A review of Burke’s theorem for Brownian motion

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.