A Simple and Complete Solution to the Stationary Queue-Length Probabilities of a Bulk-Arrival Bulk-Service Queue
A complete solution for the stationary queue-length distribution of a bulk-arrival, bulk-service (GI(X)/M(Y)/1) queue is presented. Beginning with a known expression for the probability generating function of the stationary pie-arrival-epoch queue-length distribution, the roots method is used to invert it and determine all probabilities Next, using level crossing arguments. theoretical relationships between pre-arrival and arbitrary-epoch probabilities are developed These relationships are then used to directly determine a complete set of probabilities for the arbitrary-epoch queue-length distribution Finally. selected examples are presented These demonstrate how. given arbitrary arrival time, arrival group size and service batch size probability distributions, a complete solution for the stationary queue-length probabilities can be readily determined.
queue-length roots method level crossing queueing theory bulk queues operations research