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Brownian analogues of Burke's theorem

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  • O'Connell, Neil
  • Yor, Marc

Abstract

We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of a (stable, stationary) M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burke's theorem for the Brownian queue was stated and proved by Harrison (Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985). We present several different proofs of this and related results. We also present an analogous result for geometric functionals of Brownian motion. By considering series of queues in tandem, these theorems can be applied to a certain class of directed percolation and directed polymer models. It was recently discovered that there is a connection between this directed percolation model and the GUE random matrix ensemble. We extend and give a direct proof of this connection in the two-dimensional case. In all of the above, reversibility plays a key role.

Suggested Citation

  • O'Connell, Neil & Yor, Marc, 2001. "Brownian analogues of Burke's theorem," Stochastic Processes and their Applications, Elsevier, vol. 96(2), pages 285-304, December.
    • Handle: RePEc:eee:spapps:v:96:y:2001:i:2:p:285-304
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    References listed on IDEAS

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    1. Philip M. Morse, 1955. "Stochastic Properties of Waiting Lines," Operations Research, INFORMS, vol. 3(3), pages 255-261, August.
    2. Carmona, Philippe & Petit, Frédérique & Yor, Marc, 1999. "An identity in law involving reflecting Brownian motion, derived from generalized arc-sine laws for perturbed Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 323-333, February.
    3. Eginhard J. Muth, 1979. "The Reversibility Property of Production Lines," Management Science, INFORMS, vol. 25(2), pages 152-158, February.
    4. A.J. Ganesh, 1998. "Large deviations of the sojourn time for queues in series," Annals of Operations Research, Springer, vol. 79(0), pages 3-26, January.
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    Cited by:

    1. Eric A. Cator & Sergio I. López & Leandro P. R. Pimentel, 2019. "Attractiveness of Brownian queues in tandem," Queueing Systems: Theory and Applications, Springer, vol. 92(1), pages 25-45, June.
    2. Takis Konstantopoulos, 2016. "A review of Burke’s theorem for Brownian motion," Queueing Systems: Theory and Applications, Springer, vol. 83(1), pages 1-12, June.
    3. Hambly, B. M. & Martin, James B. & O'Connell, Neil, 2002. "Concentration results for a Brownian directed percolation problem," Stochastic Processes and their Applications, Elsevier, vol. 102(2), pages 207-220, December.

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