Advertisement

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
Article
  • 17 Downloads

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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.

References

  1. 1.
    Berger, A.W., Whitt, W.: The Brownian approximation for rate-control throttles and the G/G/1/C queue. Discret. Event Dyn. Syst. 2, 7–60 (1992)CrossRefGoogle Scholar
  2. 2.
    Bonan, S.S., Clark, D.S.: Estimates of the Hermite and Freud polynomials. J. Approx. Theory 63, 210–224 (1990)CrossRefGoogle Scholar
  3. 3.
    Brigo, D., Mercurio, F.: Interest Rate Models–Theory and Practice (With Smile, Inflation, and Credit), 2nd edn. Springer, Berlin (2006)Google Scholar
  4. 4.
    D’Auria, B., Kella, O.: Markov modulation of a two-sided reflected Brownian motion with application to fluid queues. Stoch. Process. Appl. 122, 1566–1581 (2012)CrossRefGoogle Scholar
  5. 5.
    Diaconis, P.: The Markov chain Monte Carlo revolution. Bull. Am. Math. Soc. 46(2), 179–1205 (2009)CrossRefGoogle Scholar
  6. 6.
    Folland, G .B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley, New York (1999)Google Scholar
  7. 7.
    Glynn, P.W., Wang, R.J.: On the rate of convergence to equilibrium for reflected Brownian motion. Queueing Syst. 89, 165–197 (2018)CrossRefGoogle Scholar
  8. 8.
    Harrison, J.M.: Brownian Models of Performance and Control. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  9. 9.
    Iglehart, D.L.: Limiting diffusion approximations for the many-server queue and the repairman problem. J. Appl. Prob. 2(2), 429–441 (1965)CrossRefGoogle Scholar
  10. 10.
    Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, New York (1981)Google Scholar
  11. 11.
    Krasikov, I.: New bounds on the Hermite polynomials. arXiv:math/0401310 pp. 1–6 (2004)
  12. 12.
    Lachaud, B.: Cut-off and hitting times of a sample of Ornstein–Uhlenbeck processes and its average. J. Appl. Prob. 42(4), 1069–1080 (2005)CrossRefGoogle Scholar
  13. 13.
    Linetsky, V.: On the transition densities for reflected diffusions. Adv. Appl. Prob. 37, 435–460 (2005)CrossRefGoogle Scholar
  14. 14.
    Meyn, S., Tweedie, R.L.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  15. 15.
    Roberts, G.O., Rosenthal, J.S.: General state space Markov chains and MCMC algorithms. Probab. Surv. 1, 20–71 (2004)CrossRefGoogle Scholar
  16. 16.
    Wang, R.J., Glynn, P.W.: Measuring the initial transient: reflected Brownian motion. In: A. Tolk, S.Y. Diallo, I.O. Ryzhov, L. Yilmaz, S. Buckley, and J.A. Miller (eds) Proceedings of the 2014 Winter Simulation Conference, pp. 652–661 (2014)Google Scholar
  17. 17.
    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
  18. 18.
    Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)Google Scholar
  19. 19.
    Williams, R.J.: Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval. J. Appl. Prob. 29, 996–1002 (1992)CrossRefGoogle Scholar
  20. 20.
    Zhang, X., Glynn, P.W.: On the dynamics of a finite buffer queue conditioned on the amount of loss. Queueing Syst. 67(2), 91–110 (2011)CrossRefGoogle Scholar

Copyright information

© 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

Personalised recommendations