GSPN analysis of retrial systems with servers breakdowns and repairs

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Abstract

Multiserver retrial systems arise in telecommunication and computer networks areas. It is of basic importance to study performance and reliability of retrial systems with unreliable servers, because of limited ability of repairs and heavy influence of the breakdowns on the performance of the system. However, so far the repairable retrial systems are analyzed only by queueing theory and almost works assumed that service station consists of one single server. In this paper, we give a detailed analysis of finite-source retrial systems with multiple servers subject to random breakdowns and repairs using generalized stochastic petri nets model. We show how this high level model allows us to cope with the complexity of such retrial systems involving the unreliability of the servers, under the different breakdowns disciplines. The main steady-state performance and reliability indices are derived and several numerical calculations were performed to show the effect of servers number, retrial, failure and repair rates on the performability measures of the system.

Introduction

Retrial systems (or systems with repeated calls) arise in telecommunication and computer networks areas. Over the last two decades, there has been a renewed interest on the performance analysis of these systems. That is mainly explained by the advances in telecommunication technology leading to the use of new facilities as “repeat last number”, “ring back when free”, etc.

The main characteristic of retrial systems is that customers who find all servers busy or unavailable upon arrival, join the retrial group (orbit) to try again for their requests in random order and at random intervals. For a comprehensive review of the main results and literature, see for example, the surveys by Yang and Templeton [1], Falin [2], Kulkarni and Liang [3], the book by Falin and Templeton [4] and the more recent state-of-art by Artalejo [5], [6].

Since in practice some components of these systems are subject to random breakdowns, it is of basic importance to study reliability of retrial systems with servers subject to breakdowns and repairs, because of limited ability of repairs, heavy influence of the breakdowns on the performance measures of the system and because more and more applications depend on the correct and timely operation of these systems. Moreover, important users of these types of systems such as telephone, banking and airline companies, often require that suppliers give evidence that their systems can provide a particular quality of service over a certain time span, even in the presence of failures.

Although the reliability study is of great importance, there are only few works that take into consideration retrial phenomenon involving the unreliability of the server, as it can be seen in the classified bibliography of Artalejo [5], [6]. On the other hand, the repairable retrial systems are so far analyzed only by queueing theory. For related literature, interested readers may refer to the works [7], [8], [9], [10] where non-reliable single-server retrial queues were considered. However, for retrial queueing systems with non-reliable multiple servers, we have found only [11], in which the servers are asymmetric (heterogenous). Hence, the analysis of repairable multi-server retrial models are still an interesting topic.

Petri net formalisms are never seen in this area. However, using particularly the generalized stochastic petri nets (GSPNs), it is possible to obtain performance indices either with analytic means or by numerical algorithms. The GSPN formalism [12] has in the past decade shown to be a very effective mathematical model appropriate for describing and analyzing performance of parallel systems that exhibit concurrency and synchronization.

The primary aim of this paper is to give a detailed analysis of finite-source retrial systems with multiple symmetric (identical) servers subject to random breakdowns and repairs using the GSPNs model. We consider homogeneous finite-source systems, because in many practical situations, it is important to take into account the fact that the rate of generation of new primary calls decreases as the number of customers in the system increases. This can be done with the help of finite-source models [13], [14], [15].

In this paper, we show how the behavior of systems with repeated calls, servers breakdowns and repairs, can be intuitively described with a GSPN model, and how several performance and reliability indices can be derived for different breakdowns disciplines.

The proposed method allows to obtain several benefits both for the qualitative and the quantitative analysis of unreliable multiserver retrial systems. In particular, it offers the possibility of using results, methods and tools developed within the GSPNs framework.

This paper is an extension of the work [16] where a reliable single-server retrial system was considered and a generalization of the investigation done in [10] where finite-source non-reliable single server retrial queueing model was analyzed. On the other hand, a new breakdowns discipline that we called the dependent breakdowns discipline is proposed.

The paper is organized as follows. In the next section, we describe the unreliable retrial systems considered. In Section 3, the GSPN models for the different breakdowns disciplines are developed. In Section 4, the main performance and reliability measures are derived. Then, several numerical calculations are carried out using the efficient software tool GreatSPN. The results are graphically displayed. By the help of these tables and figures, we illustrate the effect of servers’ number, retrial, failure and repair rates on the mean response time and other performability measures of the system.

Section snippets

Description of retrial systems with servers subject to breakdowns and repairs

We consider multiserver retrial systems with a finite number K of homogeneous sources of primary calls. This means that the potential customers are identical and their number is finite. Each source is either free, under service or in orbit at any time.

The input stream of primary calls is the so called quasi-random input. The probability that any particular source generates a primary request for service in any interval (t, t + dt) is λdt + o(dt) as dt  0 if the source is free at time t, and zero if

Generalized stochastic petri net models of non-reliable multiserver retrial systems

In this section, we show a method of modeling and analyzing retrial systems with servers subject to breakdowns and repairs, using generalized stochastic petri nets (GSPNs).

A GSPN consists of places (round circles) which can contain tokens, timed and immediate transitions (rectangles and thin lines, respectively) that can fire, consuming tokens from its input places (places from where an arrow-headed arc leads to the transition) and putting tokens to its output places (places where an arc leads

Performance and reliability analysis

We used the GreatSPN package for performance evaluation of unreliable multiserver retrial systems modeled using GSPNs.

The GreatSPN is a software tool for modeling and analysis of parallel systems, based on the Petri Net formalism. It provides a friendly framework to experiment with GSPNs. It implements efficient analysis algorithms to allow its use on real applications, not only toy examples. The strong points of this package are: the use friendliness (in particular the availability of a

Validation of results

The results obtained in the reliable case were validated by the Pascal program given in the book of Falin and Templeton [4], since if the failure rate in non-reliable models is very low and repair rate is very high, the measures should approach the corresponding ones in reliable models.

In Table 1, we can see that the corresponding performance measures for models with active breakdowns and models with dependent breakdowns are very close to the reliable case. In fact, the derived results are the

Numerical results and discussions

In this section, we present some numerical results to investigate the performance and reliability evaluation of retrial systems with unreliable symmetric servers.

First, we illustrate the effect of the system parameters on the mean response time. Since we deal with multiserver retrial systems with breakdowns and repairs, the emphasis we will put on the influence of retrial rate, busy servers failure rate, repair rate and the number of servers on the mean response time. We consider the four

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