The goal of the paper is to study a finite-source retrial queuing system with collisions and customers' impatient behavior in the orbit. The situation when an incoming request/call/job/customer either from the orbit or from the source finds the server busy causes a collision and both requests are directed toward the orbit. It is assumed that each source can generate request whenever the server is failed but these requests immediately go into orbit and it cannot generate a new one until the previous call returns to the source. A customer after waiting for a few seconds in the orbit can depart without fulfilling its service requirement these are the so-called impatient customers. In that case, it returns to the source. The source, service, retrial, impatience, operation, and repair times are supposed to be independent of each other. The novelty of the investigation is to carry out a sensitivity analysis comparing various distributions of impatience time of customers on the main performance measures of diferent type of customers, probability of abandonment, total server utilization, etc.
AMS Subject Classification: 60K25, 68M20, 90B22
Queuing systems with repeated calls may competently describe major
telecommunication systems, such as telephone switching systems, call centers, CSMA-based
wireless mesh networks in frame level. The main feature of the retrial queueing
system is that customers remain in the system even if it is unable to find an idle
service unit and after some random time it attempts to reach the service facility
again. Customers’ impatience is a natural phenomenon and an interesting topic in
queueing theory. The process of reneging and balking has been extensively studied
by many researchers during the last years. For example in [
The novelty of this investigation is to carry out sensitivity analysis using
different distributions of impatient customers on the performance measures. Various
ifgures help us to understand the special features of the system. The model is a
generalization of [
The present paper aims to show some recent results on single server finitesource retrial queuing systems with impatient customers in the orbit. There are investigations when the server is reliable and there are models when the server is subject to random breakdowns and repairs depending on whether it is idle or busy. Tool supported, numerical, and simulation methods are considered. Several cases and examples are treated and the results of diferent approaches are compared to each other showing the advantages and disadvantages of the given method. In general, we could see that the steady-state distribution of the number of customers in the service facility can be approximated by a normal distribution with a given mean and variance. With the help of stochastic simulation, several systems are analyzed showing directions for further analytic investigations. Tables and Figures are collected to illustrate some special features of these systems.
A retrial queueing system of type // 1// is considered with a non-reliable server and impatient customers which Figure 1 represents. In the finite-source each of them can generate request towards the server with rate / so the interrequest time is exponential with parameter / . Every incoming customer has an ‘impatience’ property which determines how much time the customer spends in the orbit without fulfilling its service requirement. Exceeding this time results that the customer no longer waits for the service unit and departs without being served properly. This variable follows exponential with parameter , hypo-exponential, hyper-exponential, gamma, Pareto, and lognormal distribution with diferent parameters but with the same mean value. In absence of a waiting queue if an incoming customer finds the server in idle state its service starts immediately. The service times of the customers are exponentially distributed with parameter . After it’s successful service customers return to the source. Encountering the service unit in a busy state the incoming customer causes a collision with the call under service and both enter the orbit. After an exponentially distributed time with parameter / customers located in the orbit make another attempt to get into the service. The server is not reliable so from time to time it is supposed to break down. This is an exponentially distributed random variable with the parameter 0 in case of an idle server and 1 when the server is busy. The repair process starts immediately upon the breakdown which also follows an exponential distribution with parameter 2. If server failure takes place during the service of a customer then it is transferred to the orbit. Furthermore, in the case of a failure we can distinguish two options. Namely, the failure either stops entering new customers from the source or allows them to go to the orbit. Usually, we treat system with the latter option if the other one is not stated.
The source, service, retrial, impatience, operation, and repair times are supposed to be independent of each other.
In this section, we show the results generated by MOSEL (MOdeling, Specification
and Evaluation Language), published in [
The simulation approach is a very important method that helps us in performance
modeling when the system is too complicated to investigate with the help of other
standard methods, like analytical, numerical, or asymptotic ones. For the interested
readers we list some of the most important works dealing with stochastic simulation
and statistical investigations of output data [
In this section first, we deal with exponentially distributed impatience time which helps us to check our simulation results with the help of those we got using MOSEL. As soon as we realized that the simulation program runs correctly we can investigate the efect of the distribution of impatience time on the performance measures as we will do in the second part. One of the advantages of the simulation is that we can make a diference between abandoned and successfully served requests as we show it in the second part. 4.1. Exponentially Distributed Impatience Times In these cases, we would like to show the efect of the impatience rate on the distribution of the number of customers in the system. All the random variables mentioned above are supposed to be exponentially distributed. arbitrary customer and the probability of abandonment can be obtained. One of the advantages of the simulation that we can make diference between abandoned and successfully served requests, we can estimate the performance measures, respectively, see Figure 2. 0.1 2 1 / 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1E-10 higher the impatience rate is the fewer the number of customers is in the system see Figure 3. 4.2. Generally Distributed Impatience Times We aim to examine how the diferent distributions of impatience of customers afect the performance measure when the mean and variance are equal, respectively. The investigations are divided into two parts depending on the squared coeficient of variation.
Squared Coeficient of Variation is Greater than One In the first part, Table 4 shows the parameters of distinct distributions. The parameters are chosen in such a way that the squared coeficient of variation would be greater than one. For comparison hyper-exponential, gamma, lognormal and Pareto distributions are used besides the case when the impatience time is constant. Our simulation program is equipped with random number generators and these functions need input parameters that are diferent in each distribution. a little bit especially in the case of Pareto distribution and when the impatience time of calls is constant the mean is greater compared to the others.
The mean response time of diferent types of customer is shown in function of
arrival intensity on Figures 5, 6, 7. Figure 6 illustrates how the mean response
time of impatient customers changes. The mean waiting time in the orbit should
be constant, due to the constant impatient time of a customer. Of course, Figure 7
can be obtained with the help of the law of total expectation, too. Interestingly,
diferences can be observed even though the first two moments are equal,
respectively. Results clearly illustrate the efect of various distributions. The highest
values are experienced at gamma distribution in the case of successful customers,
but in the case of impatient calls, constant impatience time gives the greatest
values. Despite the increasing arrival intensity, the maximum property characteristic
of finite-source retrial queueing systems occurs under suitable parameter settings
as we mentioned in [
Figure 8 demonstrates how the probability of abandonment of a customer changes with the increment of the arrival intensity. By probability of abandonment we mean the probability that a customer leaves the system without getting its full-service requirement (through the orbit). After a slow increase of the value of this performance measure, it stagnates which is true for every used distribution of impatience of calls but they difer significantly from each other. At gamma distribution, the tendency of leaving the system earlier is much higher than the others especially compared to at constant mean of impatience of calls. Here the disparity is much higher among the applied distributions compared to the previous Figures. An explanation of this feature could be the following: if the squared coeficient of variation is greater than one the gamma distribution takes small values with great probability, so the customers leave the system quite early, and thus the probability of abandonment is high.
Figure 9 is related to the total utilization of server versus arrival intensity. Total utilization contains every service time including the interrupted ones no matter whether a call departed from the service unit or the orbit. By examining closely the Figure we find prominent results when gamma distribution is applied and regarding the others, the received values are almost identical. With the increment of arrival intensity, the total utilization of the service unit increases as well. Here the explanation is the same as in the previous Figure, that is there are fewer customers in the system and hence the utilization is the smallest.
We carried the simulation in the case when the squared coeficient of variation
is less than one, the mean is the same as before. The diferences are noticeable but
not as big as in the previous cases, see [
In this paper, a finite-source retrial queueing system was presented with a nonreliable server, collisions, and impatient customers in the orbit. The obtained results fully demonstrated that the distribution of impatience of customers has a great influence on the system’s characteristics even though the mean and the variance are the same. Figures in connection with the probability of abandonment assure this phenomenon. Results indicated the distinction is noticeable and significant among the performance measures having the same mean and variance of diferent distributions when the squared coeficient of variation is greater than one and moderate when it is less than one.
Acknowledgment. The authors are very grateful to the reviewers for their comments, which improved the quality and the presentation of the paper.