M/G/1//N Queues with Server Vacations and Exhaustive Service

Abstract

Queues with a finite population of customers and occasional periods (called vacations) when the server is unavailable appear in many engineering systems, but have not been studied. In this paper, we specifically consider an M/G/1//N queueing system in which the server takes repeated vacations each time it has emptied the queue, until it finds a customer waiting. For the steady state, we show that performance measures such as the system throughput and mean response time can be obtained from the known analysis of a regenerative cycle of busy and vacation periods. We then study the joint distribution of the server state, the queue size, and the remaining service or vacation time at an arbitrary point in time for certain initial conditions. In the steady state, we explicitly obtain the distributions of the unfinished work, the virtual waiting time, and the real waiting time. The mean response times in several other vacation models are also provided.