Abstract
Burke’s theorem is a well-known fundamental result in queueing theory, stating that a stationary M/M/1 queue has a departure process that is identical in law to the arrival process and, moreover, for each time t, the following three random objects are independent: the queue length at time t, the arrival process after t and the departure process before t. Burke’s theorem also holds for a stationary Brownian queue. In particular, it implies that a certain “complicated” functional derived from two independent Brownian motions is also a Brownian motion. The aim of this overview paper is to present an independent complete explanation of this phenomenon.
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Notes
To save parentheses, I decided that minimization takes precedence over addition/subtraction, so \(c\pm a\wedge b\) means \(c \pm (a\wedge b)\).
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Acknowledgments
I would like to thank Sergey Foss, Seva Shneer, and István Gyöngy for inviting me to attend a recent Maxwell Institute workshop and present this overview, and Sergey Foss and Søren Asmussen for suggesting this writeup. Takis Konstantopoulos was supported in part by the Swedish Research Council Grant 2013-4688.
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Konstantopoulos, T. A review of Burke’s theorem for Brownian motion. Queueing Syst 83, 1–12 (2016). https://doi.org/10.1007/s11134-016-9478-x
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DOI: https://doi.org/10.1007/s11134-016-9478-x