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On the rate of convergence to equilibrium for reflected Brownian motion

Author

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  • Peter W. Glynn

    (Stanford University)

  • Rob J. Wang

    (Stanford University
    Airbnb)

Abstract

This paper discusses the rate of convergence to equilibrium for one-dimensional reflected Brownian motion with negative drift and lower reflecting boundary at 0. In contrast to prior work on this problem, we focus on studying the rate of convergence for the entire distribution through the total variation norm, rather than just moments of the distribution. In addition, we obtain computable bounds on the total variation distance to equilibrium that can be used to assess the quality of the steady state for queues as an approximation to finite horizon expectations.

Suggested Citation

  • Peter W. Glynn & Rob J. Wang, 2018. "On the rate of convergence to equilibrium for reflected Brownian motion," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 165-197, June.
    • Handle: RePEc:spr:queues:v:89:y:2018:i:1:d:10.1007_s11134-018-9574-1
    DOI: 10.1007/s11134-018-9574-1
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    References listed on IDEAS

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    1. Ward Whitt, 1989. "Planning Queueing Simulations," Management Science, INFORMS, vol. 35(11), pages 1341-1366, November.
    2. Joseph Abate & Ward Whitt, 1994. "Transient Behavior of the M/G/1 Workload Process," Operations Research, INFORMS, vol. 42(4), pages 750-764, August.
    3. Søren Asmussen, 1992. "Queueing Simulation in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 84-111, February.
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    Cited by:

    1. Ward Whitt, 2018. "A broad view of queueing theory through one issue," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 3-14, June.
    2. Peter W. Glynn & Rob J. Wang, 2019. "On the rate of convergence to equilibrium for two-sided reflected Brownian motion and for the Ornstein–Uhlenbeck process," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 1-14, February.
    3. Glynn, Peter W. & Wang, Rob J., 2023. "A heavy-traffic perspective on departure process variability," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    4. Stefan Steinerberger & Aleh Tsyvinski, 2020. "On Vickrey's Income Averaging," Papers 2004.06289, arXiv.org.

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