The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

SENSING-THROUGHPUT TRADEOFF IN COGNITIVE RADIO NETWORKS:HOW FREQUENTLY SHOULD SPECTRUM SENSING BE CARRIED OUT?

Yiyang PeiNanyang Technological University

Nanyang Avenue, Singapore 639798

Anh Tuan Hoang and Ying-Chang LiangInstitute for Infocomm Research

21 Heng Mui Keng Terrace, Singapore 119613

ABSTRACT

We consider a cognitive radio (CR) network that makes op-portunistic access to a spectrum licensed to the primary users.During its operation, the CR network carries out spectrumsensing on a frame-by-frame basis to detect active primaryusers, thereby avoiding interfering with them. We formulatea collision-throughput tradeoff problem which, based on thesensing time requirement and the traffic pattern of primaryusers, finds optimal value for the frame duration of CR opera-tion so that the throughput of the CR network is maximized, yetthe collision probability of the primary users is not greater thana threshold. We derive the theoretical formula for achievablethroughput of CR network and find the optimal solution forframe duration. Computer simulations are presented to evalu-ate the performance of our approach.

I. INTRODUCTION

Recent measurements by Federal Communications Commis-sion (FCC) show that more than 70% of the allocated spectrumin US is not fully utilized [1]. Using cognitive radio technology[2, 3, 4], opportunistic spectrum access allows cognitive radio(CR) networks to opportunistically exploit the under-utilizedspectrum. However, as transmissions from CR networks cancause harmful interference to primary users of the spectrum,CR networks has to frequently carry out spectrum sensing todetect and thereby protecting active primary users [5, 6, 7, 8].To enable spectrum sensing, sensing periods have to be sched-uled within normal operation of CR networks.

In this paper, we consider a CR network that operates ona frame-by-frame basis. Each frame is divided into a sensingsubframe and a data transmission subframe. In [9], for a givenframe duration, Liang et al considered the design of optimalsensing slot duration to achieve the maximum throughput ofCR network, yet to protect the primary users by achieving cer-tain probability of detection. Intuitively, the longer time thatthe CR users spend on sensing the channel, the better that pri-mary users will be protected; however, longer sensing time willresult in a reduction in the amount of time for data transmissionand hence affect the achievable throughput of the CR users.

In this paper, we focus on another aspect of the problem,i.e., fixing the sensing duration and optimize the frame durationof CR networks. We formulate a throughput-collision tradeoffproblem which, based on the sensing time requirement and thetraffic pattern of primary users, find optimal value for the frameduration of CR operation so that the throughput of the CR net-work is maximized, yet the collision probability of the primaryusers is not greater than a threshold. We derive the theoretical

Sensing Data Transmission Sensing Data Transmission

Frame n Frame n+1τ

τ τ T τ−T τ−

Figure 1: Frame structure of a cognitive radio network.

formula for the achievable throughput of CR network and findthe optimal solution for the frame duration.

The rest of this paper is organized as follows. In Section II.,we describe our system model and the control problem. Thesystem throughput of CR network is derived in Section III.. InSection IV., the frame duration that maximizes the CR through-put is derived. We present computer simulation results in Sec-tion V.. Finally, we conclude the paper in Section VI..

II. SYSTEM MODEL AND PROBLEM DEFINITION

We consider a CR network that operates on a frame-by-framebasis as depicted in Figure 1. In each frame of duration T ,CR users sense the channel for a duration of τ . If none ofthe primary users is detected in the channel of interest, the CRusers will use the rest of frame T − τ for data transmission.Otherwise, if an active primary user is detected, the CR userswill not transmit in this frame and wait until the next frame tosense the channel again.

We assume the primary users have an exponential on-offtraffic model, with the mean durations of on and off periodsdenoted by β1 and β0, respectively. During the on period, apacket is generated every duration of tp. Generated packets aretransmitted immediately on the channel. We assume that CRusers are heavily loaded and always have packets to transmit.

Consider the operation of CR network within one frame.Since a CR user will not transmit if it senses the presenceof a primary user, we are only interested in the conditionalachievable throughput (conditional is neglected for brevity sub-sequently) of the CR user when the primary user is not activeat the time of sensing. However, the primary user may turn onduring the frame and cause transmission collision for both theprimary and the CR users.

Let us define P sc as the probability that a CR user experienc-

ing packet collision during its transmission duration T −τ . Thenormalized throughput of CR network can then be expressed as

R̂(τ ;T ) =T − τ

T(1 − P s

c )C0. (1)

where C0 is the channel capacity. Assuming that the sensingtime τ has been fixed (based on the sensing requirement), the

1-4244-1144-0/07/$25.00 c©2007 IEEE

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

Sensing Data Transmission

τ T τ−

t

0t =

Secondary user

Time

Primary user

Off On

Figure 2: A CR user’s frame when primary user is not presentduring sensing.

normalized throughput of the CR network is given by

R̃(T ) =T − τ

T(1 − P s

c ). (2)

Intuitively, for a given sensing time τ , the larger the frame du-ration, the longer the data transmission time T − τ . On theother hand, the longer the frame, the more chances that the pri-mary user will become active, thus more collisions may occurbetween the CR and primary users, which will eventually de-crease the throughput of CR users and cause more collisionsto the primary users. Therefore, there could exist an opti-mal frame duration that provides the best tradeoff between theachievable throughput and packet collision.

Our objective is thus to design the optimal frame duration Tsuch that the achievable throughput of the CR network is max-imized, subject to the constraint that the primary user’s opera-tion is sufficiently protected. Let P p

c denotes the packet colli-sion probability for the primary user, the optimization problemcan be described as

maxT

R̃(T ) = T−τT (1 − P s

c ), (3)

s.t. P pc ≤ P̄ p

c , (4)

where P̄ pc is maximum collision probability that the primary

user can tolerate.

III. DERIVATION OF P sc , P p

c , AND R̃(T )

Consider the transmission of a CR user’s frame as shown inFigure 2 when no primary user is present during the sensingslot. As the CR user has full traffic, its throughput, i.e. thenumber of transmitted collision-free packets, will depend onwhen the primary user turns on during the current frame. With-out loss of generality, we denote the end of the sensing slot asthe starting point of time t = 0 and define t as the time re-quired for the primary user to turn on from being off. Hencethe time duration within which the CR user could transmit withcollision during the current frame can be expressed as

x(t) ={

T − τ − t 0 ≤ t ≤ (T − τ)0 t > (T − τ), (5)

where the probability density function of t is given by

P0(t) =1β0

exp(− t

β0

). (6)

Therefore, the average time that the CR user will transmit withcollision can be represented as

x(T ) =∫ T−τ

0

(T − τ − t)P0(t)dt

= T − τ − β0

(1 − exp

(−T − τ

β0

)). (7)

From (7), we can calculated P sc as

P sc =

x(T )T − τ

= 1 − β0

T − τ

(1 − exp

(−T − τ

β0

)). (8)

Subsequently, the normalized throughput of the CR user is

R̃(T ) =β0

T

(1 − exp

(−T − τ

β0

)), (9)

which is a function of the the frame duration T . It should bepointed out that (9) is only an approximation for the normalizedthroughput as we have treated the transmission as a continuousprocess and assumed that for a particular frame, the primaryuser will be active until the end of the frame once it turns on.

For the packet collision probability for primary user, we needto first calculate the expected duration of collision time withineach ‘on period’ of the primary user. Denoting this value byy(T ), it can be shown that:

y(T ) =x(T )

Pb{0 ≤ t ≤ T − τ} =x(T )

1 − exp(−T−τ

β0

) . (10)

Then

P pc =

y(T )β1

=1β1

T − τ

1 − exp(−T−τ

β0

) − β0

. (11)

IV. SOLVING FOR OPTIMAL FRAME DURATION Topt

To find whether there exists an optimal value of frame durationTopt, we take the first derivative of R̃(T ) and obtain

dR̃(T )dT

=1

T 2

(−β0 + (β0 + T ) exp

(−T − τ

β0

)). (12)

Setting (12) equal 0, we have

−(

β0 + T

β0

)exp

(−β0 + T

β0

)= − exp

(−β0 + τ

β0

). (13)

Then we solve for T and obtain

T = −β0

(1 + W

(− exp

(−β0 + τ

β0

))), (14)

where W(x) is Lambert’s W function which solves the equa-tion w exp(w) = x for w as a function of x. Figure 3 showsa plot of Lambert’s W function. It is observed that W(x) is atwo-valued function when x is real in the interval x ∈ (− 1

e , 0).The branch satisfying W(x) ≥ −1 is called the principlebranch and denoted as W0(x). The other branch satisfying

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

−0.5 0 0.5 1 1.5 2−6

−5

−4

−3

−2

−1

0

1

2

x

W(x

)

Lambert’s W function W(x)

−1/e

Figure 3: Lambert’s W function.

W(x) < −1 is called the negative branch and denoted asW−1(x). However, we are only interested in the value ob-tained from negative branch of Lambert’s W because the onein the principle branch will results in T < 0, which is not validfrom the physical viewpoint of T . Hence, the possible optimalvalue of frame duration T can be expressed as

T+ = −β

(1 + W−1

(− exp

(−β0 + τ

β0

))). (15)

Next, we verify whether R̃(T ) has a maximal value at T+ bychecking if its second derivative

d2R̃(T )dT 2

=2β0

T 3

(1 − exp

(−T − τ

β0

)(1 +

T

β0+

T 2

2β20

))(16)

is negative for T > 0. From the second condition d2R̃(T )dT 2 < 0,

we have

τ > T − β0 ln(

1 +T

β0+

T 2

2β20

). (17)

The right hand side of (17) is a monotonously increasing func-tion of T in the interval (0,∞). If we denote it as f(T ), fromits inverse function, we can have

T < f−1(τ). (18)

Furthermore, from the physical meaning of frame duration, wemust have

T > τ. (19)

Hence, the optimal frame duration can be found as

Topt = −β0

(1 + W−1

(− exp

(−β0 + τ

β0

))), (20)

if this Topt satisfy both (18) and (19).

0 20 40 60 80 100 120 140 160 1800

0.05

0.1

0.15

0.2

0.25

Frame duration T

Pro

babi

lity

of c

ollis

ion

Pc

Primary userSecondary user

Figure 4: Probability of collision for both primary user and CRuser in the heavy traffic scenario.

V. SIMULATION RESULTS

We consider a CR operation that involves one primary user andone CR user. The sensing slot duration is set to τ = 1 msec andthe mean of off and on periods duration of the primary user areset to β0 = 650 and β1 = 352 msec, respectively. During theon period, the primary user’s packet is generated and transmit-ted every tp = 20 msec. Note that the parameters of primaryusers represent VoIP traffics. The CR user’s slot duration is 1msec and its transmission rate is n = 1 packet/slot. Any packetwill be considered lost due to collision if its transmission timeoverlaps with that of another.

A. Heavy Traffic Scenario

In this subsection, simulation results for heavy traffic are pre-sented and compared with those of theoretical analysis.

Figure 4 shows the average collision probabilities of both theprimary and the CR users. The calculation is from the simula-tion as the ratio of the collided packets to the total transmittedpackets. It is observed that both of them incur higher packetloss due to collision as the frame duration increases. When theframe duration is larger, the CR user senses the channel less of-ten. Since the primary user has the priority in using the channeland its traffic load is fixed, it happens with a higher probabilitythat he may turn on during the CR user’s transmission. Thatis why we observe the almost linearly increased probability ofcollision as the frame duration becomes longer.

Figure 5 compares the simulated average probability of col-lision of CR user and that of the analytic results calculated from(8). It is observed that the two curves exhibit the same trend.However, the theoretical results is slightly higher than the sim-ulated one. The small discrepancy can be explained by the fol-lowing two reasons. One reason is that in deriving (8), we as-sume that the primary user is active until the end of currentframe ones he turns on. However, in reality, this is not true. Aswe model the on-period of the primary user as a random num-ber, there is probability that he may turn off after it turns on for

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

0 20 40 60 80 100 120 140 160 1800

0.05

0.1

0.15

0.2

0.25

Frame duration T

Pro

babi

lity

of c

ollis

ion

Pc

TheoreticalSimulated

Figure 5: Simulated and theoretical Pc of CR user.

0 20 40 60 80 100 120 140 160 1800.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Frame duration T

Nor

mal

ized

Ach

ieva

ble

Thr

ough

put

ConditionalAbsolute

Figure 6: Simulated achievable throughput.

a very short period of time. Hence, it is desirable that the theo-retical result is higher than the simulated one. Another reasonis that we use continuous distribution for analytical derivationbut simulation is carried out in a discrete matter. This maycause the difference due to some round-off effect. Neverthe-less, the theoretical derivation can still be considered as a goodapproximation of the transmission scenario.

We make use of the collision probability obtained and plotthe normalized achievable throughput following (3) as shownby the upper curve in Figure 6. Noted that this curve in factshows only the conditional throughput as pointed out in Sec-tion II.. In the simulation, we also find the absolute normal-ized throughput. This can be obtained by directly countingthe collision-free packets over the entire simulation time andit is plotted as the lower curve in Figure 6. We can actuallyconvert this value in to conditional value. Since the sensingslot duration is very short (1 msec), the probability that pri-mary user is not present during sensing can be approximatedby β0/(β0 + β1) = 0.6487.

0 20 40 60 80 100 120 140 160 1800.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

Frame duration T

Nor

mal

ized

Ach

ieva

ble

Thr

ough

put

Simulated 1Simulated 2

Figure 7: Comparison of two simulated achievable throughput.

0 20 40 60 80 100 120 140 160 1800.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

Frame duration T

Nor

mal

ized

Ach

ieva

ble

Thr

ough

putSimulatedTheoretical

Figure 8: Comparison of simulated and theoretical results.

The plots of the two simulated normalized throughput whenscaled to conditional value are shown in Figure 7. As can beseen, this two curves is actually very close to each other. Wewill use the original conditional curve as the simulation for thefollowing comparison.

Figure 8 shows the comparison of simulated and theoreticalconditional normalized throughputs. It is observed that the twocurves exhibit the same trend although the deviation betweenthem becomes larger as frame duration increases. The differ-ence can be explained using the same arguments as for theprobability of collision. We can see that the simulated curvehas a maximum value at frame duration T around 40 msec,while the calculated value (based on (15)) is 36.7. It revealsthat there does exist such an optimal frame duration such thatthe system throughput is maximized.

Furthermore, to ensure that the primary user is sufficientlyprotected, the collision probability should be kept below cer-tain threshold value. The exponential on-off traffic can be con-sidered as what a VoIP user generates. The performance of

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

0 20 40 60 80 100 120 140 160 1800.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

Frame duration T

Nor

mal

ized

Ach

ieva

ble

Thr

ough

put

Figure 9: Achievable throughput for λ = 0.4.

this kind of traffic is considered acceptable within 10 percentof collision. At the optimal frame duration, it is observed fromFigure 4 that the collision is about 8 percentage which is withinthe target collision probability. If in any case, the frame dura-tion at the target collision probability is less than optimal frameduration seen by the CR user, the CR user will not be able toachieve its maximum throughput. Instead, a small frame has tobe chosen to ensure the sufficient protection to primary user.

B. Normal Traffic Load Scenario

The CR user’s slot duration is 1 msec and its transmission rateis n = 1 packet/slot. Since we can approximate the percentageof time for CR user’s transmission as β1/(β0 + β1) = 0.6487,the effective transmission rate is thus approximately 0.6487packet/slot. To be considered as a low-traffic scenario, we setλ = 0.4 and λ = 0.6 for simulation.

Figures 9 and 10 show the simulated achievable throughputfor the two settings. It should be pointed out that the results arenot conditional values and they are the absolute throughput.In both figures, the tradeoff phenomenon can be observed andthe optimal frame duration is observed shorter than the heavytraffic scenario. This can be explained as follows. In the heavytraffic case when the CR user always has packet to transmit, theincrease in the number of packet transmitted (either collided ornot) will be linearly proportional to the increase in the frameduration. However, the relationship may not be linear in thelow traffic scenario as the CR user does not always have packetto transmit. Besides, the increase in collision probability for thetwo case will be the same since this only depends on the trafficof the primary user. At a certain frame duration, the CR useris less incentive to increase his frame duration. Therefore, theoptimal frame duration is shorter in the low traffic scenario andthe lighter the traffic, the smaller the optimal frame duration.

VI. CONCLUSIONS

In this paper, we studied the design of transmission frame du-ration to achieve the maximum throughput for the CR network,

0 20 40 60 80 100 120 140 160 180

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Frame duration T

Nor

mal

ized

Ach

ieva

ble

Thr

ough

put

Figure 10: Achievable throughput for λ = 0.6.

yet to ensure that the collision probability for primary users isless than a threshold. Theoretical formulae for the achievablethroughput and optimal frame duration are derived. We provedthat for a given properly designed sensing slot duration, thereindeed exists an optimal frame duration to obtain the best trade-off. Simulation results validated this theoretical derivation.

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