An Mx/G/1 queueing system with a setup period and a vacation period

This paper deals with an M^x/G/1 queueing system with a vacation period which comprises an idle period and a random setup period. The server is turned off each time when the system becomes empty. At this point of time the idle period starts. As soon as a customer or a batch of customers arrive, the setup of the service facility begins which is needed before starting each busy period. In this paper we study the steady state behaviour of the queue size distributions at stationary (random) point of time and at departure point of time. One of our findings is that the departure point queue size distribution is the convolution of the distributions of three independent random variables. Also, we derive analytically explicit expressions for the system state probabilities and some performance measures of this queueing system. Finally, we derive the probability generating function of the additional queue size distribution due to the vacation period as the limiting behaviour of the M^x/M/1 type queueing system.