available and waiting times for cognitive radios

Wireless Pers Commun (2012) 65:319–334DOI 10.1007/s11277-011-0450-0

Available and Waiting Times for Cognitive Radios

Seyed M. Hosseini · Mahdi Teimouri ·Saralees Nadarajah

Published online: 9 November 2011© Springer Science+Business Media, LLC. 2011

Abstract Cognitive radios (CRs) have been recently proposed for the problem of spec-trum scarcity. The principle of CRs’ operation is based on the opportunistic access to thefrequency spectrum mainly dedicated to primary users (PUs). The statistical time pattern ofPUs’ channel usage and arrival can affect the usability of specific frequency bands for CRs.In this note, the effect of the arrival rate and channel holding time of PUs on the availabletimes for CRs is analyzed. To this end, first, based on Poissonian arrivals, the available timefor CRs is calculated. Then, assuming a gamma distribution for the inter-arrival times and auniform distribution of channel holding time of PU in these intervals, the probability densityfunction and moments of the available time for CRs are derived. Next, the effect of PUsstatistical parameters on the average number of packets and the average symbol rate thata CR can transmit is analyzed. Also, taking that CR needs at least T seconds, the averagewaiting time is calculated.

Keywords Available times · Cognitive radios · Primary users · Waiting times

1 Introduction

The idea of cognitive radios (CRs) was first defined by Joseph Mitola [1] to represent anintelligent wireless mobile radio unit capable of adapting its environment towards its userapplications. Based on the requirements for spectrum frequency bands, later works changedthe definition to a wireless unit that can benefit either from the dedicated frequency bands

S. M. HosseiniDepartment of Engineering, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran

M. TeimouriDepartment of Statistics, Gonbad Kavous University, Gonbad Kavous, Iran

S. Nadarajah (B)School of Mathematics, University of Manchester, Manchester M13 9PL, UKe-mail: [email protected]

123

320 S. M. Hosseini et al.

when they are not used by their primary users (PUs) or from coexistence with PUs providedthat their interferences for PUs do not exceed a predefined threshold called the maximumInterference Temperature (IT) [2]. For more information on CRs, readers are referred to[3–9]. Statistical analysis of users’ behavior within a communication system has long beenan interest for telecommunication companies, since it can provide them with useful infor-mation for planning, design and utilization. In this respect, for example, in [10–19], readerscan find useful information on arrivals, channel holding time, and inter-arrival probabilitydistributions. In [20–24], authors have analyzed the spectrum utilization of cellular com-munications systems. However, none of these papers provide an analytical treatment of theeffect of PUs parameters on the available time intervals for CRs.

In this note, we analyze the effect of statistical characteristics of PUs on the availabletimes for CRs to transmit. To this end, first, we take the arrivals of PUs as a Poisson processand derive the available empty times between arrivals. Then we put aside the Poissonianarrival and assume, according to [10,16,25,26], that the inter-arrival times are independentgamma random variables. Since we have no knowledge of the statistical distribution of thechannel usage by PUs, we assume that it is a uniform random variable within the intervalsconditioned on the distribution of inter-arrivals being gamma. Based on this, we derive theprobability density function of the available time within an interval. Then, we analyze theeffect of PUs statistical parameters on the average number of packets that a CR can transmit,the average symbol rate and the time that CR must wait until it finds an opportunity to obtainan empty time of length T .

The contents of this note are organized as follows. In Sect. 2, assuming a Poissonian dis-tribution for arrivals, the empty times between PUs’ arrivals are calculated. Then, in Sect. 3,the probability density function of empty times is derived based on a uniform distribution ofchannel holding time within gamma intervals. Section 4 shows the effect of PUs statisticalparameters on the average number of packets, average symbol rate and the average waitingtime. Section 5 shows the results of computer simulation for some scenarios. Conclusionsare mentioned in Sect. 6. Some technical lemmas needed are mentioned in Appendix A.

2 Empty Time Intervals Between Poisson Arrivals

Within a communications system, users enter, use the communication channel for transmittingtheir data and then leave. One common candidate for modeling the arrivals in communica-tion systems is Poisson [27,28]. The probability mass function of a Poisson random variablewithin a time interval t is:

Pr(X = k) = exp(−λt)(λt)k

k!for k = 0, 1, 2, . . ..

The channel holding time, the time that PU uses its channel, is stochastic. Let it be arandom variable with mean μ. Theorem 2.1 expresses the empty time intervals in term of λ

and μ.

Theorem 2.1 For Poisson arrivals with rate λ, assume that the channel usage by PUs is arandom variable with mean μ. Then the mean of empty time intervals, Taverage, within a timeperiod of length TA is equal to TA(1 − λμ).

Proof Each time PU arrives, it uses the channel for μ seconds on average and leaves. So, fork arrivals, the time usage is kμ and the empty time interval is TA − kμ. So, we have:

123

Available and Waiting Times 321

Taverage =∞∑

k=0

(TA − kμ) Pr(X = k) =∞∑

k=0

(TA − kμ)exp (−λTA) (λTA)k

k! = TA(1 − kμ).

The proof is complete. ��

So, the average time available to CR is equal to:

ρ = Taverage

TA= 1 − λμ.

This implies that CRs must basically hope for operations, where μλ (the product of meanchannel usage and mean arrival rate) is smaller than unity.

3 Probability Density Function of Empty Time Intervals Between PU’s Arrivals

Figure 1 shows the relationship between PU and CR within a communication channel. In thisfigure, TPU, TCR and TC are, respectively, the time when PU uses the channel, time when CRuses the channel after PU has already left and the time between PU’s arrivals.

Since there is no knowledge of the distribution of TPU, we assume that it is uniformlydistributed in the interval (0, TC ). So, TCR is also uniformly distributed in the interval (0, TC ).The distribution of TCR is stated by Theorem 3.1.

Theorem 3.1 Suppose TC R is a uniform random variable in the interval (0, TC ), where TC

itself is a gamma random variable with parameters λ and n. Then the moments, moment gen-erating function, Laplace transform and the probability density function of TCR are given by:

E[T r

CR

] = �(n + r)

λr�(n)(r + 1),

MTCR (t) = −λ + (−λ)nλ(−λ + t)1−n

t (n − 1),

L (fTCR (t)

) = λ + (−λ)nλ(−λ − t)1−n

t (n − 1),

TCR

TPU

TC

Fig. 1 Relationship between PU and CR for Poisson arrivals: TPU and TCR are the respective times when PUand CR use the channel with inter-arrivals Tc

123


and

fTCR (x) = λ�(n − 1, λx)

�(n),

respectively, for r > −n and 0 < x < ∞, where �(·, ·) denotes the incomplete gammafunction defined by [29]

�(a, x) =∞∫

x

ya−1 exp(−y)dy.

Proof The moments of TCR can be calculated using conditional expectation:

E[T r

CR

] = ETC

[ETCR|TC

[T r

CR | TC]]

=∞∫

0

tC∫

0

trCR

tCdtCR

λntn−1C exp (−λtC )

�(n)dtC

= λn

�(n)(r + 1)

∞∫

0

tn+r−1C exp (−λtC ) dtC

= 1

λr�(n)(r + 1)

∞∫

0

un+r−1 exp (−u) du

= �(n + r)

λr�(n)(r + 1).

So, the moment generating function is:

MTCR (t) =∞∑

r=0

�(n + r)tr

λr�(n)(r + 1)r !

=∞∑

r=0

(n + r − 1)!tr

λr (n − 1)!(r + 1)r !

=∞∑

r=0

1

r + 1

(n + r − 1

r

) (t

λ

)r

=∞∑

r=0

(−1)r

r + 1

(−n

r

)(t

λ

)r

=∞∑

r=0

1

r + 1

(−n

r

) (− t

λ

)r

= 1

t

t∫

0

(1 − y

λ

)−ndy

= −λ + (−λ)nλ(−λ + t)1−n

t (n − 1),

123


where we have applied Lemmas A.1, A.2 and A.3. So, the Laplace transform is:

L (fTCR (t)

) = λ + (−λ)nλ(−λ − t)1−n

t (n − 1).

Finally, using the inverse Laplace transform, we obtain the probability density function ofTCR as:

fTCR (t) = L−1(

λ + (−λ)nλ(−λ − t)1−n

t (n − 1)

)

= λ

n − 1− λn

�(n)

t∫

0

yn−2 exp(−λy)dy

= λ

n − 1− λn

�(n)[�(n − 1) − �(n − 1, λt)]

= λ

�(n)�(n − 1, λt).

The proof is complete. ��

Figure 2 shows possible shapes of the probability density function of TCR for selectedvalues of λ and n.

4 Average Number of Packets and Waiting Time

4.1 Average Number of Packets

In digital communications, data are defined and sent in terms of frames and packets. So,when one packet is transmitted safely, it does not need to be retransmitted even if the rest ofthe data are not communicated safely. In this section, the effect of PU’s parameters on theaverage number of packets, the average symbol rate that a CR can transmit and the time thatCR must wait is analyzed.

When PU leaves the channel, TCR starts and so CR starts to transmit. If TCR > T thenCR can transmit at least one packet without causing any interference for PUs. Accordingly,if 2T < TCR < 3T then CR can transmit two packets. If N is the number of packets that CRcan transmit then we have:

Pr(N = k) =(k+1)T∫

kT

fTCR (t)dt

= λ

�(n)

(k+1)T∫

kT

∞∫

λt

yn−2 exp(−y)dydt

= λ

�(n)

⎡

⎢⎣∞∫

kT

θ/λ∫

kT

yn−2 exp(−y)dtdy +∞∫

(k+1)T

(k+1)T∫

kT

yn−2 exp(−y)dtdy

⎤

⎥⎦

123


a b

dc

Fig. 2 a Probability density function of TCR for n = 1. b Probability density function of TCR for n = 2. cProbability density function of TCR for n = 5. d Probability density function of TCR for n = 10

123


Fig. 3 The shaded area is thedomain of integration

= λ

�(n)

[ (k+1)λT∫

kλT

θ/λ∫

kT


+∞∫

(k+1)λT

(k+1)T∫

kT


]

= λ

�(n)

[1

λ

(k+1)λT∫

kλT

yn−1 exp(−y)dy − kT

(k+1)λT∫

kλT

yn−2 exp(−y)dy

+T

∞∫

(k+1)λT

yn−2 exp(−y)dy

]

= λ

�(n)

[1

λ

∞∫

kλT

yn−1 exp(−y)dy − 1

λ

∞∫

(k+1)λT

yn−1 exp(−y)dy

− kT

∞∫

kλT

yn−2 exp(−y)dy + kT

∞∫

(k+1)λT

yn−2 exp(−y)dy

+ T

∞∫

(k+1)λT

yn−2 exp(−y)dy

]

= 1

�(n)[� (n, λkT ) − � (n, λ(k + 1)T )]

+ λT

�(n)[(k + 1)� (n − 1, λ(k + 1)T ) − k� (n − 1, λkT )] .

Note that we have changed the order of integration and broken the integral in two parts, asshown in Fig. 3.

So, the average number of packets is:

Naverage =∞∑

k=0

k Pr(N = k)

123


= 1

�(n)

∞∑

k=0

k [� (n, λkT ) − � (n, λ(k + 1)T )]

+ λT

�(n)

∞∑

k=0

k [(k + 1)� (n − 1, λ(k + 1)T ) − k� (n − 1, λkT )] .

For n = 1, 2, 3, we have the particular cases:

Naverage = exp(−λT ) − exp(−2λT )

{1 − exp(−λT )}2 − λT Ei(λT ),

Naverage = exp(−λT ) − exp(−2λT )

{1 − exp(−λT )}2 ,

and

Naverage = exp(3λT )

2 {−2 − 2λT + 2 exp(λT ) + λT exp(λT )} {exp(λT ) − 1}3

{4λT − 12 exp(−λT )

+ 4 − 12λT exp(−λT ) + 12 exp(−2λT ) + λ2T 2 − 3λ2T 2 exp(−λT )

+ 12λT exp(−2λT ) − 4 exp(−3λT ) + 2λ2T 2 exp(−2λT ) − 4λT exp(−3λT )},

where Ei(·) denotes the exponential integral defined by [30]

Ei(x) =∞∫

x

exp(−y)

ydy.

Set T = TS , the duration of one symbol. Then, (1) gives the average number of symbols NS

that CR is able to transmit in each interval. Since the mean of each gamma interval is n/λ,the average symbol rate RS is:

RS = NS

n/λ

= λ

n�(n)

∞∑

k=0

k [� (n, λkTS) − � (n, λ(k + 1)TS)]

+ λ2TS

n�(n)

∞∑

k=0

k [(k + 1)� (n − 1, λ(k + 1)TS) − k� (n − 1, λkTS)] .

= λ

�(n + 1)

∞∑

k=0

k [� (n, λkTS) − � (n, λ(k + 1)TS)]

+ λ2TS

�(n + 1)

∞∑

k=0

k [(k + 1)� (n − 1, λ(k + 1)TS) − k� (n − 1, λkTS)] .

Figures 4, 5 illustrate the variation of Naverage and RS versus λ and T , respectively, forn = 1, 2, 3, 4. It can be seen that when either λ or T increases, Naverage decreases almostexponentially. However, within RS , the parameter T (or TS in this case) plays the moreimportant role; that is, sensitivity of RS with respect to T is much higher than that of λ. It isworth nothing here that, for n = 1, the parameter λ is exactly the mean arrival rate, since thegamma distribution reduces to the exponential distribution. Accordingly, for n not equal tounity, the λ can be thought of as playing the role of arrival rate.

123


ba

c d

Fig. 4 a RS for n = 1. b RS for n = 2. c RS for n = 3. d RS for n = 4

4.2 Average Waiting Time

When CR wants to send its data, it must be sure that its packets do not interfere with PUtransmissions. If PU starts to transmit, CR must stop and wait until PU leaves the channel.In this case, it is important for CR to know how much time it must wait until it finds anopportunity for transmitting its data. Theorem 4.1 derives the average waiting time.

Theorem 4.1 If CR uses packets of length T then the waiting time for CR to find a timeinterval of length T or greater is:

W T = T λa + n − na

λa,

where

a = �(n, λT ) − λT �(n − 1, λT )

�(n).

123


a

c

b

d

Fig. 5 a Naverage for n = 1. b Naverage for n = 2. c Naverage for n = 3. d Naverage for n = 4

Proof When PU leaves the channel, if tCR > T then CR has already found the time it needs.Otherwise, it must wait until the next leaving of PU. So,

W T = T Pr (tCR > T ) + τ1 Pr (tCR < T ) = T a + τ1(1 − a),

where τ1 is a time greater than T . In the next interval, when CR has already waited as long asn/λ (the mean time between PU arrivals), again, if tCR > T then CR would find T , otherwiseit must wait for the next intervals. So,

τ1 =(n

λ+ T

)a + τ2(1 − a).

Repeating this process, we have

W T = T a + τ1(1 − a)

= T a +((n

λ+ T

)a + τ2(1 − a)

)(1 − a)

= T a +(n

λ+ T

)a(1 − a) + τ2(1 − a)2

= T a +(n

λ+ T

)a(1 − a) +

((2n

λ+ T

)a + τ3(1 − a)

)(1 − a)2

123


a b

dc

Fig. 6 a W T for n = 1. b W T for n = 2. c W T for n = 3. d W T for n = 4

= T a +(n

λ+ T

)a(1 − a) +

(2n

λ+ T

)a(1 − a) + τ3(1 − a)3

= · · ·= a

∞∑

k=0

(kn

λ+ T

)(1 − a)k

= T λa + n − na

λa.

The proof is complete. ��Figure 6 shows how W T varies with respect to T and λ. It can be seen that W T increases

exponentially as T increases. This dictates using packets as small as possible. Also, when λ

grows up, the probability of finding any opportunity to send data in the empty times decreases.This implies that CR must hope for channels, where their corresponding PU arrives rarely. Onthe other hand, if CR still tries the channels with large λ, its interference on the correspondingPU may violate the spectrum legacy mainly dedicated to PU. To this not to occur, CR mighthave to use packets of very small length. In this case, it might be better off not to transmitsince the corresponding packets may not be capable of carrying any data.

5 Computer Simulation

In order to justify the validity of analytical outcomes, in this section, results of computersimulation for a typical communication system will be shown. For practical reasons, we

123


−30 −20 −10 0 10 20 30

101

SNR (dB)

Wai

tng

Tim

e (s

ec)

T=1, Lambda=0.1T=1, Lambda=1T=2, Lambda=0.1T=2, Lambda=1T=3, Lambda=0.1T=3, Lambda=1

−30 −20 −10 0 10 20 30

101

SNR (dB)

Wai

ting

Tim

e (s

ec)

T=1,Lambda=0.1T=1,Lambda=1T=2,Lambda=0.1T=2,Lambda=1T=3,Lambda=0.1T=3,Lambda=1

a

b

Fig. 7 a W T for n = 2. b W T for n = 4

should consider such fundamental limitations of a communication channel as bandwidthand noise. Since the duration of symbol is inversely related to the communication band-width, this parameter will be lower bounded within our simulation. On the other hand, as itis expected, noise would affect the results adversely. In this respect, it is expected that theaverage throughput either in terms of average number of packets and symbols or symbolrate be decreased while the waiting time should be increased. The scenario of simulation isas follows. Within each empty time interval, CR, first, modulates its signals using a binaryPulse Amplitude Modulation (PAM) scheme. Then the modulated signals travel through anAdditive White Gaussian Noise (AWGN) channel. Within the receiver, symbols are detectedusing a match filter detector. To show the band-limited nature of channels, we have lowerbounded the symbol duration by seconds.

Figures 7, 8, 9 and 10 show the results of simulation for average waiting time, averagenumber of packets in each empty time interval and average packet and symbol rate versusSignal-to-Noise Ratio (SNR). It should be noted here that since the length of symbols isusually fixed due to the band-limited nature of channels, RS must be interpreted as packetrate. The following observations can be made from the figures:

• As SNR grows up, more desirable results are obtained in terms of higher average numberof packets, higher symbol and packet rate and lower average waiting time.

• For larger n, since the average empty time is larger, all the results become better thanthose for smaller n.

• The waiting time is more sensitive to packet length than λ, especially for large λ, whereit increases almost exponentially as the length of packets grows up.

123


−30 −20 −10 0 10 20 300

20

40

60

80

100

SNR (dB)

Ave

rage

Num

ber

of P

acke

ts

T=0.1,Lambda=0.1

T=0.1, Lambda=1T=1, Lambda=1 T=1, Lambda=0.1

−30 −20 −10 0 10 20 300

50

100

150

200

SNR (dB)

Ave

rage

Num

ber

of P

acke

ts

T=0.1, Lambda=0.1

T=1, Lambda=0.1

T=0.1, Lambda=1T=1, Lambda=1

a

b

Fig. 8 a Naverage for n = 2. b Naverage for n = 4

• Average number of packets within each empty time interval is sensitive to the multipli-cation of λ and packet length, rather than to each of them individually. In this respect, thelarger the multiplication is the larger the average number of packets would be.

• Average symbol rate is almost the same for the given range of parameters. This is dueto the fact that as λ increases, the number of PUs arrivals per second increases and thenumber of empty times increases too. In other words, the average empty time wouldremain the same.

• Average packet rate is very sensitive to the length of packets but has minor sensitivity toλ.

6 Conclusion

In this note, the effect of PUs’ statistical characteristics including arrival rate and channelusage on the available time resources as well as waiting time of CRs was analyzed. Theresults of analysis as well as computer simulation suggest that when CR uses longer packets,its average waiting time increases almost exponentially. Since the scenario analyzed here wasfor a uniform random preference between PU and CR, it can be seen that symbol rate is veryinsensitive to variation of either CRs packet length or PUs arrival rate. On the other hand, theaverage number of packets a CR can transmit within each empty time interval is mostly deter-mined by the multiplication of PUs arrival rate and CRs packet length, where any increasein either of the parameters results in a corresponding decrease in average number of packets.

123


−30 −20 −10 0 10 20 300

1

2

3

4

5

SNR (dB)

Ave

rage

Pac

ket R

ate

T=0.1, Lambda=0.1

T=1, Lambda=0.1T=1, Lambda=1

T=0.1, Lambda=1

−30 −20 −10 0 10 20 300

1

2

3

4

5

6

SNR (dB)

Ave

rage

Pac

ket R

ate

T=0.1, Lambda=0.1

T=0.1, Lambda=1

T=1, Lambda=1T=1, Lambda=0.1

a

b

Fig. 9 a RS for n = 2. b RS for n = 4

−30 −20 −10 0 10 20 302

2.5

3

3.5

4

4.5

5

5.5 x 106

SNR (dB)

Ave

rage

Sym

bol R

ate

T=0.1, Lambda=0.1T=0.1, Lambda=1T=1, Lambda=0.1T=1, Lambda=1

−30 −20 −10 0 10 20 302.5

3

3.5

4

4.5

5

5.5 x 106

SNR (dB)

Ave

rage

Sym

bol R

ate

T=0.1, Lambda=0.1T=0.1, Lambda=1T=1, Lambda=0.1T=1, Lambda=1

a

b

Fig. 10 a Average symbol rate for n = 2. b Average symbol rate for n = 4

123


The real scenario might be more complicated for CR. In one hand, CR might prefer shorterpackets than longer ones to make the results better, but in this case, since shorter packetscarry less data, it makes the situation worse. On the other hand, in a real environment, CRmust sense the spectrum in order to detect the spectrum holes, something which complicatesthe situation especially when PUs arrival rate is large. In this case, the interference of CR forPU, resulted from misdetection of spectrum holes, might render the scenario as unacceptable.Since statistical parameters of PU might not be fixed all the time, it seems that the analysisof real environments might be better done through real time optimization methods.

Acknowledgments The authors would like to thank the Editor and the referee for careful reading and fortheir comments which greatly improved the paper.

Appendix A

Lemma A.1 The Laplace Transform of a function, say f (·), is defined by [31]:

L ( f (x)) =∞∫

−∞f (x) exp(−sx)dx .

If X is a random variable then its moment generating function is defined by:

M (t) =∞∫

−∞f (x) exp(t x)dx,

where f (·) denotes the probability density function of X .

Lemma A.2 For integers n and k, we have [30]:(

n + k − 1

k

)= (−1)k

(−n

k

).

Lemma A.3 For real a, b and integer n, we have [30]:

(a − b)−n =∞∑

r=0

(−n

r

)a−n−r (−b)r .

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Author Biography

Saralees Nadarajah is a Senior Lecturer in the School of Mathematics, University of Manchester, UK. Hisresearch interests include climate modeling, extreme value theory, distribution theory, information theory,sampling and experimental designs, and reliability. He is an author/co-author of four books, and has over 300papers published or accepted. He has held positions in Florida, California, and Nebraska.

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