The optimal batch size of M/DN/1 queue

A single-server, infinite-capacity queue with Poisson input and deterministic bulk service rule is considered. The service time of a batch of a given size is a + b × N, where a is the common service time for all customers in a batch such as a set-up time, b is the individual service time for each customer, and N is the batch size. The queue is analysed using embedded Markov chain techniques. An explicit expression for the mean steady-state waiting time and an estimate of the optimal batch size minimising the mean steady-state waiting time are obtained. Our results are consistent with numerical computations.