Queueing Systems

, Volume 89, Issue 1–2, pp 165–197 | Cite as

On the rate of convergence to equilibrium for reflected Brownian motion

  • Peter W. Glynn
  • Rob J. Wang


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.


Reflected Brownian motion Queueing theory Total variation distance Rate of convergence to equilibrium Large deviations Steady-state simulation Diffusion processes 

Mathematics Subject Classification

60F05 60F10 60G05 60J60 60K25 



This paper honors the fundamental contributions of Ward Whitt to the applied probability community, particularly to queueing theory, weak convergence, and diffusion approximations. The authors would also like to thank Vadim Linetsky for a helpful personal communication, regarding how to directly obtain the spectral representation for RBM, and the referee for a careful reading of this paper and for related comments on improving the exposition. Rob J. Wang is grateful to have been supported by an Arvanitidis Stanford Graduate Fellowship in memory of William K. Linvill, the Thomas Ford Fellowship, as well as NSERC Postgraduate Scholarships.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Management Science and EngineeringStanford UniversityStanfordUSA
  2. 2.AirbnbSan FranciscoUSA

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