Bounds for Some Generalizations of theGI/G/1 Queue

Abstract

Expressions are derived for the expected number and expected wait in queue for the following generalizations of the GI / G /1 queue: (1) arrivals in batches of random size, (2) service in batches of fixed size, (3) queues with added delay for the first customer in a busy period. These expressions contain the mean and variance of the idle time distribution. Results on the idle period moments, which were derived by the author in a previous paper, are used to obtain bounds for the expected wait and expected number in queue under two different assumptions on A ( t ), the interarrival time distribution. As an example of (3) we mention the queue where the server, once he becomes idle, does not restart until m customers have arrived (called the GI m / G /1 queue). Such a queue arises when one considers a class of optimization problems in single server queues.