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
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 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.