An MX/G/1 energetic retrial queue with vacations and control

This paper deals with analysis and control of an M^X/G/1 retrial queue with vacations where batches of customers arrive according to a compound Poisson process. Retrial times are governed by an arbitrary probability law which is independent of the number of customers in the retrial group. We consider an energetic interpretation in the sense that the service of a customer requires not only a random time but also a random amount of work (or energy) with an arbitrary probability distribution. The server is turned off and takes a vacation when the system becomes empty. The random work required for each vacation is also arbitrarily distributed. We show that the evolution of the queue can be described at some regenerative points by a stochastically recursive relation. This gives a discrete event simulation algorithm for our queue for which we derive an ergodicity condition. We obtain an explicit formula for the generating function of the number of customers in orbit in steady state and demonstrate a stochastic decomposition property. Finally, we obtain performance measures and consider optimal control of vacation and retrial policies.