Queueing Systems

, Volume 91, Issue 1–2, pp 1–14 | Cite as

On the rate of convergence to equilibrium for two-sided reflected Brownian motion and for the Ornstein–Uhlenbeck process

  • Peter W. Glynn
  • Rob J. WangEmail author


This paper studies the rate of convergence to equilibrium for two diffusion models that arise naturally in the queueing context: two-sided reflected Brownian motion and the Ornstein–Uhlenbeck process. Specifically, we develop exact asymptotics and upper bounds on total variation distance to equilibrium, which can be used to assess the quality of the steady state as an approximation to finite-horizon performance quantities. Our analysis relies upon the simple spectral structure that these two processes possess, thereby explaining why the convergence rate is “pure exponential,” in contrast to the more complex convergence exhibited by one-sided reflected Brownian motion.


Two-sided reflected Brownian motion Ornstein–Uhlenbeck process Queueing theory Total variation distance Rate of convergence to equilibrium 

Mathematics Subject Classification

60F05 60F10 60G05 60J60 60K25 



The authors would like to thank 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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