The Unreliable M/M/1 Retrial Queue in a Random Environment

Abstract

We examine an M/M/1 retrial queue with an unreliable server whose arrival, service, failure, repair, and retrial rates are all modulated by an exogenous random environment. Provided are conditions for stability, the (approximate) orbit size distribution, and mean queueing performance measures which are obtained via matrix-analytic methods. Additionally, we consider the problem of choosing arrival and service rates for each environment state with the objective of minimizing the steady state mean time spent in orbit by an arbitrary customer, subject to cost and revenue constraints. Two numerical examples illustrate the main results.

Keywords:

• LDQBD process
• Retrial queue
• Unreliable server

Mathematics Subject Classification:

• 60K25
• 90B22

ACKNOWLEDGMENTS

We thank Dr. Srinivas Chakravarthy and two anonymous referees for their helpful comments. This research was sponsored in part by a grant from the U.S. Air Force Office of Scientific Research (FA9550-08-1-0004). The views expressed in this paper are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

This article is part of the Special Issue in honor of Marcel F. Neuts, published in volume 27(4) of Stochastic Models.