Optimal control of batch service queues with finite service capacity and linear holding costs

We consider the optimal control problem of certain batch service queueing systems with compound Poisson arrivals and linear holding costs. The control problem involves the determination of the epochs at which the service is initiated as well as the sizes of the batches served. The service times are assumed to be independent and identically distributed, however, with a general distribution. A quite natural operating policy is to start the service as soon as the number of customers reaches some threshold and serve always as many customers as possible. Assuming infinite service capacity Deb proved that under some mild conditions the optimal operating policy is of this type. In this paper we show that a similar result is valid even if the service capacity is finite. In this case the threshold is never greater than Q, the service capacity (the maximum number of customers that can be served at the same time).