Bounds for Some Generalizations of the GI/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 GIm/G/1 queue). Such a queue arises when one considers a class of optimization problems in single server queues.

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