The interchangeability of $\cdot/M/1$ queues in series

R.R. Weber, J. Appl. Prob. 16(9) 690-695, 1979.


A series of queues consists of a number of $\cdot/M/1$ 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 queueing 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.

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