The Relation between Customer and Time Averages in Queues

Daniel P. Heyman and Shaler Stidham
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Daniel P. Heyman: Bell Laboratories, Holmdel, New Jersey
Shaler Stidham: North Carolina State University, Raleigh, North Carolina

Operations Research, 1980, vol. 28, issue 4, 983-994

Abstract: Brumelle has generalized the queueing formula L = λ W to H = λ G , where λ is the arrival rate and H and G are respectively time and customer averages of some queue statistics which have a certain relationship to each other but are otherwise arbitrary. Stidham has developed a simple proof of L = λ W for each sample path, in which the only requirement is that λ and W be finite. In this note it is shown that Stidham's proof applies directly to the more general case of H = λ G , provided λ and G are finite and a simple technical assumption is satisfied. The result is used to obtain time average probabilities in the queue GI / M / c / K . Finally, a counterexample is given to demonstrate that the technical assumption is not superfluous, even in the special case where H and G can be interpreted, respectively, as the time average number of units in the system and the average time spent by a unit in the system, as is the case with both L = λ W and the application to the queue GI / M / c / K .

Date: 1980
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