M/G/1/N queue with vacation time

Abstract

This paper studies an M/G/1 queueing system with a finite waiting room and with server vacation times consisting of periods of time that the server is away from the queue doing additional work. Service at the queue is exhaustive, in that a busy period at the queue ends only when the queue is empty. At each termination of a busy period, the server takes an independent vacation. In the case that the system is still empty upon return from vacation the server waits for the first customer to arrive when an ordinary M/G/1 busy period starts. The queue length process is studied using the embedded Markov chain. Using a combination of the supplementary variable and sample biasing techniques, we derive the general queue length distribution of the time continuous process, as well as the blocking probability of the system, due to the finite waiting room in the queue. We also obtain the busy period and waiting time distributions. The results for this model are compared with the those for queueing system with vacation studied by Tony T. Lee, which has different vacation policy; if the server finds the system empty at the end of a vacation, he immediately takes another vacation.