The Gated-Service M/GI/1 Queue with Single Vacations and Its Application to Batch-Service Queues

In this paper, we consider the M/GI/1 queue with single vacations under the gated service discipline. We obtain the probability generating function of the stationary queue length, the Laplace-Stieltjes transform of the system delay distribution in steady state, and the joint transform of the busy cycle length and the number of customers served in the busy cycle. Furthermore, as an application, we consider a batch-service M/G/1 queue, where service times depend on the number of customers in batch.