Some time-dependent queuing problems with batch arrivals and departures

In this thesis a queue with infinitely many states, compound Poisson arrivals, bulk service, and batch departures is investigated. It is shown that the queue-length probabilities satisfy an infinite system of differential-difference equations (the \birth and death equations\") which are solved in various special cases. A closely allied system of equations is found for the probability distribution function (which may be defective) of the server's busy period. In chapter 1 the queue is specified in terms of the arrival process the service discipline and the departure process. The birth and death equations are then derived in their most general form. In chapter 2 queues with batch arrivals and departures are investigated. This particular case arises by taking the general queue and making the arrival and departure processes independent of the state of the queue. It is here that most of the original work appears as the finite-time behaviour of queues with batch departures does not seem to have been studied in the literature. Chapter 3 embodies an exposition of two papers each by Karlin and McGregor which the author has studied in detail. In this case the arrival and departure processes depend upon the state of the queue. In the final chapter the above special cases are considered \\({t ‚Äövúv¿ ‚Äöv†vª}\\) expressions for the probability distribution of a customer's waiting time are also found."