Burke's theorem

In queueing theory, a discipline within the mathematical theory of probability, Burke's theorem (sometimes the Burke's output theorem[1]) is a theorem (stated and demonstrated by Paul J. Burke while working at Bell Telephone Laboratories) asserting that, for the M/M/1 queue, M/M/c queue or M/M/∞ queue in the steady state with arrivals is a Poisson process with rate parameter λ: Burke first published this theorem along with a proof in 1956.

[2] The theorem was anticipated but not proved by O’Brien (1954) and Morse (1955).

[3][4][5] A second proof of the theorem follows from a more general result published by Reich.

[6] The proof offered by Burke shows that the time intervals between successive departures are independently and exponentially distributed with parameter equal to the arrival rate parameter, from which the result follows.

[8] An analogous theorem for the Brownian queue was proven by J. Michael Harrison.