Departures from Many Queues in Series

Abstract : This paper considers a queueing model that could be used to represent the start-up behavior of a long production line or the transient flow of messages over a long path in a communication network. In particular, we consider a series of n single-server queues, each with unlimited waiting space and the first-in first-out service discipline. Initially, the system is empty; then k sub n customers are placed in the first queue. The service times of all the customers at all queuses are i.i.d. with a general distribution having mean 1 and finite positive variance delta squared. Our object is to describe the departure process from the n to the th power queue as n gets large. (Equivalently, since customers are served in order of arrival, we can consider infinitely many queues in series with infinitely many customers in the first queue; we are still interested in the departure times of the first k sub n customers from the n to the th power queue as n yields infinity.) We may have k sub n constant, independent of n, or k sub n yields infinity as n yields infinity.

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