A DISCRETE-TIME Geo/G/1 RETRIAL QUEUE WITH SERVER BREAKDOWNS

Abstract

This paper discusses a discrete-time Geo/G/1 retrial queue with the server subject to breakdowns and repairs. The customer just being served before server breakdown completes his remaining service when the server is fixed. The server lifetimes are assumed to be geometrical and the server repair times are arbitrarily distributed. We study the Markov chain underlying the considered queueing system and present its stability condition as well as some performance measures of the system in steady-state. Then, we derive a stochastic decomposition law and as an application we give bounds for the proximity between the steady-state distributions of our system and the corresponding system without retrials. Also, we introduce the concept of generalized service time and develop a recursive procedure to obtain the steady-state distributions of the orbit and system size. Finally, we prove the convergence to the continuous-time counterpart and show some numerical results.

The M/G/1 retrial queue: New descriptors of the customer’s behavior

“…In addition, the efficiency of the approximations can be measured in terms of the convergence of the approximate expected values to the true expectations given by formulas (4)- (8). In this sense, in Table 4 we supplement the results in Table 3 ]) can be easily derived by multiplying y b k j (r ) (respectively P{R b = r } and x b i (r )) by r and adding from r = 0 to ∞.…”

“…This analysis of the customer's behavior provides qualitative insight in the characteristics of the retrial group. The methodology can be extended to more general queuing models combining retrials and other queuing phenomena: unreliable queues [8,14,[18][19][20], discrete-time models [8,20], etc. The consideration of the new descriptors in a time dependent context [16] could be the subject matter of a forthcoming study.…”

“…We also remark the paper by Artalejo and Falin [3] where two structural characteristics of the orbit, namely, the orbit busy period and the orbit idle period, were investigated. To illustrate the active role of the retrial queues over the last few years, we mention a recent paper published in this journal [21] as well as some other recent publications [1,[8][9][10][11][12][14][15][16][18][19][20].…”

A discrete-time retrial queue with negative customers and unreliable server

“…Recently, numerous papers, for instance, Atencia and Moreno (2004a), Atencia and Moreno (2004b), Atencia and Moreno(2006), Moreno (2006) investigated the stochastic decomposition property of the discrete-time retrial queueing systems. In this section we analyze the stochastic decomposition law on discrete-time retrial G-queues.…”

A discrete-time retrial queueing system with starting failures, Bernoulli feedback and general retrial times

“…In some applications, arrivals, service times and other queueing activities, if any, are synchronized, making discrete time models more appropriated. However, in spite of the importance of the discrete-time retrial queues, the related literature is scarce (Alfa, 2006;Artalejo, Atencia, & Moreno, 2005;Atencia & Moreno, 2004a, 2004b, 2006a, 2006bChoi & Kim, 1997;Li & Yang, 1998Moreno, 2006;Takahashi, Osawa, & Fujisawa, 1999& Yang & Li, 1995.…”