I remember a question asked of me in college. I believe it was something like this: Imagine a toll booth. Suppose people arrive at a rate of an average of 1 per minute, and suppose that it takes exactly 1 minute to process each person at the toll booth. After an extended amount of time, what is the length of the line? Have you heard this question, is the answer an infinite line?

Well Frank your memory is accurate. If fact, if you make some quite standard assumptions about the distributions of arrival and service times, and x represents the average arrival rate and y the average service rate then the expected length of the queue is x/(y-x). So, when y and x are the same, the expected length of the queue is infinite.

A reference is

Christos G. Cassandras, Discrete Event Systems, Irwin, 1993, pp. 349 356.


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