International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 1994 / Article

Open Access

Volume 7 |Article ID 434957 | 18 pages | https://doi.org/10.1155/S1048953394000171

A finite capacity queue with Markovian arrivals and two servers with group services

Received01 Feb 1994
Revised01 May 1994

Abstract

In this paper we consider a finite capacity queuing system in which arrivals are governed by a Markovian arrival process. The system is attended by two exponential servers, who offer services in groups of varying sizes. The service rates may depend on the number of customers in service. Using Markov theory, we study this finite capacity queuing model in detail by obtaining numerically stable expressions for (a) the steady-state queue length densities at arrivals and at arbitrary time points; (b) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. The stationary waiting time distribution is shown to be of phase type when the interarrival times are of phase type. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures are discussed. A conjecture on the nature of the mean waiting time is proposed. Some illustrative numerical examples are presented.

Copyright © 1994 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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