The current paper presents a finite-source retrial queuing system that models a Cognitive Radio Network (CRN) dealing with two types of requests (primary and secondary) attached to two interconnected, non-independent frequency bands. An orbit and a First In First Out queue are assigned to the second and first service units, respectively. The first server is meant to handle requests coming from Primary Users (PUs), the second one is built for the requests of Secondary Users (SUs). The newly generated primary requests are directed to the Primary Service Unit. In case of an idle status, the service process of these jobs can start immediately. If it is busy with a licensed request, the last generated packet is routed to the FIFO queue. However, if the channel is busy with an unlicensed request, its service is discontinued and the latter request must be returned to the Secondary Service Unit. Depending on the status of the units, the suspended jobs are added to either the server or the orbit. If an SU request discovers an idle status in the Secondary Channel Service (SCS), the service can be resumed right
away, however, If the SCS is busy, the new request might attempt joining the Primary Channel Service (PCS). Assuming the PCS is idle, the low priority packet can opportunistically be hosted by the high-priority server, otherwise, it needs to join the orbit. From orbit, the deferred requests retry to be served after a random time. It should be noted that the secondary service unit is non-reliable, which means that the server is subject to random failures depending on whether it is busy or idle. We decided to assume that only the secondary server of our system is unreliable, since having both servers unreliable is another case study and due to the limitation of pages we have focus our analysis on the secondary part of the system to demonstrate the positive efect of sharing the channels cognitively. The novelty of this work is to investigate the impact of the failure time distribution of the SCS, distinguishing its state (idle or busy). Our approach is based on simulation to study the efects of the mentioned two scenarios on diferent performance measures of the system, using several distributions (Pareto, gamma, log-normal, hyperexponential and hypo-exponential). Multiple figures illustrate the problem in the question.
The main objective of our model “Cognitive Radio Network” is to utilize the free
portions of the licensed frequency bands for the benefit of secondary customers.
Further details are given in [
However, in this investigation, we are assuming that only the secondary unit is unreliable and diferentiating the failures (idle or busy state).
After a deep immersion in many similar investigations and studies, we did not ifnd any papers that dealt with the unreliability in such model, distinguishing the failures in a busy and idle state. Consequently, several figures will illustrate the impact of the various distributions on the performance measures of the second part of the system, using stochastic simulation.
Figure 1 shows a queueing cognitive radio system with the following assumptions. Consider two interconnected subsystems, where the licensed requests are generated by a finite number of sources 1. These sources generate primary calls corresponding to an exponentially distributed time with an average value of 1 which are sent to the primary service unit. If the server is idle, the service starts immediately. If the server is busy, the call joins a preemptive priority queue. The primary service time is supposed to be exponentially distributed random variable with a mean 1.
For the secondary part, the number of sources is denoted by 2. Each source generates low priority calls according to an exponentially distributed time with a mean value of 2/ 2. The secondary service time is exponentially distributed with a parameter 2. We assume that the secondary service unit is non-reliable, which means that the server is subject to random failures depending on whether it is in a busy or idle state. The secondary service unit may fail after a time, which is generally distributed with a rate 2 during idle state and 2 during a busy state. The operating time (or inter-failure time) during the busy or idle state is supposed to be hyper-exponential, hypo-exponential, gamma, lognormal and Pareto distributed random variables. Similarly, for repair time, the rate is 2. The retrial time of the secondary customers is supposed to be exponentially distributed random variable with a parameter /
2.
Assuming that all random variables included in the system are exponentially
distributed except the failure and repair times which are generally distributed
random variables, we created a stochastic simulation program written in C coding
language with SimPack [
Performance modelling and analysis of systems with non-reliability has been
investigated using asymptotic methods in [
This paper deals with a more general situation allowing non-exponential distributions. Therefore, the simulation approach is the most eficient method which helps us in performance modelling and analysis when the steady-state equations are not solvable.
For a better understanding of numerical results, we presume that in our system the disrupted secondary service from the PCS due to the arrival of PUs or from the SCS due to server failure will be repeated from the beginning (non-intelligent). Furthermore, the failure of the service unit will not obstruct the system and free sources will continue to create new jobs.
In this paper, we have performed several simulations runs in order to investigate the impact of the operating and the repair times distribution on the behaviour of the system. Mainly, we have investigated separately the cases when the server fails during idle or busy state. Many scenarios might be treated using our introduced model, however, due to the limitation of pages, we have assumed only the following three scenarios: Scenario 1: SCS repair time is generally distributed, assuming that the operating time is Exponentially distributed.
Scenario 2: SCS operating time is generally distributed when server fails during idle state, assuming that the repair time is Exponentially distributed. Scenario 3: SCS operating time is generally distributed when server fails during busy state, assuming that the repair time is Exponentially distributed.
All the numerical values of parameters for the statistical methods are defined
in Table 2 and were collected based on some previous works dealing with the same
model, such as [
Firstly, our aim is to investigate how the various distributions of repair times, where the mean and variance are equal, efect the performance analysis. Depending on the square coeficient of variation, the investigation is split into two parts.
Table 3 represents the input parameters for the distributions of the repair times. These parameters are chosen according to the squared coeficient of variation.
Figure 2, Figure 3 and Figure 4 illustrate the impact of the server’s repair time distribution on the mean sojourn time of secondary customers, total utilization of secondary server and mean service time of secondary customers, respectively. These measures are displayed in function of the arrival intensity of the secondary customers. Figures show an important impact of the repair times distribution on these features. In Figure 2, the log-normal distribution gives a higher value of the mean response time while the gamma distribution provides the smallest value of the mean. However, since the gamma distribution generates small values using the mentioned input parameters, the server is recovered faster, thus, the greater value of the utilization is shown in Figure 3. Similarly, in Figure 4 the highest value of the mean service time is provided by the gamma distribution since the system is most of the time in operational mode.
The explanation of the results might depend on the observed random variable. For instance, the behaviour of mean total of primary and secondary service time will change in case of changing the distributions mean and variance. The analysis of the performance measures of such a system using simulation allows us to investigate the characteristics that are almost impossible to analyse analytically. However, our explanation of the results is as follows:
In Figure 2 the repair time is generally distributed. If we see the graph of the Probability Density Function of the lognormal and gamma distribution, the relative likelihood of the random variable x (repair time) is clearly greater in the case of the lognormal distribution than in the case of the gamma which means that the repair time will take more likely greater values in case it is lognormally distributed. Thus, it involves greater response time of the customers considering that the server takes longer time to be repaired. On the other side, Figure 3 shows that the utilization of the server while the repair time is lognormally distributed has smaller values than when it is gamma distributed because the server is most of the time down and not occupied by a customer.
In this part, let us investigate and compare the efect of the repair time distribution on the same features shown above, but in this situation, we replace the hyper-exponential distribution by the hypo-exponential distribution and set new parameters that their coeficient of variation is less than one.
Similarly, Figure 5, Figure 6 and Figure 7 show the efect of the repair time distribution on the mean response/service time of secondary calls and utilization of
SCS versus secondary request generation rate. In this case where the distribution have their 2 < 1, the diference in the values of the performance measures is between two groups of distributions. Pareto and hypo-exponential give similar values of the estimations. These values are greater than the one resulted from the log-normal and gamma distribution. The explanation of the illustrated impact of the distributions is similar as the case of squared coeficient of variation greater than 1.
In this scenario, the expected phenomenon was obtained such the property of having a maximum value of the mean. Also, increasing the arrival intensity involves higher utilization of the server and lower mean service time.
In this scenario, we analyze the distributions of operating times depending on the square coeficient of variation, this part is divided into two subsections.
Table 4 represents the input parameters for the distributions of the inter-failure times. Figure 11,12,13,17,18,19 Figure 8,9,10,14,15,16
Hyper N/A N/A
N/A Mean Variance Parameters Let’s consider the inter-failure time of the server during an idle state generally distributed and all the other involved random variables are exponentially distributed.
Figure 8, Figure 9 and Figure 10 illustrate the impact of the failure time distribution during idle state on the mean response time, utilization of the SCS and PCS respectively while the running parameter is 2/ 2. The distributions on the ifgures have their 2 > 1. Despite the gamma distribution, which involves a big diference in the estimation of the mean sojourn time and the utilization, the other distributions have no efect on the performance. An impact of the gamma distribution as well on the utilization of the primary channel can be seen in Figure 10. The generated inter-failure times from gamma distribution make the server more often non-operational and the impact can be seen even on the utilization of the primary service channel.
Figure 11, Figure 12 and Figure 13 illustrate the same features as the figures above in this section, but in this time, we replaced the hyper-exponential distribution by the hypo-exponential distribution in order to investigate the case of distinguished distributions with 2 < 1. With this set of parameters, the relative diference between gamma and the other distributions is very small comparing with the corresponding figures above. Still a small impact can be seen on the mean residence time and utilization of the SCS but there no efect on the PCS.
In the last scenario of our investigation, we suppose the failure time of the server during a busy state generally distributed and all the other involved random variables exponentially distributed. Similarly, as the above investigation, we will start with the hyper-exponential distribution and set the gamma, Pareto and log-normal with 2 > 1.
Figure 14, Figure 15 and Figure 16 display the efect of the inter-failure time distribution during busy state on the mean sojourn time, utilization of the SCS and PCS respectively versus 2/ 2. Figure 14 shows the impact of the distribution where the gamma gives a higher value of the mean response but the log-normal gives the smallest value. In Figure 15, the gamma distribution provides the lowest utilization of the server as expected.
In Figure 17, Figure 18 and Figure 19. The hypo-exponential distribution takes place instead of the hyper-exponential. The figures illustrate the same estimation as Figure 14, Figure 15, and Figure 16, respectively. In this case, we notice an opposite behaviour of scenario 2 where the inter-failure time during idle state was generally distributed. In this later, the relative diference of the estimations caused by the gamma distribution was smaller in the case of the hypo-exponential distribution. However, in this scenario, we see the figures that the relative diference between the estimations is greater in the case of the hypo-exponential distribution and smaller in the case of the hyper-exponential distribution.
We notice from Figure 16 and Figure 19 that whether the squared coeficient of variation is greater or less than one, the distribution of the inter-failure time during busy state has no efect on the primary service channel.
However, our explanation regarding the significant diference is shown in the ifgures is the following: let us consider the sample examples illustrated by Figure 8 and Figure 14 that show respectively the efect of the operating time distribution on the SU mean response time while the server breaks down during idle and busy state. The diference is significant when the arrival rate 2/ 2 ∈ [0.1, 1]. At this low arrival intensity, the server breaks down more likely during idle state, and the arrival customers during the repair time cannot join the service unit (Continues case study) which it involves greater response time. The diference can be seen while the operating time is gamma distributed because of the smallest values of that gamma distribution generates (Short operation time).
In this paper, we presented a finite-source retrial queueing system with two nonindependent sub- systems to model cognitive radio network with primary and secondary service units subject to random break-downs and repairs. With the help of simulation, a detailed analysis has been performed in order to investigate the impact of the inter-failure time distribution separately during idle and busy state on the main performance measures of the system. In this paper, we have shown that the performance of the system depends on the state of the server at a random breakdown.