Many models of two-way communication queueing systems have been studied in recent years, they can be utilized in many fields of life like in [7, 28, 30]. Customers have always been characterized by the phenomena of impatience due to the long waiting time for service [4, 14, 15, 27]. In this paper, we consider two-way communication systems with a non-reliable server where primary customers may decide to leave the system after spending a considerable amount of time in the system before getting its proper service. The service unit can break down during its operation or in an idle state, too. Whenever the server becomes idle it may generate requests towards the customers' residing in an infinite source. These requests, the so-called secondary customers, can enter the system after a random time if the service unit is available and functional upon their arrivals. Otherwise, they return to the source without coming into the system. Every primary customer has a property of impatience meaning that an arbitrary request has the ability to
quit the system after some time while its demand remains unsatisfied. During server failure, every individual may generate requests but these will be forwarded immediately towards the orbit. The source, service, retrial, impatience, operation, and repair times are supposed to be independent of each other.
Nowadays due to the rapid technology development and the increase of trafic
growth, the study of communication systems is crucial and inevitable especially
in the topic of optimization. Not just companies or associations have some kind
of networking infrastructure but our homes as well, which results in a load of
communication processes. Therefore, the creation and modelling of new or
existing telecommunication systems are needed. With the help of queuing systems
with repeated calls are perfectly useful for modelling such arising issues in main
telecommunication systems, such as telephone switching systems, call centers, or
computer systems. In the last years, a great number of articles dealt with such
scenarios like in [
The fundamental characteristic of retrial queueing systems is that when every service unit is busy an incoming customer remains in the system in a virtual waiting room called the orbit. The customers who are located in the orbit try to enter the service unit after a random time.
Impatience is a natural phenomenon and models with this property are nearer to
reality and lead us to more precise analysis. Because of the relatively great number
of situations that can occur in healthcare applications, call centers,
telecommunication networks, it is no wonder that many papers were devoted to examining the
efect of impatience like [
In connection with the communication systems users (or sources) typically are
coerced to fight for the available channels or facilities. The possibility of conflict
is relatively high when several sources initiate random attempts on the channel
producing collisions and the loss of transmissions. Developing eficient methods is
important for avoiding such collisions and corresponding message delay. The efect
of collisions have been published in numerous papers for example in [
The examination of the availability of the service unit is vital because a lot of
studies suppose that the service unit never fails and it is accessible on a permanent
basis. But these assumptions are quite unrealistic and in practice, on many
occasions, we find the service stations in failed state. This unfortunate behaviour has
a considerable influence on the system’s characteristics as well on the performance
measures thus it is worth investigating such systems with server breakdowns like
in the following papers [
Recently in a relatively high number of papers make an efort to study two-way
communication designs thanks to their usefulness to model real-life examples in the
various application field. This is especially true for call centers where the service
unit not only processes incoming calls but also operates in idle state executing
certain other works like advertising and promoting products. In such systems
utilization of the service unit is always pivotal, see for example in [
The novelty of the present paper is to achieve a sensitivity analysis using various distributions of service time of customers on the performance measures like the mean waiting time of an arbitrary, successfully served and impatient customer, the total utilization of the service unit, the probability of abandonment, etc. To compare the efect of the diferent distributions on distinct metrics a stochastic simulation program is developed based on SimPack. In this collection, you may perform various algorithms in connection with discrete event simulation, continuous simulation, and combined (multi-model) simulation. So we built a simulation model to implement every feature of the system and to calculate and estimate the desired measure using various values of input parameters. The obtained results demonstrate the importance of utilized distribution under diferent parameter settings represented by numerous figures and highlight some interesting specialties of these types of systems.
In this paper, a two-way communication retrial queuing model is considered with a non-reliable server (Figure 1). In this model, customer populates the finitesource implying that the system will be stable at every moment. Each customer generates a request (primary customers) towards the server with rate / , so the inter-request time follows an exponential distribution with parameter / . Our model does not have a queue at all, thus in the case of an idle server the service of a primary customer starts immediately. The service time of these requests follows gamma, hypo-exponential, hyper-exponential, Pareto, and lognormal distribution with diferent parameters but with the same mean value. After departing from the system, successful requests go back to the finite-source. When the server is occupied, the incoming primary customers are forwarded to the orbit where they may retry to attain the service unit after an exponentially distributed random time with parameter / . It is assumed that server failures take place during its operation or in idle state according to exponential distribution time with the rate 0 when it is busy and with rate 1 if it is idle. After server breakdown the repair process begins right away and the repair time of the server is also an exponentially distributed random variable with parameter 2.
Every primary customer is characterized by an impatience property and in our investigated model two diferent types are distinguished: 1. a primary customer after waiting for some time in the orbit leaves the system without receiving its appropriate service, 2. a primary customer after waiting for some time in the system leaves the system without receiving its appropriate service.
This decision is made after a random time which is exponential with rate . After the server is becoming idle it has the possibility to produce outgoing calls towards the customers (secondary) from an infinite source which is performed after an exponentially distributed period with parameter . The requirement of the service of these customers is that the server will not be in a failed or busy state upon their arrivals, otherwise, they are cancelled and turn back to the infinite source without entering the system. The service time of this type of customer follows a gamma distribution with parameters 2 and 2. Whenever the server breaks down during the service of a customer the primary ones are forwarded immediately towards the orbit and the secondary ones depart the system without continuing their service.
In Table 1 the various values of input parameters are shown for the simulation.
A statistics package is utilized to estimate the desired performance measures in
our program which was developed by Andrea Francini in [
Achieving our objective we use hyper-exponential, gamma, lognormal and Pareto
distributions to investigate how these distributions of service time of primary
customers alter the performance measures. In the first scenario, the squared coeficient
of variation is greater than one. The parameters are selected that the mean and
variance would be identical and to accomplish an accurate comparison a fitting
process has been done. From this process, you can find detailed information in the
following paper [
The mean waiting time is presented in function of arrival intensity of primary customers on Figure 4, 5 and 6. First, I compared the efect of impatience. Interestingly, the results are almost identical between the impatient modes whether we talk about the mean waiting time of an arbitrary, successfully served, or impatient customer. However, pronounced diferences appear when impatience is not present vs. when there it is.
After that we are curious to see how the efect of diferent used distributions develop, so Figure 7, 8 and 9 demonstrates that. Even though the mean and variance are the same, results clearly illustrate the efect of various distributions. The highest values are experienced in the case of Pareto distribution and the lowest in the case of the gamma distribution. With suitable parameter settings, we experience the maximum property characteristic of a finite-source retrial queueing system.
Figure 10 demonstrate how the probability of abandonment of a customer changes with the increment of the arrival intensity. By a 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 the other distributions.
The question arises whether it is true for using other parameter settings for example when the squared coeficient of variation of the service time of primary customers is less than one. We use the same parameters as in the previous section (Table 1), and Table 3 contains the changed parameters of service time of primary customers. Instead of hyper-exponential, we use this time hypo-exponential distribution because the squared coeficient of variation is always less or equal to one. We will show some figures, Figure 11 is connected to the steady-state distribution of the number of customers in the orbit. Analyzing the curves in more detail they completely overlap each other. As regards the shape of the curves they correspond to normal distribution. The mean number of customers is higher in the case of every distribution compared to the previous section.
The Figure 12 and 13 are related to the mean waiting of an arbitrary and impatient customer. As you can see very slight diferences occur in the case of Pareto distribution the values are a little bit higher. With this parameter setting the interesting maximum value of the mean waiting time appears as in the previous section.
Figure 14 demonstrates the probability of abandonment of a customer versus arrival intensity. Not surprisingly after seeing the previous figures the diference of achieved values are relatively close to each other. It can be stated that with these parameters customers are more tend to leave the system from orbit.
We investigated a queueing system of type // 1// with impatient customers in the orbit and in the system and an unreliable server capable of calling in requests from an infinite source in this paper. Simulation has been implemented, it is shown that stationary probability distribution of the number of customers in the orbit and in the system correlates to the Gaussian distribution regardless of the used distribution of service time of the primary customers. Under diferent scenarios, it was displayed when the squared coeficient of variation is greater than one the applied distributions of service time have a significant influence on the performance measures like the mean waiting time of an arbitrary, successfully served and impatient customers or the probability of abandonment even though the mean and variance are equal in the case of every distribution. Results also indicate that there is almost no gap between the obtained values of measures when the squared coeficient of variation is less than one. The authors plan to continue their research work, examining the obtained phenomenon in more detail and expand their model with other features like collisions, outgoing calls toward the customers from the orbit, or carrying out other sensitivity analysis on other random variables.