A General Class of Bulk Queues with Poisson Input

We assume that customers arrive at a counter according to a homogeneous Poisson process and are served in groups, according to the following policy: If there are less than L customers waiting at the time of a departure, the server must wait until there are L customers present, whereupon he serves them together. If there are L or more, but less than K(K ? L) customers waiting, all are served together. If there are K or more customers waiting, a group of K customers are served and the others must wait. The service times of successive groups are assumed to be conditionally independent given the bulk sizes, but may depend on their magnitude. We obtain 1. a description of the output process, 2. the queue length in discrete time, 3. the distribution of the busy period, 4. the queue length in continuous time and 5. some limit theorems for the number of customers served over a long period of time. The order of service is irrelevant in this paper. The method used throughout is that of the imbedded semi-Markov process.