Journal of Applied Mathematics and Stochastic Analysis
Volume 8 (1995), Issue 2, Pages 151-176

A discrete single server queue with Markovian arrivals and phase type group services

Attahiru Sule Alfa,1 K. Laurie Dolhun,1 and S. Chakravarthy2

1University of Manitoba, Department of Mechanical and Industrial Engineering, Manitoba, Winnipeg R3T 5V6, Canada
2GMI Engineering and Management Institute, Department of Science and Mathematics, Flint 48504-4898, MI, USA

Received 1 October 1994; Revised 1 March 1995

Copyright © 1995 Attahiru Sule Alfa et al. 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.


We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.