We consider a batch service queue which is controlled by switching the server on and off, and by controlling the batch size and timing of services. These batch sizes cannot exceed a fixed number Q, which we call the service capacity. Costs are charged for switching the server on and off, for serving customers and for holding them in the system. Viewing the system as a semi-Markov decision process, we show that the policies which minimize the expected continuously discounted cost and the expected cost per unit time over an infinite time horizon are of the following form: at a review point if the server is off, leave the server off until the number of customers x reaches an optimal level M, then turn the server on and serve min (x, Q) customers; and when the server is on, serve customers in batches of size min(x, Q) until the number of customers falls below an optimal level m(m ≦ M) and then turn the server off. An example for computing these optimal levels is also presented.

A single-server queue with batch arrivals in a non-homogeneous Poisson process and with balking is studied with respect to the busy period, using supplementary variables. A system of integral equations is obtained on the base of which the transforms are expressed in series. For the homogeneous case, assuming finite waiting room, the solutions are obtained via Cramer's rule. This gives asymptotic expressions for the expectations for large arrival intensity. An efficiency measure giving the long run loss probability is given. For a special case contour integral representations are given as solutions.