Light tailed asymptotics in an unreliable $M/G/1$ retrial queue

Abstract

Abstract. We consider the standard unreliable M/G/1 retrial queuing system with active and passive breakdowns. The explicit expressions of the probability generating functions of distribution of server state and orbit size are well known from early works. However, some problems particularly related to Cybernetic and Artificial Intellect need to save computational effort. So, we give here another look to solve this problem more simply, but under some light tailed assumptions. IntroductionQueueing Systems with breakdowns have been the subject of many investigations since they are interesting modeling tools in many modern systems: production, and telecommunication, computer science, finance and so on. Such studies are interesting since they create a gap between Queueing Theory and Reliability theory in the context of probabilistic framework, which is one of the logics for uncertainty problems [7], [6].In recent years, particular attention has been devoted to Retrial Queueing Systems for their interest in many modern systems (call centers, mobile systems, CSMA/CD protocol with star topology, random access protocols. . . ); see, for example, [11] and the references in the didactical book [9], or bibliographical surveys of [3,4].Early works were concerned with analytical or algorithmic resolution of the underlying problems; see, for example, [6,5]. Asymptotic methods are interesting in the sense that we can save computational effort for the resolution of the problem under study; see [2]. We refer also to some alternative approaches to study more qualitative properties such as stability (see [10]) or monotonicity and comparability methods; see [12]. The last paper gives also a short survey about recent advances in Unreliable Retrial Queues.In this paper, we consider the approximation of the queue size of the unreliable M/G/1 retrial queue under light tailed assumptions. In the next section we describe the mathematical model and in section 3 we provide some preliminary results. The light tailed approximation is given in section 4.2010 Mathematics Subject Classification. Primary 60K25; Secondary 68M20, 90B22.