Analysis of M/G/1 queues with delayed vacations and exhaustive service discipline

We consider M/G/1 queues with exhaustive service and delayed vacations, where at the end of every busy period the server stays idle in the system for a period of time called changeover time and then follows a mixed vacation policy from a given vacation policy set if there is no arrival during the changeover time. The successive vacations are assumed to be independent but not necessarily identical. This vacation policy includes many existing vacation policies as special cases. We derive the Laplace transform of the joint distribution of the queue length and the remaining service (or vacation) time at arbitrary time as well as that of the virtual waiting time distribution. It results in the stochastic decomposition property immediately. We also provide an explicit formula for the Laplace transform of the additional delay for our model. It enables us to analyze and unify many new and existing vacation policies easily.