Burke's theorem

In probability theory, Burke's theorem (sometimes the Burke's output theorem[1]) is a theorem in queueing theory by Paul J. Burke while working at Bell Telephone Laboratories that states for an M/M/1, M/M/m or M/M/∞ queue in the steady state with arrivals a Poisson process with rate parameter λ then:

  1. The departure process is a Poisson process with rate parameter λ.
  2. At time t the number of customers in the queue is independent of the departure process prior to time t.

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]

An analogous theorem for the Brownian queue was proven by J. Michael Harrison.[3][7]

References

  1. ^ Walrand, Jean (November 1983). "A Probabilistic Look at Networks of Quasi-Reversible Queues". IEEE Transactions on Information Theory 29 (6). doi:10.1109/TIT.1983.1056762. 
  2. ^ Burke, Paul J. (December 1956). "The Output of a Queuing System". Operations Research 4 (6): 699–704. doi:10.1287/opre.4.6.699. JSTOR 166919. 
  3. ^ a b O'Connell, N. (December 2001). "Brownian analogues of Burke's theorem". Stochastic Processes and their Applications 96 (2): 285–298. doi:10.1016/S0304-4149(01)00119-3.  edit
  4. ^ O'Brien, G. G. (September 1954). "The Solution of Some Queueing Problems". Journal of the Society for Industrial and Applied Mathematics 2 (3): 133–142. JSTOR 2098899.  edit
  5. ^ Morse, P. M. (August 1955). "Stochastic Properties of Waiting Lines". Journal of the Operations Research Society of America 3 (3): 255–261. JSTOR 166559.  edit
  6. ^ Reich, E. (1957). "Waiting Times when Queues are in Tandem". The Annals of Mathematical Statistics 28 (3): 768–773. doi:10.1214/aoms/1177706889.  edit
  7. ^ Harrison, J. Michael (1985). Brownian Motion and Stochastic Flow Systems. New York: Wiley. http://faculty-gsb.stanford.edu/harrison/Documents/BrownianMotion-Stochasticms.pdf. 

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