Analysis of Departure Process in Batch Arrival Queue with Multiple Vacations and Exhaustive Service

A batch arrival single-server queueing system of the MX/G/1 type is considered. At the end of each busy period the server begins a multiple vacation period when the service process is stopped. A new method of the study of departure process in such a system in transient state is proposed. Using the formula of total probability, we direct the analysis to that in the system without vacations. General results are obtained using the renewal-theory approach. An explicit representation for the probability generating function of Laplace transform of departure process is derived. The formula is written down by means of transforms of distributions describing the arrival and service processes, vacation times, and components of certain factorization identity of Wiener-Hopf type connected with these distributions.