Time-Dependent Behaviour of an Alternating Service Queue

Abstract

We consider a model describing the waiting time of a server alternating between two service points. This model is described by a Lindley-type equation. We are interested in the time-dependent behaviour of this system and derive explicit expressions for its time-dependent waiting-time distribution, the correlation between waiting times, and the distribution of the cycle length. As our model is closely related to Lindley's recursion, we compare our results to those derived for Lindley's recursion.

Keywords:

• Busy period
• Covariance function
• G/M/1 queue
• Generalized Wiener-Hopf equation
• Lindley's recursion
• Phase-type distributions
• Waiting time

Mathematics Subject Classification:

• 60K25
• 90B22

ACKNOWLEDGMENTS

We are indebted to a referee for the careful reading of the manuscript and for pointing out numerous typos and unclear points. For the completion of this work we gratefully acknowledge the hospitality and support offered by EURANDOM, The Netherlands.