A Queueing Model with Multiple Types of Server Breakdowns

Abstract

In this paper we consider a general queueing model in which the server may experience several different types of breakdowns. Each type of breakdown requires a finite random number of stages of repair. We obtain a necessary and sufficient condition for the stationary queue length distribution to exist. We then show how to compute the queue length distribution by matrix geometric methods, and we find an explicit expression for its mean. We also find properties of several random variables associated with the model, such as the repair time for a server breakdown, a customer’s completion time, the number of stages of repair required to restore service, and some relationships among these variables and between them and the number of breakdowns that occur during a customer’s completion time. We then discuss some interesting special cases of the model.

Key Words:

• M/PH/1 model
• mean queue length
• queue length distribution
• queue server breakdowns

Additional information

Notes on contributors

William J. Gray

William J. Gray Senior Professor of the Department of Mathematics, University of Alabama. He has published numerous papers in the area of topology, and topological dynamics. His current research interest is the queueing theory.

P. Patrick Wang

P. Patrick Wang Professor of the Department of Mathematics, University of Alabama. His research interests are in the areas of stochastic processes and applications including queueing theory, stochastic modeling and optimization. Recent publications of his have appeared in Computers & Operations Research, European Journal of Operations Research, Journal of Applied Probability and Statistical Science.

Meckinley Scott

MecKinley Scott Professor Emeritus of both the Mathematics Departments of the University of Alabama, and more recently, Western Illinois University. He has served as chair of the department at Western Illinois. His entire research career has been devoted to the study of queueing theory.