Cognitive Radio Networks with the RESTART Retransmission Strategy and Limited Reconnections

S. Lirio Castellanos-Lopez, Felipe A. Cruz-Pérez

Elect. Eng. Dept., CINVESTAV-IPN Mexico City, Mexico

lirio@cinvestav.mx, facruz@cinvestav.mx

Genaro Hernandez-Valdez Electronics Department, UAM-A

Mexico City, Mexico ghv@correo.azc.uam.mx

Abstract— In cognitive radio networks (CRNs) based on

spectrum overlay sharing, the connection of a secondary user fails when a primary user arrives and there is not idle channels. Retransmissions of interrupted secondary sessions may lead the system towards an unstable condition. Therefore, it is important to develop mechanisms to alleviate performance degradation due to reconnected secondary sessions. To the best of the authors’ knowledge, mechanisms based on limiting the number of reconnections for reducing failure rate have been neither analyzed nor evaluated in the literature. In this paper, limiting the maximum number of reconnections in CRNs with RESTART retransmission strategy for delay tolerant traffic is evaluated as a mechanism to reduce the negative impact of retransmissions and to widen the conditions under which system stability can be achieved. Numerical results show that limiting reconnections is an effective mechanism to alleviate performance degradation due to connection failure. Also a tradeoff between the call completion probability and the supported offered traffic load that can be controlled by varying either the number of reserved channels or the number of allowed retransmission is observed.

Keywords— Cognitive radio networks, RESTART retransmission, teletraffic and performance analysis, delay tolerant traffic, completion probability, successful carried traffic.

I. INTRODUCTION Cognitive radio networks (CRNs) have been proposed to

address the issue of spectrum efficiency by enabling opportunistic access to the licensed spectrum band by secondary users (SUs). This paper focuses on overlay dynamic spectrum sharing approach among infrastructure-based CRNs. That is, when an idle primary channel is detected, SUs may temporally occupy this unused channel. If a PU decides to access the primary channel, all SUs using this channel must relinquish their transmission immediately. These unfinished secondary calls may be either simply blocked [1]-[2] or directed to other available cannel (this process is called spectrum handoff). Moreover, for delay tolerant services, to reduce the impact of service interruption, if no vacant channels are available, interrupted secondary sessions may be queued in a buffer to wait for the releasing of an occupied channel [3]-[4]. In this case, when a queued SU finds a new available channel, it is allowed to continue transmitting its information. Two simple retransmission strategies to handle with queued SUs in CRNs have been commonly employed [3]-[4], [13]-[15]: RESTART and Resume. In the RESTART strategy, when an ongoing secondary session is interrupted, all the information transmitted (i.e., packets, frames, or other data units) until the interruption point is lost and the SU must start over again its

session when allowed to occupy an available primary channel [5], [8]-[9], [13]-[15]. On the other hand, in the Resume (RR) strategy, when an ongoing secondary session is interrupted, it knows where it exactly stops and can resume at that point when it leaves the queue to be reconnected to the system [3]-[4], [8], [13]-[15].

In relation to the RESTART strategy, authors in [8]-[9] showed that, if the distribution of the unencumbered service time (i.e., total time a task spends executing without failures) has an exponential tail, the total time to complete the job (total time a task spends executing because of the failures, but not including the time it spends waiting) is power tailed. In this research direction, authors in [10]-[11] showed that, under the assumption that the packet size distribution has infinite support; all retransmission-based protocols could cause heavy-tailed behavior. These heavy tails can result entirely from retransmissions, even when the service time and channel characteristics are light-tailed. This in turn can yield unstable system operation (i.e., zero throughput). The conditions under which unstable system operation takes place closely depend on the connection failure process. To the best of our knowledge, mechanisms used to reduce failure rate have been neither analyzed nor evaluated in the literature1. Exception of this is the channel reservation strategy proposed in [15] to prioritize ongoing over new secondary-session requests in CRNs.

In the present paper, limiting the maximum number of reconnections in CRNs with delay tolerant traffic and the RESTART strategy is proposed as a mechanism to reduce the negative impact of retransmissions. Our numerical results show that limiting retransmissions is an effective mechanism (that performs as well as the channel reservation approach proposed in [15]) to reduce the negative impact of retransmissions and to widen the conditions under which system stability can be achieved. In a related work [13]-[14], we compare the performance between the RESTART and RR strategies; however, no mechanism to reduce the negative impact of retransmission is employed. Even more, in [13]-[14], the system performance under the RESTART strategy was not mathematically analyzed, its performance was evaluated by simulation. The methodology developed in [15] to mathematically analyze CRNs under RESTART strategy is adopted here.

1 In [12], power control at the physical layer is mentioned as a way to change the relationship between the channel dynamics and the units in which packets should be transmitted in order to achieve the best network performance. However, its performance it is neither analyzed nor evaluated.

978-1-4673-6187-3/13/$31.00 ©2013 IEEE

II. SYSTEM MODEL The CRN model used in [3]-[4] and [15] is adopted here. It

is assumed that there are M channels. When an ongoing SU detects or is informed (by its base station –BS- or other SUs) of an arrival of PT call in its current channel, it immediately stops transmitting data, releases the channel, and switches to an idle channel, if one is available, to continue its transmission. If at that time all the channels are occupied, the secondary traffic (ST) call is interrupted and placed into a buffer located at its BS. The queued ST calls are served in first-come first-served order. We propose a mechanism to alleviate performance degradation due to interruptions of secondary sessions. This mechanism consists in limit the number of reconnections up to a maximum number of R times. That is, secondary sessions that are interrupted after their R-th reconnection are forced to terminate. In this paper, it is assumed that ST calls can wait indefinitely to be reconnected. Clearly, the maximum number of queued ST calls is M, which corresponds to the limiting case that all the M ongoing calls are ST calls and are eventually preempted to the queue due to the arrivals of primary traffic (PT) calls. Thus, a finite queue of length M is considered.

Arrivals of the PT and ST calls are assumed to form independent Poisson processes with rates λ(P) and λ(S), respectively. The random variable (RV) used to represent the PUs’ inter-arrival time is Xa. Service time for PUs is considered exponentially distributed with rate μ(P). The RV used to represent this time variable is XS

(P). The corresponding unencumbered service time (UST) (that is, the time a SU uses a channel without interruptions) for SUs is modeled either as a negative exponential and 2nd order Coxian distributed RV. The RV used to represent this time is XS

(S). Fig. 1 shows a diagram of phases of a n-th order Coxian distribution. Notice that βi (for i = 1, 2, …, n-1) represents the probability that the absorbing state is reached after the i-th phase. For a 2nd order Coxian distribution, β2=1 and β1= β. The i-th phase of this distribution is an independent exponential RV with parameter μi

(S) for (i = 1, 2). The RVs used to represent these times are Xi

(S) (for i = 1, 2). The mean of the UST for SUs is denoted by 1/μs

(S). Notice that ( )( )( ) ( ) ( ) ( )

1 1 21 1 1 1 .S S S Ssμ = β μ + −β μ + μ

( )1

Sμ ( )2Sμ

1β( )Snμ( )

1S

n−μ2β 1n−β

Fig. 1. Diagram of phases of a n-th order Coxian distributed service time.

0

1

2 1

0 1 3

0 2 3

0 3

0

1

2

3

k + ek + ek + e - ek + e - e + ek + e - e + ek + e - ek - ek - ek - ek - e

k

2 1

0 1 3

0 2 3

0 3

2

3

++++

++

k - e ek - e e - ek - e e - ek - e ek ek e

Fig. 2. Transition diagrams for the Markov chain of the CRN.

III. TELETRAFFIC ANALYSIS OF THE RESTART STRATEGY WITH A SINGLE RECONNECTION

In this section, an approximate teletraffic analysis for the performance evaluation of CRNs with the RESTART strategy and a single reconnection allowed is developed. To this end,

the RESTART strategy is analyzed using the queuing model of the RR strategy considering that the service time of preempted SUs can be approximated by an exponential distribution but with different mean value respect to that of the UST of SUs. The mean service time of SUs that have been interrupted once is denoted by

1( )1 .Ssμ As in [15], to reduce the interruption of

ST calls upon the arrival of PT calls, a number r (<M) of channels is precluded to be used by ST calls. Contrary to [15], in this paper we consider a real number of reserved channels. As such, when the total number of users (primary users plus secondary users) in the system equals M r− ⎢ ⎥⎣ ⎦ , a new

secondary session is accepted with probability rp r r= − ⎢ ⎥⎣ ⎦ .

First, let us consider the RR strategy. In the RR strategy, when a queued SU is reconnected to the system, it transmits its information starting at the point it was preempted. As the permanence time in the different phases of service time is exponentially distributed, it is sufficient to know the phase where the interruption occurred because of the Markovian property. Thus, it is possible to keep track in a single state variable the number of users in each phase of the service time for both ongoing and queued SUs. Then, a four-dimensional birth and death process is required for modeling this system. Each state variable is denoted by ki (for i = 0, 1, 2, 3). k0 represents the number of ongoing PUs, 1k the number of ongoing SUs in phase 1, 2k the number of ongoing and queued SUs in phase 2, and 3k the number of SUs in the queue that have been interrupted once in phase 1. To simplify mathematical notation the following vectors are defined: k=(k0, k1, k2, k3), and ei is a unit vector of four elements, whose all entries are 0 except the i-th position which is 1 (for i = 0,…,3). Considered the state k as the reference one, a detailed description of all the possible transition rates from and toward state k are represented in Fig. 2 and given below.

The call birth rate for PUs or SUs generating the transitions i i⇒ ⇒k - e k k + e is given by

( )( )

3( )

00

3( )

10

3( )

10

; ;0 ; 0

; ; 0; 1

1 ; ; 0; 1

0 ;otherwise

Pj

j

Sj

ji

Sr j

j

k M k M i

k M r k ia

p k M r k i

=

=

=

⎧λ < ≤ < =⎪⎪⎪λ < − ≥ =⎡ ⎤⎪ ⎢ ⎥= ⎨⎪

− λ = − ≥ =⎡ ⎤⎢ ⎥⎪⎪⎪⎩

∑

∑

∑k

The call death rate for PUs or SUs (in phase 1, 2, and those that are retransmitting for first time) generating the transitions

i i⇒ ⇒k + e k k - e is given by

( )

1

2

1

2( )

0 00

3 2( )

11 0

3 2( )

21 0

3 3 2( )

30 1 0

; ; ; 0

; ; ; 1

; ; ; 2

max 0, ; ; ; 3

0 ;otherwise

Ps j

j

Sj j

j j

Sij j

j j

Sj s j j

j j j

k k M k M i

k k M k M i

b k k M k M i

k k M k M k M i

=

= =

= =

= = =

⎧ μ ≤ ≤ =⎪⎪⎪β μ ≤ ≤ =⎪⎪⎪= ⎨ μ ≤ ≤ =⎪⎪⎪⎛ ⎞⎛ ⎞

− − μ ≤ ≤ =⎜ ⎟⎪ ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎝ ⎠⎪⎩

∑

∑ ∑

∑ ∑

∑ ∑ ∑

k

The transition rate of the service time of a SU from phase 1 to phase 2 generating the transitions

⇒ ⇒1 2 1 2k + e - e k k - e + e is given by

( ) ( ) ( )1 1 2 11 ; 0;

0 ;otherwise

Sk k k Mc

⎧ − β μ ≥ ≤⎪= ⎨⎪⎩

k

The transition rate that interrupts a secondary user in phase 1 due to the arrival of a primary user generating the transitions

⇒ ⇒0 1 3 0 1 3k - e + e - e k k + e - e + e is given by

( )3

( )10 3

01 0

;0 ; 0;

0 ;otherwise

Pi

i

k k M k k Md M k =

⎧ λ ≤ < ≥ ≥⎪= −⎨⎪⎩

∑k

The transition rate that interrupts a secondary user in phase 2 due to the arrival of a primary user generating the transitions

⇒ ⇒0 2 3 0 2 3k - e + e - e k k + e - e + e is given by

( )3

( )20 3

02 0

;0 ; 0;

0 ;otherwise

Pi

i

kk M k k M

d M k =

⎧ λ ≤ < ≥ ≥⎪= −⎨⎪⎩

∑k

The transition death that interrupts for second time a SU due to the arrival of a PU generating the transitions

⇒ ⇒0 2 3 0 2 3k - e + e - e k k + e - e + e is given by

( )

3

3 3 30 ( )3 0

0 10

max 0,

;0 ; ; ;

0 ;otherwise

jj P

i ii i

k k Md k M k M k M

M k=

= =

⎧⎛ ⎞⎛ ⎞− −⎪⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎝ ⎠= ⎨ λ ≤ < ≥ ≤

−⎪⎪⎩

∑∑ ∑k

The valid space state is given by

3

00

| 0, 2 ,i i ii

k k M k M=

⎧ ⎫Ω = ≥ ≤ ≤⎨ ⎬⎩ ⎭

∑k

The steady state probabilities balance equation is given by

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 3 3 1

0 0 1 0

2

1 2 1 20

2

0 3 0 3 3 0 3 0 31

i i i i i ii i i i

i i ii

i i ii

a b c d P a P

b P c P

d P d P

= = = =

=

=

⎡ ⎤+ + + =⎢ ⎥⎣ ⎦

+ + +

+

∑ ∑ ∑ ∑

∑

∑

k k k k k k - e k - e

k + e k + e k + e - e k + e - e

k - e + e - e k - e + e - e k - e + e k - e + e

An arrival of a new secondary call is blocked when there is not idle channel. That is, new call blocking probability for secondary users when the total number of retransmissions is limited to 1 (denoted by Pb

(1)), can be computed as follows ( ) ( ) ( )

3 3

0 0

1

| 2 |i ii i

b r

M r k M k M r

P P p P

= =− < ≤ = −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

= +

∑ ∑∑ ∑

k k

k k (1)

On the other hand, during the lifetime of a SU, a certain number of PU arrivals may occur. For each PU arrival, the SU may be interrupted with probability PInt. It is straightforward to show that PInt can be computed as follows [2]

( )

( )

3 3 30

0 1 0

3 30

1 0

02 ; max 0, 0;

max 0, 0;0

1

i i ii i i

i ii i

M k M k k M k M

Int

k k M k M

PM k

PP

= = =

= =

⎧ ⎫⎛ ⎞⎪ ⎪∈Ω ≤ ≤ − − > <⎜ ⎟⎨ ⎬⎪ ⎪⎝ ⎠⎩ ⎭

⎧ ⎫⎛ ⎞⎪ ⎪∈Ω − − > ≤ <⎜ ⎟⎨ ⎬⎝ ⎠⎪ ⎪⎩ ⎭

∑ ∑ ∑

∑ ∑

−

=

∑

∑k

k

k

k (2)

Similarly, the interruption probability (due to PU’s arrival) of active SUs that have been interrupted once, is given by [13]

( )

( )

2 20

0 1

3 30

0 1

3

30

02 ; 0;

_ int1

2 ; 0;0

max 0,

i ii i

i ii i

jj

M k M k k M

Int

k M k k M

k k MP

M k

PP

= =

= =

=

⎧ ⎫⎪ ⎪∈Ω ≤ ≤ > <⎨ ⎬⎪ ⎪⎩ ⎭

⎧ ⎫⎪ ⎪∈Ω ≤ > ≤ <⎨ ⎬⎪ ⎪⎩ ⎭

∑ ∑

∑ ∑

⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

−

=

∑∑

∑k

k

k

k(3)

Also, the interruption probability (due to PU’s arrival) of active SUs that have not been interrupted at all, is [13]

( )

( )

3 3 30

0 1 0

3 30

1 0

1 2

02 ; max 0, 0;

_

max 0, 0;0

i i ii i i

i ii i

M k M k k M k M

Int new

k k M k M

k k PM k

PP

= = =

= =

⎧ ⎫⎛ ⎞⎪ ⎪∈Ω ≤ ≤ − − > <⎜ ⎟⎨ ⎬⎪ ⎪⎝ ⎠⎩ ⎭

⎧ ⎫⎛ ⎞⎪ ⎪∈Ω − − > ≤ <⎜ ⎟⎨ ⎬⎝ ⎠⎪ ⎪⎩ ⎭

∑ ∑ ∑

∑ ∑

+−

=

∑

∑k

k

k

k(4)

As such, the probability _Call IntP that an active secondary session be interrupted is given by [13]

( )

( )

IntCall Int S

IntP

PP

P=

μ +λ

(5)

Finally, we employ the methodology developed in Section IV of [15] to mathematically analyze the CRN under the RESTART retransmission strategy. For completeness, we reproduce this methodology below.

The RESTART strategy is evaluated using the just above developed queuing model for the RR strategy considering the total transmission time in RESTART as the UST of SUs. Fig. 3 shows that a SU call with UST xs

(S) has a total transmission time x under the RESTART strategy, which consists of a number of preempted (interrupted) secondary service times xi and one successful UST xs

(S). Also, Fig. 3 shows that if x is considered as the UST in the RR strategy, then total channel occupancy is the same in both RESTART and Resume strategies. The problem is to determine the pdf of the total transmission time in the RESTART strategy and approximate it by a suitable distribution. As the pdf of the total transmission time depends on the failure rate (i.e., call arrival rate of PUs, λ(P), and the interruption probability upon the arrival of primary users, PInt) and vice versa, it is necessary to employ a fix point iteration method. The approximated mathematical analysis approach develop in [15] works as follows.

Step 0: Set PInt = 0 and assume a predefined pdf for the unencumbered service time of SUs. Go to Step 1.

Step 1: Calculate the moments of the transmission time of SUs that have been interrupted once considering the actual value of PInt. For this calculation, the expressions obtained in [8]-[9] are used, considering that the failure rate is PIntλ(P) and the predefined pdf of the UST for SUs. Go to Step 2.

Step 2: Approximate the pdf of the transmission time of SUs that have been interrupted once by means of a phase-type pdf (in the numerical results section, negative exponential and order 2 Coxian distributions are used) with the moments calculated in Step 1. Go to Step 3.

Step 3: The RESTART strategy is analyzed using the teletraffic model developed in this section to analyze the resume retransmission strategy. The key aspect is to use in that queuing model the pdf of the transmission time of SUs that have been interrupted once instead of the secondary UST. Go to Step 4.

Step 4: Calculate PInt (i.e., for the case when a 2nd order Coxian is considered to approximate the pdf of the transmission time of SUs that have been interrupted once, it is given by eq. (2)). If the just calculated value of PInt differs (for more than a predefined value) from its previous considered value, then go to Step 1 considering the calculated value of PInt. Otherwise, go to Step 5.

Step 5: Obtain performance metrics and end.

The call completion probability when i-retransmissions are allowed is defined as follows

( )( )( )( ) ( )1

_

when the maximum number of1 ;

retransmissions is not limited.

1 1 ; for <

b

Cii

b Call Int

PP

P P i

∞

+

⎧−⎪⎪= ⎨

⎪ − − ∞⎪⎩

where ( )bP ∞ and ( )i

bP represent, respectively, the blocking probability when the total number of retransmissions is not limited and when it is limited to i. As such, the effective carried traffic is defined as follows

( ) ( )( )( ) ( )( )( )1

_

when the maximum number of1 ;

retransmissions is not limited.

1 1 ; for <

b

Cii i

b Call Int

a Pa

a P P i

∞ ∞

+

⎧−⎪⎪= ⎨

⎪ − − ∞⎪⎩

where ( )a ∞ and ( )ia represent, respectively, the secondary offered traffic when the total number of retransmissions is not limited and when it is limited to i.

t

x

t

1x

2x

3x( )Ssx( )

1 2 3S

sx x x x x= + + +

( )Ssx

( )Ssx

( )Ssx

Fig. 3. Equivalence of the CHT in the RESTART strategy with unencumbered service time xs

(S) and the RR strategy with unencumbered service time equal to the total transmission time x in RESTART.

IV. NUMERICAL RESULTS The goal of the numerical evaluations is both to validate

our proposed analytical models and quantify the benefits of limiting the total number of retransmissions to reduce the negative impact of retransmissions. Numerical results for negative-exponential and 2-nd order Coxian are obtained. For the 2-nd order Coxian three different cases in terms of its first two standardized moments (i.e., coefficient of variation –CV- and Skewness –SK-) are considered. These cases are

summarized in Table I. Unless otherwise specified, the following values of the system parameters were used in the plots of this section: M = 6, 1/μ(P)=1/0.06 s.

TABLE I. PARAMETERS OF THE 2-ND ORDER COXIAN DISTRIBUTIONS USED TO MODEL THE SECONDARY CHT.

CV SK γ ( )1

Sμ ( )2Sμ

1 2 1 0.82 - 9 30 0.955 1.48178 0.009054

15 30 0.995 3.2745 0.0055

System performance is evaluated in terms of SU call completion probability, PC, supported offered traffic2, and SU mean normalized transmission delay. The mean normalized transmission delay is the difference of the mean value of the elapsed time between the epoch the SU arrives to the system to the epoch it finally leaves the system minus the mean service time, normalized to the mean service time. Results for the mean normalized delay are obtained by discrete-event computer simulation.

For a given value of the primary traffic load and for a fair basis of comparison, all the analyzed strategies are compared under the same value of the effective carried traffic. To this end, the strategy that allows a single reconnection is taken as the reference one (in the plots of this section, the strategy of reference is denoted by “A 1ReTx r=0”). In this respect, for a given value of the primary traffic load and under the strategy of reference, the secondary traffic load is found in order to obtain a call completion probability equals 0.9 (this particular value for the secondary traffic load is referred to as the supported offered traffic). After that, the corresponding effective carried load is obtained and this value is used as input parameter to obtain the call complete probability of all other evaluated strategies.

Figs. 4, 5 and 6 (7, 8 and 9) {10, 11, 12} show secondary completion probability (mean normalized transmission delay) {secondary offered traffic} as function of primary mean traffic load for the case when unencumbered service time for SU is exponentially distributed, 2-nd order Coxian distributed with CV=9 and SK=30, and 2-nd order Coxian distributed with CV=15 and SK=30, respectively. In the plots of this section, the labels “A”, “S”, “iReTx”, and “r=x”, stand, respectively, for analytical results, simulation results, i allowed reconnections, and x number of reserved channels for exclusive use of primary users. If the symbol ∝ is used instead of “iReTx”, it means that the total number of reconnections is not limited.

From Figs. 4-6, perfect agreement is observed between analytical (represented by the strategy “A 1ReTx r=0”) and simulation (represented by the strategy “S 1ReTx r=0”) results, which validates our proposed analytical model. A relevant result that can be extracted from Figs. 4-6 is that, for the same effective carried traffic, limiting retransmissions is a more effective way to improve call completion probability than reserving channels. In fact, from Figs. 4-6 it is evident that as the number of channels reserved increases, call completion

2 The supported offered traffic is defined as the value of the secondary offered traffic for which the target effective carried traffic is achieved.

probability decreases, indicating a detrimental effect of channel reservation on call completion. This is an expected result due to the fact that, for a given target effective-carried traffic, as the number of channels reserved for exclusive use of PT calls increases less channels are available for the ST stream. Consequently, call completion probability is degraded. However, this is the price that the channel reservation mechanism has to pay in order to improve mean normalized delay and to support higher secondary traffic loads, as it is observed in Fig. 7-12 and explained below. On the other hand, analyzing Figs. 4-12, revels that the mechanism based on limiting retransmissions trades secondary traffic load for call completion probability and mean normalized delay.

Another interesting result that can be extracted from Fig. 4 and 10 is as follows. For the case when the secondary service time is modeled by a negative-exponential distribution, both call completion probability and secondary traffic load are insensitive to the number of allowed retransmissions. This is due to the fact that, as shown in Table III of [13], for the exponential case, the mean value of the residual service time of the interrupted calls does not significantly differ from that of the unencumbered service time. On the other hand, from Figs. 7-9, it is observed that either limiting the number of reconnections or reserving channels for exclusive use of PUs has the beneficial effect of improving mean normalized delay. This behavior can be explained as follows. Limiting the number of reconnections or reserving resources for exclusive use of PUs, decreases the traffic of SUs inside the system. Referring to the mechanism based on limiting retransmissions, the traffic of secondary users is reduced due to the fact that ongoing secondary sessions that are interrupted certain number of times are forced to terminate. On the other hand, the mechanism based on reserving channels for exclusive use of PUs limits the secondary traffic that enters the system by precluding new secondary sessions to access if the number of available channels is less than r. As such, less number of retransmissions is needed and, therefore, the mean normalized delay of SUs is reduced.

Figs. 10-12 show the supported offered traffic (i.e., secondary offered traffic for which the effective carried traffic of the different strategies equals that of the reference strategy).

From these figures, it is observed that the supported offered traffic increases as the number of reserved channels is increased. However, as stated before, this improvement is achieved by reducing call completion probability, as shown in Figs. 4-6. Also from Figs. 10-12, it is observed that, as expected, increasing the number of allowed retransmission increases supported offered traffic. Then, there is a tradeoff between the call completion probability and the supported offered traffic load that can be controlled by varying either the number of reserved channels or the number of allowed retransmission.

Let us, now, analyze the impact of the CV of the unencumbered service time on system performance. Notice that in Figs. 4, 7, and 10 (5, 8, and 11) {6, 9, and 12} the value of the CV is 1 (9) {15}. In this sense, from Fig. 4-12 it is observed that the advantage of limiting retransmissions increases as the value of the coefficient of variation of the unencumbered service time increases. In general, it can be concluded that limiting retransmission is an effective mechanism to widen the conditions under which system stability is achieved (i.e., conditions under which completion probability or effective carried traffic are greater than zero).

V. CONCLUSIONS In this paper, limiting the maximum number of

reconnections in cognitive radio networks with delay tolerant traffic and the RESTART retransmission strategy was proposed as a mechanism to reduce the negative impact of retransmissions and to widen the conditions under which system stability can be achieved. It was observed that limiting reconnections is an effective mechanism to alleviate performance degradation due to connection failure. The benefit of limiting reconnections increases as the value of the coefficient of variation of the unencumbered service time increases. A tradeoff between the call completion probability and the supported offered traffic load that can be controlled by varying either the number of reserved channels or the number of allowed retransmission was observed. The consideration of, on average, a real number of allowed retransmissions and the determination of the optimum number of allowed retransmission are topics of future research.

0.2 0.4 0.6 0.8 1 1.20.3

0.4

0.5

0.6

0.7

0.8

0.9

1

aP [Erlangs]

PC

EXP-NEG

S 0ReTx r=0

S 1ReTx r=0A 1ReTx r=0

S 2ReTx r=0

S ∞ r=0

S ∞ r=0.5

S ∞ r=1.0

S ∞ r=1.5

0.2 0.4 0.6 0.8 1 1.20.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

aP [Erlangs]

PC

CV=9, SK=30

S 0ReTx r=0

S 1ReTx r=0

A 1ReTx r=0S 2ReTx r=0

S ∞ r=0

S ∞ r=0.5

S ∞ r=1.0

0.2 0.4 0.6 0.8 1 1.20.65

0.7

0.75

0.8

0.85

0.9

0.95

1

aP [Erlangs]

PC

CV=15;SK=30

X: 0.456Y: 0.7212

S 0ReTx r=0

S 1ReTx r=0A 1ReTx r=0

S 2ReTx r=0

S ∞ r=0

S ∞ r=0.5

S ∞ r=1.0

S ∞ r=1.5

Fig. 4. Call completion probability with exponentially distributed service time and total transmission time approximated by an exponential distribution.

Fig. 5. Call completion probability with 2-nd order Coxian distributed service time with CV=9 and SK=30.

Fig. 6. Call completion probability with 2-nd order Coxian distributed service time with CV=15 and SK=30.

0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3 EXP-NEG

aP [Erlangs]

Mea

n n

orm

aliz

ed d

elay

∞ r=0

2ReTx r=01ReTx r=0

∞ r=0.5

∞ r=1.0

∞ r=1.5

∞ r=2.00ReTx r=0

0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7x 10

-3 CV=9 SK=30

aP [Erlangs]

Mea

n no

rmal

ized

del

ay

∞ r=0

2ReTx r=0

∞ r=0.51ReTx r=0

∞ r=1.00ReTx r=0

0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

8

9x 10

-3 CV=15 SK=30

aP [Erlangs]

Mea

n no

rmal

ized

del

ay

∞ r=0

2ReTx r=0

1ReTx r=0

∞ r=0.5

∞ r=1.0

∞ r=1.50ReTx r=0

Fig. 7. Transmission delay with exponentially distributed service time and the total transmission time approximated by an exponential distribution.

Fig. 8. Transmission delay with 2-nd order Coxian distributed service time with CV=9 and SK=30.

Fig. 9. Transmission delay with 2-nd order Coxian distributed service time with CV=15 and SK=30.

0.2 0.4 0.6 0.8 1 1.22

3

4

5

6

7

8

9

10EXP-NEG

aP [Erlangs]

a S [

Erl

angs

]

S ∞ r=2.0

S ∞ r=1.5

S ∞ r=1.0

S ∞ r=0.5

S ∞ r=0S 2ReTx r=0

A 1ReTx r=0

S 1ReTx r=0A 0ReTx r=0S 0ReTx r=0

0.2 0.4 0.6 0.8 1 1.2

2

2.5

3

3.5

4

4.5CV=9; SK=30

aP [Erlangs]

a S [

Erl

ang

s]

S ∞ r=1.0

S ∞ r=0.5

S ∞ r=0S 2ReTx r=0

A 1ReTx r=0

S 1ReTx r=0A 0ReTx r=0

S 0ReTx r=0

0.2 0.4 0.6 0.8 1 1.2

2.5

3

3.5

4

4.5

CV=15; SK=30

aP [Erlangs]

a S [

Erl

ang

s]

S ∞ r=1.5

S ∞ r=1.0

S ∞ r=0.5

S ∞ r=0S 2ReTx r=0

A 1ReTx r=0

S 1ReTx r=0A 0ReTx r=0

S 0ReTx r=0

Fig. 10. Secondary offered traffic with exponentially distributed service time and total transmission time approximated by an exponential distribution.

Fig. 11. Secondary offered traffic with 2-nd order Coxian distributed service time with CV=9 and SK=30.

Fig. 12. Secondary offered traffic with 2-nd order Coxian distributed service time with CV=15 and SK=30.

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spectrum access with optimal channel reservation,” IEEE Commun. Lett., vol. 11, pp. 1-3, Apr. 2007.

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[8] P. M. Fiorini, R. Sheahan, and L. Lipsky, “On unreliable computing systems when heavy-tails appear as a result of the recovery procedure,” ACM SIGMETRICS Perf. Evaluation Review, vol. 33, no. 2, pp.15–17, Sep. 2005.

[9] P.M. Fiorini, R. Sheahan, and L. Lipsky, “On the completion time distribution for tasks that must restart from the beginning if a failure occurs,” ACM SIGMETRICS Perf. Evaluation Review, vol. 34, no. 3, Dec. 2006.

[10] P.R. Jelenkovic and J. Tan, “Can retransmissions of superexponential documents cause subexponential delays?,” in Proc. IEEE INFOCOM’2007, Anchorage, AK, May 2007.

[11] P.R. Jelenkovic and J. Tan, “Characterizing heavy-tailed distributions induced by retransmissions,” Dept. of Electrical Engineering, Columbia University, New York, NY, Tech. Rep. EE2007-09-07, Sep. 2007.

[12] J. Tan and N.B. Shroff, “Transition from heavy to light tails in retransmission durations,” in Proc. IEEE INFOCOM’2010, San Diego, CA, Mar. 2010.

[13] S.L. Castellanos-López, F.A. Cruz-Pérez, and G. Hernández-Valdez, “Performance of cognitive radio networks under resume and restart retransmission strategies,” in Proc. IEEE WiMob’2011, Shangai, China, Oct. 2011.

[14] S. L. Castellanos López, F.A. Cruz Pérez, and G. Hernández Valdez (2011). Resume and Starting-Over-Again Retransmission Strategies in Cognitive Radio Networks, Advanced Trends in Wireless Communications, Mutamed Khatib (Ed.), INTECH.

[15] S.L. Castellanos-López, F.A. Cruz-Pérez, and G. Hernández-Valdez, “Channel reservation in cognitive radio networks with the RESTART retransmission strategy,” in Proc. IEEE CROWNCOM’2012, Stockholm, Sweden, June 2012.