The interchangeability of ·/M/1 queues in series

Abstract

A series of queues consists of a number of · /M/1 queues arranged in a series order. Each queue has an infinite waiting room and a single exponential server. The rates of the servers may differ. Initially the system is empty. Customers enter the first queue according to an arbitrary stochastic input process and then pass through the queues in order: a customer leaving the first queue immediately enters the second queue, and so on. We are concerned with the stochastic output process of customer departures from the final queue. We show that the queues are interchangeable, in the sense that the output process has the same distribution for all series arrangements of the queues. The ‘output theorem' for the M/M/1 queue is a corollary of this result.