Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 2, Pages 159-170

GIX/MY/1 systems with resident server and generally distributed arrival and service groups

Alexander Dukhovny

San Francisco State University, Department of Mathematics, San Francisco 94132, CA, USA

Received 1 December 1995; Revised 1 March 1996

Copyright © 1996 Alexander Dukhovny. 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.


Considered are bulk systems of GI/M/1 type in which the server stands by when it is idle, waits for the first group to arrive if the queue is empty, takes customers up to its capacity and is not available when busy. Distributions of arrival group size and server's capacity are not restricted. The queueing process is analyzed via an augmented imbedded Markov chain. In the general case, the generating function of the steady-state probabilities of the chain is found as a solution of a Riemann boundary value problem. This function is proven to be rational when the generating function of the arrival group size is rational, in which case the solution is given in terms of roots of a characteristic equation. A necessary and sufficient condition of ergodicity is proven in the general case. Several special cases are studied in detail: single arrivals, geometric arrivals, bounded arrivals, and an arrival group with a geometric tail.