Transient solution of an M/G/1 queueing system with group arrivals and vacations to the server

In this paper the authors consider a single server queueing system in which the server takes vacation under the exhaustive service discipline. The arrivals of customers follow a compound Poisson process. The service times and vacation periods are distributed according to general distributions. Using renewal theoretic arguments, the authors derive explicit expressions for the system size probabilities at arbitrary epochs. Also they obtain expressions for the probability distribution of the virtual waiting time in the queue at any time t.