Threshold-policy analysis of an M/M/1 queue with working vacations

Abstract

In this paper, we consider two M/M/1 queues with working vacations and two policies, $m$-policy and $(m,N)$-policy, respectively. The server begins to take the vacation when the number of customers is below $m$ after a service. The server also works in a slow speed in the vacation rather that stoping work completely. We establish a system with two operation periods, higher speed and lower speed periods. First, we study pure $m$-policy where the server continues another vacation if a vacation is completed and there are less than $m$ customers, otherwise he comes back to regular work. Another $(m,N)$-policy is the generalization of $m$-policy where if a vacation is completed and there are less than $N$ customers, the server continues another vacation. Using the quasi birth–death process and matrix-geometric solution method, we give the distributions for the number of customers and some indices of the system, including expected sojourn time and state probabilities of the server. Finally, some numerical examples are presented to verify the validity of the model.