Skip to main content

On the rate of beta-mixing and convergence to a stationary distribution in continuous-time Erlang-type systems

  • Communication Network Theory
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

We establish sufficient conditions for polynomial rate of convergence to a stationary distribution and of beta-mixing in continuous-time Erlang-type systems. Our results are a natural complement both to results of Erlang himself, dating back to the beginning of the 20th century, and to exponential estimates established later.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Asmussen, S., Applied Probability and Queues, New York: Springer, 2003, 2nd ed.

    MATH  Google Scholar 

  2. Lindvall, T., Lectures on Coupling Method, New York: Wiley, 1992.

    MATH  Google Scholar 

  3. van Doorn, E.A. and Zeifman, A.I., On the Speed of Convergence to Stationarity of the Erlang Loss System, Queueing Systems, 2009, vol. 63, no. 1–4, pp. 241–252.

    Article  MATH  MathSciNet  Google Scholar 

  4. Erlang, A.K., The Theory of Probabilities and Telephone Conversations, Nyt Tidsskrift for Matematik B, 1909, vol. 20, pp. 33–39.

    Google Scholar 

  5. Veretennikov, A.Yu., On the Mixing Rate and Convergence to the Stationary Distribution in the Discrete Erlang problem, Avtomat. i Telemekh., 2009, no. 12, pp. 59–70 [Remote Control (Engl. Transl.), 2009, vol. 70, no. 12, pp. 1992–2002].

  6. Veretennikov, A.Yu., On Polynomial Mixing and the Rate of Convergence for Stochastic Differential and Difference Equations, Teor. Veroyatnost. i Primenen., 1999, vol. 44, no. 2., pp. 312–327 [Theory Probab. Appl. (Engl. Transl.), 2000, vol. 44, no. 2, pp. 361–374].

    MathSciNet  Google Scholar 

  7. Has’minskii, R.Z., Ustoichivost’ sistem differentsial’nykh uravnenii pri sluchainykh vozmushcheniyakh ikh parametrov, Moscow: Nauka, 1969. Translated under the title Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Alphen aan den Rijn: Sijthoff & Noordhoff, 1980.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. Yu. Veretennikov.

Additional information

Original Russian Text © A.Yu. Veretennikov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 4, pp. 122–129.

Supported in part by the Russian Foundation for Basic Research, project no. 08-01-00105.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Veretennikov, A.Y. On the rate of beta-mixing and convergence to a stationary distribution in continuous-time Erlang-type systems. Probl Inf Transm 46, 382–389 (2010). https://doi.org/10.1134/S0032946010040083

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946010040083

Keywords