The Relation between Customer and Time Averages in Queues

Abstract

Brumelle has generalized the queueing formula L = λW to H = λG, 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 = λW 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 = λG, 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 = λW and the application to the queue GI/M/c/K.