A G/M/1 retrial queue with constant retrial rate

Chesoong Kim (), Valentina Klimenok () and Alexander Dudin ()

TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, 2014, vol. 22, issue 2, 509-529

Abstract: In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented. Copyright Sociedad de Estadística e Investigación Operativa 2014

Keywords: Retrial queue; Renewal input; Constant retrial rate; Stationary distribution; 60K25; 68M20 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s11750-012-0267-3

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