Bounds for Some Generalizations of the GI / G /1 Queue

K. T. Marshall
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K. T. Marshall: Bell Telephone Laboratories, Incorporated, Holmdel, New Jersey

Operations Research, 1968, vol. 16, issue 4, 841-848

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.

Date: 1968
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