On the rate of convergence to equilibrium for reflected Brownian motion
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.
KeywordsReflected Brownian motion Queueing theory Total variation distance Rate of convergence to equilibrium Large deviations Steady-state simulation Diffusion processes
Mathematics Subject Classification60F05 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.
- 5.Artin, E.: The Gamma Function. Holt, Rinchart, and Winston Inc, New York (1964)Google Scholar
- 9.Chung, K.L.: A Course in Probability Theory, 3rd edn. Academic Press, San Diego (2001)Google Scholar
- 10.Cohen, J.W.: The Single Server Queue, 2nd. Revised edn. Elsevier, Amsterdam (1982)Google Scholar
- 13.Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence, 2nd edn. Wiley, New York (2005)Google Scholar
- 14.Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley, New York (1999)Google Scholar
- 15.Glynn, P.W., Meyn, S.P.: A Liapounov bound for solutions of the Poisson equation. Ann. Probab. 2(2), 916–931 (1996)Google Scholar
- 16.Glynn, P.W., Wang, R.J.: On the rate of convergence to equilibrium for two-sided reflected Brownian motion and for the Ornstein–Uhlenbeck process, pp. 1–10 (2018) (Submitted for Publication)Google Scholar
- 18.Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes, 3rd edn. Oxford University Press, Oxford (2001)Google Scholar
- 27.Wang, R.J., Glynn, P.W.: Measuring the initial transient: reflected Brownian motion. In: Tolk, A., Diallo, S.Y., Ryzhov, I.O., Yilmaz, L., Buckley, S., Miller, J.A. (eds.) Proceedings of the 2014 Winter Simulation Conference, pp. 652–661 (2014)Google Scholar
- 28.Wang, R.J., Glynn, P.W.: On the marginal standard error rule and the testing of initial transient deletion methods. ACM Trans. Model. Comput. Simul. 27(1), 1–30 (2016)Google Scholar