The goal of the paper is to study a M/M/1//N finite-source retrial queuing system with collisions and customers' impatient behavior in the orbit. The server is not reliable, breakdown can happen either in busy or in idle states. The situation when an incoming customer from the orbit or from the source ifnds the server busy causes a collision and both requests are directed toward the orbit (including the customer under service, too). It is assumed that every request in the source is eligible to generate customers whenever the server is not working but these requests immediately get into the orbit. A customer after some waiting for the server to be served can depart from the orbit without fulfilling its service requirement these are the so-called impatient customers. In that case it goes back to the source. All random variables involved in the model construction are supposed to be independent of each other. The novelty of the investigation is to carry out a sensitivity analysis comparing various distributions of impatient time of customers on the performance measures such as mean number of customers in the orbit, mean waiting time of an arbitrary customer, mean waiting time of customers who leave the system without service, probability of abandonment, server utilization, etc. By the use of a self-developed simulation program several graphical results and comparisons of the investigated systems are illustrated.
At present with the rapid increase of trafic growth balks the analysis and optimal
designing of communication systems. With regard to exchanging of information,
a lot of communication process are created by companies from day to day.
Consequently, construction of new applicable models of telecommunication systems or
modifying of existing ones are necessary. 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.
Numerous papers and books are devoted to study their importance like [
Speaking of communication systems where the available channels or other
facilities are very limited thus users (sources) usually need to fight for these resources.
This results a high possibility of conflict because several sources may launch
uncoordinated attempts producing collisions. In these cases the loss of transmission
takes place and it is necessary to ensure of the process of retransmission. So
evolving eficient procedures for preventing conflict and corresponding message delay is
essential. Some results on retrial queues with collision that have been published in
[
In many papers of retrial queueing literature the service unit is assumed to
be available steadily. But these assumptions are quite unrealistic because in real
life applications of these systems can break down, diferent types of problems can
arise like power outage, human error or other failures. Various factors have efect
on the transmission rate of the wireless channel in a a wireless communication
scenario and these are apt to sufer transmission failure, interruptions throughout
transferring the packets. Investigating retrial queueing systems with random server
breakdowns and repairs has a great importance as the operation of non-reliable
systems modifies system characteristics and performance measures. Finite-source
retrial queues with server breakdowns have been investigated in several papers, for
example in [
In this paper we study the operation of the system where the service unit is
subject to breakdown and waiting customers can leave the system without spending
time at the service unit. The novelty of this investigation is to carry out sensitivity
analysis using assorted distributions of impatience of calls on performance measures
like mean waiting time of an incoming call or total utilization of the server. To
accomplish this goal a simulation program is evolved based on SimPack toolkit [
A retrial queueing system of type M/M/1//N is considered with a non-reliable server and impatient customers which Figure 1 represents. In the finite-source
customers reside and each of them is able to generate calls towards the server with rate / so the inter-request time is exponential with parameter / . Every incoming customer has an “impatience” property which determines how much time the customer spend in the orbit without fulfilling its service requirement. Exceeding this time result that the customer no longer waits for the service unit and departs without being served properly. This variable follows gamma, hypo-exponential, hyper-exponential, Pareto and lognormal distribution with diferent parameters but with the same mean value. In absence of waiting queue if an incoming customer ifnds the server in idle state its service starts immediately. The service times of the customers is exponentially distributed with parameter . After its successfully service customers return to the source.
Encountering the service unit in busy state the incoming customer remains in the system entering the orbit. After an exponentially distributed time with parameter / customers located in the orbit make an other 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 parameter 0 in case of a busy server and 1 when the server awaits a customer.
which also follows exponential distribution with parameter 2. If server failure takes place during the service of a customer then it is transferred to the orbit.
All the random variables involved in the model construction are assumed to be totally independent of each other.
performance measures a statistics package is involved in our simulation program,
written by Andrea Francini in [
level of 99.9%. Relative half-width of the confidence interval required to stop the simulation run is 0.00001.
Our aim is to examine how the diferent distributions of impatience of calls
have an efect on the performance measure when the mean and variance are equal,
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
utilized besides the case when the mean value is constant. Our simulation program
is equipped with random number generators and these functions need input
parameters which are diferent in every distribution. To achieve valid comparison in
case of every distribution fitting process is needed to be done which can be viewed
in [
N 100 / 0.01 0 0.1 1 0.1 2 1 / 0.01 1
Figure 2 shows the comparison of steady-state distribution of the customers. It
represents the probability of how many customers residing in the system. Taking
a closer look on the results all the curves correspond to normal distribution and
in the following papers the proof of this phenomenon can be found in [
Figure 4 demonstrate how the probability of abandonment of a customer changes with the increment of the arrival intensity. Under probability of abandonment we mean the probability of 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 distributions 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 ifgures.
The last figure 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 from 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. 3.1. Diferent parameter setting After viewing the above outcomes and figures we are intrigued to know how the operation of the system changes if another parameter setting is used. To do so we modify the parameters in order the squared coeficient of variation to be less than one so hyper-exponential is exchanged for hypo-exponential distribution.
Distribution Parameters
Mean
Variance Squared coeficient of variation
Gamma among the applied distributions with this parameter setting, too. As regard to the values with these parameters the mean number of customer is higher in case of every distribution.
The next figure shows the mean response time of a successfully served customer in function of arrival intensity. Examining Figure 7 the same tendency can be seen as on the previous figure but diferences can still be discovered especially in case of gamma distribution. This figure also reveals that customers averagely spend less time in the system compared to the previous parameter setting. Figure 8 demonstrates the probability of abandonment of a customer versus arrival intensity. Not surprisingly after seeing the previous two figures the diference of achieved values are relatively far from each other, disparity is still present among the applied distributions. It can be stated that with these parameters customers are more tend to leave the system from the orbit.
Lastly, on Figure 9 the running parameter (value of x-axis) is the arrival intensity and value of y-axis is the total utilization of the server. Among the lines there are not so significant diferences, they coincide with each other meaning that the utilization is almost the same except in case of Pareto and gamma distribution where the utilization of service unit is significantly less.
In this paper a finite-source retrial queueing system is presented with a non-reliable server, collisions and impatient customers. The obtained results fully demonstrated how pivotal selecting a distribution of impatience of calls because it has a great influence on the system characteristics despite the fact that the mean and the variance are exactly the same. Figure in connection with probability of abandonment clearly assure this phenomenon. Results evidently 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. In the future we would like to deal with more distributions to expand our investigation and examine the performance measures when the distribution of service time is not exponential. We also would like to analyze systems of two-way communication in order to carry out a valid collating.
Acknowledgements. The research work of Ádám Tóth, János Sztrik was supported by the construction EFOP-3.6.3-VEKOP-16-2017-00002. The project was supported by the European Union, co-financed by the European Social Fund.