Abstract
This study considers a <V, p>-policy Geo/G/1 queue with startup and closedown times. The < V, p>-policy operates as follows. On completion of the vacation, if there are customers in the queue, the server is either to activate with probability p or to stay dormant in the system with probability 1-p for waiting a new arrival; and if no customer presents in the queue, the server obeys classical single vacation policy to stay dormant in the system until one arrival. We give analytic expressions for the stationary system size distributions of the various states of the server, the length distributions of various state periods and the queue waiting time distribution of an arbitrary customer. Furthermore, we also demonstrate the stochastic decomposition property of the system size and waiting time in the queue. Some numerical examples of the mean system sizes and the mean waiting times in the queue with respect to startup/closedown times, startup times, closedown times, and both no startup and no closedown times, are presented. Finally, with the vacation of fixed length time (say T), the long run average cost function per unit time is analytically developed to determine the joint optimal values of T and p at a minimum cost.
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