Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 2, Pages 161-178

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

S. Chakravarthy1 and Attahiru Sule Alfa2

1GMI Engineering & Management Institute, Department of Science and Mathematics, Flint 48504-4898, MI, USA
2University of Manitoba, Department of Mechanical and Industrial Engineering, Winnipeg R3T 2N2, Canada

Received 1 February 1994; Revised 1 May 1994

Copyright © 1994 S. Chakravarthy and Attahiru Sule Alfa. 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.


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.