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Attractiveness of Brownian queues in tandem

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Radboud University, Nijmegen, The Netherlands

    Eric A. Cator

  2. Facultad de Ciencias, UNAM, Mexico City, Mexico

    Sergio I. López

  3. Instituto de Matemática, UFRJ, Rio de Janeiro, Brasil

    Leandro P. R. Pimentel

Authors
  1. Eric A. Cator
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  2. Sergio I. López
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  3. Leandro P. R. Pimentel
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Correspondence to Sergio I. López.

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Cator, E.A., López, S.I. & Pimentel, L.P.R. Attractiveness of Brownian queues in tandem. Queueing Syst 92, 25–45 (2019). https://doi.org/10.1007/s11134-019-09609-y

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  • DOI: https://doi.org/10.1007/s11134-019-09609-y

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