An M-X / G/ 1 Energetic Retrial Queue with Vacations and it's Control

In this note, we present some results about the MX/G/1 retrial queue with vacations. 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 energy with arbitrarily probability distribution. The server is turned off and takes a vacation when the system becomes empty. The random energy required for each vacation is also arbitrary distributed. We derive a stochastically recursive relation which can be used as a discrete-event simulation algorithm for our queue. Next, we give an explicit formula for the generating function of the number of customers in orbit in steady state and exhibit explicit forms of stochastic decomposition property. Finally, we show how to obtain performance measures of interest and optimal control parameters for vacation and retrial policies.