A Finite Source Retrial Queue: Number of Retrials

Abstract

We consider a single-server queuing system with a finite number of sources, where customers are not allowed to queue; instead of that, they make repeated attempts, or retrials, in order to enter service after some time. This queuing system and its variants are widely used to model disk memory systems, star-like local area networks, and other communication systems. The article extends previous works on this topic and deals with the number of retrials, produced by a tagged customer, until he finds the server available. An algorithm for determination of all moments of this number is obtained, and the results of numerical experiments are presented.

Asymptotic Waiting Time Analysis of a Finite-Source M/M/1 Retrial Queueing System

“…In this paper, we will investigate the distribution of the number of retrials of a customer together with the distribution of the waiting time of a request in a RQ-system since they are connected to each other. Related results can be found, for example in Alfa and Isotupa [1], Dragieva [10], Dragieva [11], Falin and Artalejo [13], Gharbi and Dutheillet [14], Kvach and Nazarov [20], Nazarov, Kvach and Yampolsky [25], Nazarov and Sudyko [22], Wang, Zhao and Zhang [29], Wang, Zhao and Zhang [30], Zhang and Wang [32]. We will use the method of asymptotic analysis under limiting condition of a growing number of sources as it has been applied, for example in Kvach and Nazarov [20], Nazarov, Kvach and Yampolsky [25], Nazarov and Moiseeva [22].…”

“…Let us denote the prelimit probability distribution by π(r) = P {R res = r} and since it is an essential contribution to the comparison, we show how it can be obtained by numerical methods. In Dragieva [10], a similar problem was treated by a rather complicated way that is, we did not want to use that method. Instead, we propose our own way which is also standard but has not been used frequently since the investigation of a number of retrials is not so popular due to its complexity.…”

“…Although the state space is finite in the case of prelimit situation, that is when N is finite, an exact analysis of the steady-state waiting time distribution, see Falin and Artalejo [13] and the distribution of the number of retrials, see Dragieva [10] is very complicated, for that reason, our main contribution is to propose an asymptotic analysis of these measures by sending the number of sources N to infinity and using of course some proper form of scaling or normalizing. The main advantage of the proposed method is that it directly works on the asymptotic solution to the system of balance equations without deriving explicit expressions of the exact generating and characteristic functions.…”

Steady state analysis of the M/G/1//N queue with orbit of blocked customers

“…a single server finite retrial queue, the failed customers, instead of being blocked, repeat their attempts in exponentially distributed time intervals. The model is useful for performance analysis of many real queueing systems and has been extensively studied in a number of papers (Amador 2010;De Kok 1984;Dragieva 2013;Falin and Artalejo 1998;Ohmura and Takahashi 1985).…”

Inventory Control in Retrial Service Facility System – Semi Markov Decision Process

“…For example, Elcan [8], Arivudainambi et al [1], Dragieva [6], Dudin et al [7] and Artalejo et al [3,5] discussed a single server retrial queue with returning customers examined by balking or Bernoulli vacations and derived the analysis part and solution technique using Matrix method or generating function or Truncation method using level dependent quasi-birth-and-death process (LDQBD).…”