Attractiveness of Brownian queues in tandem
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Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one-sided reflection.
KeywordsBrownian queue Tandem queues Last-passage percolation Exclusion process
Mathematics Subject Classification60K25 60K35
The authors would like to thank an anonymous referee for her helpful comments that greatly improved the presentation and clarity of this work.
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