of added SUs for block i, recorded in the first lineof the table above each corresponding block. The second line of each table is the corresponding gain.

with the same channel idle probability 0.48. We obtainanalytical results by using (35). For simulation, we calculatethe average per-SU throughput for 10,000 time slots fordifferent values of N , and obtain the one with maximumper-SU throughput.

In both correlated PU channel scenarios, we observeanalytical results are a little larger than those of simulationswhen K is small, which validates our analytical results onper-SU throughput in the correlated PU channel scenario isan upper bound. We can observe that per-SU throughput ofthe setting �l ¼ 0:05, �h ¼ 0:2 is the highest, followed by thatof the setting �l ¼ 0:34, �h ¼ 0:085, and the throughput ofindependent channels is the lowest. In other words, highercorrelation leads to larger per-SU throughput. When K islarge, there is no difference among different scenarios,because N� ¼ 1 in this case, where per-SU throughput is notaffected by channel correlation. We further study howchannel correlation affects N�. By both our analytical results(by (34)) and simulation results (the same simulation settingas in Fig. 7), we found that N� is always the same given thesame channel idle probability P , despite the ratio between�l and �h. Recall that P ¼ ð1� �l � �hÞð1� �hÞ2. Thisobservation is very interesting and useful. In practice, theSUs prefer to access more correlated bands given the samechannel idle probability on each individual channel. But itmay be hard to identify the �l and �h for calculating N�.The above observation implies that an SU can use (16) andthe observed P to calculate N�, which is also optimal in thecorrelated channel case. Note this does not mean (34) is lessimportant. Per-SU throughput equation in (34) shows theessence of channel correlation.

By Figs. 5 and 7, we conclude that UNC leads to a higherper-SU throughput than UAC. However, the benefits ofUNC is small when K is large or P is large or small. WhenPU channels are correlated, the gap between UNC and UACis smaller, since UAC has a better performance in this case.

In Fig. 8, we compare the results of our optimalbandwidth solution for the UAC case both with andwithout CREC. In the figure, we plot curves using bothreal trace data described in Section 2.3, and simulated datatraces (EXP). Recall Fig. 1 visualizes the real trace data usedin our simulations. Note that in this simulation setting, eachchannel in the real trace data is assumed to have bandwidthof 5M, which is different from the actual bandwidth of thechannels. However, the importance of the real trace datalies in the statistical heterogeneity of PU activity in realenvironments, such as different idle probabilities, mean idletimes, and channel correlations.

Fig. 8 represents the per-SU throughput versus numberof SUs in the system for a given overhead C. We alsogenerate simulated channel traces to compare the perfor-mance over real trace data with known distributions. In thesimulated traces, we generate 60 channels with exponen-tially distributed idle times, with each channel’s mean equalto the mean of the corresponding channel in the real tracedata. The simulated channels are generated independently.For our analytical results (labeled “Analysis” in the figure),we use the mean of the idle probabilities and the mean ofthe idle time over all 60 channels as the input parameters toour closed-form expression. The average idle time mean ofour real trace data is 13.3 s, and the average idle probabilityis 0.695. For simulation results, we let each SU choose N idlechannels to access, and wait a time C before transmitting.We implement the CREC scheme following Algorithm 1.

From Fig. 8, we can see that when C is small (0 to 1 s),our analytical results are very close to our RT and EXPresults. It also appears that RT leads to a higher per-SUthroughput than EXP. This is most likely caused by thelikely correlations that appear in real channel traces.Correlations between channels allow SUs to have morechannel access opportunities, as we discussed in Section 4.


Fig. 8. Comparison of performance among Theory, Simulation on real channel trace (RT) and channel trace with exponentially distributed idle time(EXP). P ¼ 0:695 and EðTIÞ ¼ 13:3 s for the curve of Analysis. M ¼ 60, K ¼ 1; . . . ; 40.

We also observe that CREC increases the optimalthroughput of each SU. However, notable gains are onlymade in cases where the optimal number of channels is 2 or3, where we observe frequent reconfigurations of SUs.Generally, the gain from CREC is not significant for tworeasons. First, opportunities to use CREC are fairly limited.Second, CREC introduces its own overhead since SUreconfiguration disrupts SU communication and requiresSUs to undergo communication setup overhead. From thefigure, we also see that as C increases, the benefit of CRECdecreases, and in the extreme case that C ¼ 10 s, there is nothroughput gain from CREC. This is because when C ¼ 10s, the overhead C is too large compared to the expectedchannel holding time, EðThðN�ÞÞ. In this case, our channelholding time is generally much smaller than 13.3 s, theaverage idle time of a channel.

Finally, the figure also shows that in cases when C is verylarge, i.e., C ¼ 10 s, our analytical model appears to beinapplicable. This is expected, since our model is based onthe assumption that overhead C is much smaller thanaverage channel holding time. As we argued before, it isimpractical for an SU network to operate in a givenspectrum if its overhead is comparable to or even largerthan the channel holding time. We also note that the optimalnumber of channels N� is very similar in the analyticalmodel, the RT, and EXP simulations. This indicates that inpractical systems, our model can be used to derive theoptimal bandwidth, which leads to optimal throughput.

7 RELATED WORK

Research in cognitive radio networks has manifested inmany areas, including spectrum pooling [2], [3], channelsensing [13], [14], [15], [16], [17], and coexistence of SUs[18], [19].

Centralized and distributed schemes for spectrumallocations or multichannel dynamic spectrum access havebeen considered [4], [5], [6], [20], [21], [24], [26], [27]. Amongthem, there is some work studying the dynamic multi-channel channel access or allocation problems based on agiven set of available channels. For example, in [4], Houet al. consider users’ heterogenous channel availability indetermining bandwidth and frequency location. In [5], Caoand Zheng propose a distributed algorithm to sharespectrum among users with various fairness considerations.In [6], Yuan et al. propose B-SMART, a spectrum-time blockallocation scheme targeted for TV band unlicensed usage.

DSA in dynamic multi-PU channel environments havebeen studied. For example, in [20], [21], [22], [23], and [24],the authors study the multichannel probing and accessproblems. The works are based on the model of slotted PUactivity. They mainly apply POMDP or multiarmed banditmethods to address the sensing and transmitting strategiesof a SU, which is assumed to have limited sensing abilityand partial PU channel states information. In [25], Hoanget al. characterize the tradeoff between maximizing the sumthroughput of PUs and SUs while minimizing PUs’interference. The authors find the optimal number of SUs.When the SUs adapt the traffic arrival probability, theauthors find optimal transmission probability of SUs, andthe optimal number of SUs.

There are also some other works on DSA. For example, in[27], Yang et al. consider the hardware constraint ofcognitive radio, i.e., the limited capability of sensing andtransmitting. The authors design a multichannel MACprotocol for cognitive radio networks. However, the authorsdo not study the impact of PU activity in the paper. Thebandwidth allocation problem is not addressed either. Abrief of some other works includes dynamic traffic drivenspectrum allocation [28], combinatorial multiarmed bandit-based spectrum allocation [29], and distributed spectrumallocation for sequentially arriving SUs [30]. Neither ofthese works addresses the impact of PU activity, thetradeoff between instantaneous transmission rate andswitching overhead, and optimal bandwidth selection.

Spectrum-agile access and different MAC manners havealso been studied. In a recent work [31], Park et al. show thebenefits of agile radios by accessing noncontinuous bands.They characterize the performance improvement of k-agileradio over 1-agile radio in dynamic demand and conflicttopology scenarios, respectively. In [32], Bahl et al.analytically study link layer throughput of different MACprotocols for spectrum access. They characterize thedifference between whether an SU buffers a connection orswitches when PUs return.

Our work differs from all the above works because itfocuses on the impact of PU behavior on the SUperformance and addresses the optimal channel bandwidthallocation issue considering switching overhead and multi-ple SU competition, in both the independent and correlatedPU channels scenarios. Note we do not assume slotted PUand SU systems.

There are significant advances in channel allocationschemes in multichannel wireless networks scenarios, e.g.,in [33] and [34]. These works handle channel allocationdynamically based on traffic conditions, but cannot beapplied in cognitive radio networks directly, because theydo not explicitly address the challenge of channel dynamicsdue to PU activity.

8 CONCLUSIONS

In this paper, we study the optimal bandwidth allocationfor SUs in accessing dynamic PU channels. We derive theoptimal per-SU throughput for a set of SUs which canaccess a number of channels. The result depends on variousfactors, including PU channel idle probability, PU channelcorrelation, and SU access scheme and overhead. Thehardware capability of the SU, i.e., whether it has to use aset of adjacent channels or can use nonadjacent channels,has significant impact on the SU performance. Usingnonadjacent channels results in higher performance thanusing adjacent channels, but the gap is small when thechannel idle probability is very small or large.

We also study the impact on the SU performance ofchannel correlation caused by adjacent channel interference,by both the metrics of the channel holding time and the per-SU throughput. Given the same channel idle time, higherchannel correlation leads to longer channel holding time foran SU. In the multi-SU case, when SUs use adjacent channels,higher correlation allows the spectrum to support more SUsgiven the same channel idle probability. These resultsindicate that it is important to identify channel correlation.


The SUs prefer to access more correlated channels given thesame channel idle probability. We note that in correlated PUchannels, the gap between using nonadjacent channels andusing adjacent channels is smaller.

Numerical simulations and real channel activity tracesare used to validate our analysis. In general, thesesimulations show that our analytical models are robustand applicable to real world models.

ACKNOWLEDGMENTS

The work was in part supported by the US National ScienceFoundation through CAREER Award #0448613 and Grant#0520126, and by Intel through a gift grant. The authorswould like to thank the anonymous reviewers for theirvaluable comments and suggestions. This work waspartially presented at IEEE DySPAN 2008 [35].

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Dan Xu is currently working toward the PhDdegree in the Department of Computer Scienceat the University of California, Davis. Hisresearch interests include the area of networkresource allocation and optimization, specificallyin cognitive radio networks, data center net-works, and 3G/4G mobile networks. He is astudent member of the IEEE and the IEEEComputer Society.

Eric Jung received the BS and MS degrees inelectrical engineering from the same universityin 2006 and 2009, respectively. Currently, he isworking toward the PhD degree in the Electricaland Computer Engineering Department at theUniversity of California, Davis. His researchinterests include wireless networking and com-munications with an emphasis on dynamicspectrum access. He is a student member ofthe IEEE.

Xin Liu received the PhD degree in electricalengineering from Purdue University in 2002.Currently, she is working as an associateprofessor in the Computer Science Departmentat the University of California, Davis (UC Davis).Before joining UC Davis, she was a postdoctoralresearch associate in the Coordinated ScienceLaboratory at the University of Illinois at Urbana-Champaign. Her research interest include wire-less communication networks with a focus on

resource allocation and dynamic spectrum management. She receivedthe best paper of the year award from the Computer Networks Journal in2003 for her work on opportunistic scheduling. She received the USNational Science Foundation (NSF) CAREER award in 2005 for herresearch on SmartRadio-Technology-Enabled Opportunistic SpectrumUtilization. She received the Outstanding Engineering Junior FacultyAward from the College of Engineering, University of California, Davis, in2005. She is a member of the IEEE.

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