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.
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Abate, J., Whitt, W.: Transient behavior of regulated Brownian motion, I: starting at the origin. Adv. Appl. Probab. 19(3), 560–598 (1987a)
Abate, J., Whitt, W.: Transient behavior of regulated Brownian motion, II: non-zero initial conditions. Adv. Appl. Probab. 19(3), 599–631 (1987b)
Abate, J., Whitt, W.: Transient behavior of the M/M/1 queue: starting at the origin. Queueing Syst. 2, 41–65 (1987c)
Abate, J., Whitt, W.: Transient behavior of the M/G/1 workload process. Oper. Res. 42(4), 750–764 (1994)
Artin, E.: The Gamma Function. Holt, Rinchart, and Winston Inc, New York (1964)
Asmussen, S.: Queueing simulation in heavy traffic. Math. Oper. Res. 17(1), 84–111 (1992)
Asmussen, S., Glynn, P.W., Thorisson, H.: Stationarity detection in the initial transient problem. ACM Trans. Model. Comput. Simul. 2(2), 130–157 (1992)
Borovkov, A.A.: Some limit theorems in the theory of mass service II. Theor. Probab. Appl. 10, 375–400 (1965)
Chung, K.L.: A Course in Probability Theory, 3rd edn. Academic Press, San Diego (2001)
Cohen, J.W.: The Single Server Queue, 2nd. Revised edn. Elsevier, Amsterdam (1982)
Diaconis, P.: The Markov chain Monte Carlo revolution. Bull. Am. Math. Soc. 46(2), 179–1205 (2009)
Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1(1), 36–61 (1991)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence, 2nd edn. Wiley, New York (2005)
Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley, New York (1999)
Glynn, P.W., Meyn, S.P.: A Liapounov bound for solutions of the Poisson equation. Ann. Probab. 2(2), 916–931 (1996)
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)
Grassmann, W.K.: Factors affecting warm-up periods in discrete event simulation. Simulation 90(1), 11–23 (2014)
Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes, 3rd edn. Oxford University Press, Oxford (2001)
Harrison, J.M.: Brownian Models of Performance and Control. Cambridge University Press, Cambridge (2013)
Iglehart, D.L., Whitt, W.: Multiple channel queues in heavy traffic. I. Adv. Appl. Probab. 2(1), 150–177 (1970)
Kingman, J.F.C.: The single server queue in heavy traffic. Proc. Camb. Philos. Soc. 57, 902–904 (1961)
Linetsky, V.: On the transition densities for reflected diffusions. Adv. Appl. Probab. 37, 435–460 (2005)
Meyn, S., Tweedie, R.L.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge (2009)
Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25(3), 518–548 (1993)
Roberts, G.O., Rosenthal, J.S.: General state space Markov chains and MCMC algorithms. Probab. Surv. 1, 20–71 (2004)
Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000)
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)
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)
Whitt, W.: Planning queueing simulations. Manag. Sci. 35(11), 1341–1366 (1989)
Acknowledgements
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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Glynn, P.W., Wang, R.J. On the rate of convergence to equilibrium for reflected Brownian motion. Queueing Syst 89, 165–197 (2018). https://doi.org/10.1007/s11134-018-9574-1
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DOI: https://doi.org/10.1007/s11134-018-9574-1
Keywords
- Reflected Brownian motion
- Queueing theory
- Total variation distance
- Rate of convergence to equilibrium
- Large deviations
- Steady-state simulation
- Diffusion processes