Waiting Times in the B/G/1 Queue with Server Vacations

We consider a B/G/1 queueing with vacations, where the server closes the gate when it begins a vacation. In this system, customers arrive according to a Bernoulli process. The service time and the vacation time follow discrete distributions. We obtain the distribution of the number of customers at a random point in time, and in turn, the distribution of the residence time (queueing time + service time) for a customer. It is observed that solutions for our discret time B/G/1 gated vacation model are analogous to those for the continuous time M/G/1 gated vacation model.