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Queueing Systems

, Volume 67, Issue 2, pp 91–110 | Cite as

On the dynamics of a finite buffer queue conditioned on the amount of loss

  • Xiaowei Zhang
  • Peter W. Glynn
Article

Abstract

This paper is concerned with computing large-deviation asymptotics for the loss process in a stylized queueing model that is fed by a Brownian input process. In addition, the dynamics of the queue, conditional on such a large deviation in the loss, is calculated. Finally, the paper computes the quasi-stationary distribution of the system and the corresponding dynamics, conditional on no loss occurring.

Keywords

Large deviations Reflected Brownian motion Local time Quasi-stationary 

Mathematics Subject Classification (2000)

60F10 60J55 60J65 60K25 60F05 34L99 90B15 

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References

  1. 1.
    Anantharam, V.: How large delays build up in a GI/G/1 queue. Queueing Syst., Theory Appl. 5, 345–368 (1989) CrossRefGoogle Scholar
  2. 2.
    Asmussen, S.: Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the G/G/1 queue. Adv. Appl. Probab. 14, 143–170 (1982) CrossRefGoogle Scholar
  3. 3.
    Asmussen, S., Pihlsgård, M.: Loss rates for Lévy processes with two reflecting barriers. Math. Oper. Res. 32, 308–321 (2007) CrossRefGoogle Scholar
  4. 4.
    Berger, A.W., Whitt, W.: The Brownian approximation for rate-control throttles and the G/G/1/C Queue. Discrete Event Dyn. Syst. Theory Appl. 2, 7–60 (1992) CrossRefGoogle Scholar
  5. 5.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998) Google Scholar
  6. 6.
    Duffield, N.G., O’Connell, N.: Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Camb. Philos. Soc. 118, 363–374 (1995) CrossRefGoogle Scholar
  7. 7.
    Echeverria, P.E.: A criterion for invariant measure of Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 61, 1–16 (1982) CrossRefGoogle Scholar
  8. 8.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986) Google Scholar
  9. 9.
    Glynn, P.W., Meyn, S.P.: A Lyapunov bound for solutions of the Poisson equation. Ann. Probab. 24, 916–931 (1996) CrossRefGoogle Scholar
  10. 10.
    Glynn, P.W., Thorisson, H.: Two-sided taboo limits for Markov processes and associated perfect simulation. Stoch. Process. Appl. 91, 1–20 (2001) CrossRefGoogle Scholar
  11. 11.
    Glynn, P.W., Whitt, W.: Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. J. Appl. Probab. 31A, 131–156 (1994) CrossRefGoogle Scholar
  12. 12.
    Harrison, J.M.: Brownian Motion and Stochastic Flow Systems. Krieger, Melbourne (1985) Google Scholar
  13. 13.
    Kuczek, T., Crank, K.N.: A large-deviation result for regenerative processes. J. Theor. Probab., 4, 551–561 (1991) CrossRefGoogle Scholar
  14. 14.
    Lalley, S.P.: Limit theorems for first-passage times in linear and non-linear renewal theory. Adv. Appl. Probab. 16, 766–803 (1984) CrossRefGoogle Scholar
  15. 15.
    Meyn, S.P., Tweedie, R.L.: Stability for Markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Probab. 25, 487–517 (1993) CrossRefGoogle Scholar
  16. 16.
    Sweet, A.L., Hardin, J.C.: Solutions for some diffusion processes with two barriers. J. Appl. Probab. 7, 423–431 (1970) CrossRefGoogle Scholar
  17. 17.
    Williams, R.: Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval. J. Appl. Probab. 29, 996–1002 (1992) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Management Science and EngineeringStanford UniversityStanfordUSA

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