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Teor. Veroyatnost. i Primenen., 2009, Volume 54, Issue 1, Pages 18–38 (Mi tvp2497)  

This article is cited in 25 scientific papers (total in 25 papers)

Bounds and Asymptotics for the Rate of Convergence of Birth-Death Processes

E. van Doorna, A. I. Zeifmanb, T. L. Panfilovac

a University of Twente
b Vologda State Pedagogical University
c Ryazan State Pedagogical University

Abstract: The first part of the paper is a review; it describes the proposed approach and gives the general basis of the method constructed by one of the authors in the 1990's in order to obtain estimates and explicit representations for the rates of convergence for birth-death processes. The second part of the paper presents new results obtained with the described method, which has been applied to specific classes of birth-death processes related to mean-field models and the $M/M/N/N+R$ queueing system related to the asymptotic behavior of the rate of convergence in the case when the number of states of the process tends to infinity.

Keywords: rate of convergence, birth-death processes, mean-field models, Charlier polynomial, queueing system

DOI: https://doi.org/10.4213/tvp2497

Full text: PDF file (223 kB)
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English version:
Theory of Probability and its Applications, 2010, 54:1, 97–113

Bibliographic databases:

Received: 17.09.2008

Citation: E. van Doorn, A. I. Zeifman, T. L. Panfilova, “Bounds and Asymptotics for the Rate of Convergence of Birth-Death Processes”, Teor. Veroyatnost. i Primenen., 54:1 (2009), 18–38; Theory Probab. Appl., 54:1 (2010), 97–113

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. I. Zeifman, “On the nonstationary Erlang loss model”, Autom. Remote Control, 70:12 (2009), 2003–2012  mathnet  crossref  mathscinet  zmath  isi
    2. van Doorn E.A., Zeifman A.I., “On the speed of convergence to stationarity of the Erlang loss system”, Queueing Syst., 63:1-4 (2009), 241–252  crossref  mathscinet  zmath  isi  scopus
    3. E. B. Yarovaya, “Models of branching walks and their use in the reliability theory”, Autom. Remote Control, 71:7 (2010), 1308–1324  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. van Doorn E.A., “Rate of convergence to stationarity of the system system $M/M/N/N+R$”, TOP, 19:2 (2011), 336–350  crossref  mathscinet  zmath  isi  scopus
    5. A. I. Zeifman, A. V. Korotysheva, T. L. Panfilova, S. Ya. Shorgin, “Otsenki ustoichivosti dlya nekotorykh sistem obsluzhivaniya s katastrofami”, Inform. i ee primen., 5:3 (2011), 27–33  mathnet
    6. Ya. A. Satin, A. I. Zeifman, A. V. Korotysheva, “Convergence rate and truncations for one class of Markov queueing systems”, Theory Probab. Appl., 57:3 (2013), 529–539  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. A. I. Zeifman, A. V. Korotysheva, Ya. A. Satin, S. Ya. Shorgin, “Otsenki v nul-ergodicheskom sluchae dlya nekotorykh sistem obsluzhivaniya”, Inform. i ee primen., 6:4 (2012), 27–33  mathnet
    8. Zeifman A. Korotysheva A., “Perturbation bounds for $M_t/M_t/N$ queue with catastrophes”, Stoch. Models, 28:1 (2012), 49–62  crossref  mathscinet  zmath  isi  elib  scopus
    9. Gamarnik D., Goldberg D.A., “On the Rate of Convergence to Stationarity of the M/M/n Queue in the Halfin-Whitt Regime”, Ann. Appl. Probab., 23:5 (2013), 1879–1912  crossref  mathscinet  zmath  isi  scopus
    10. Zeifman A., Satin Ya., Panfilova T., “Limiting Characteristics for Finite Birth-Death-Catastrophe Processes”, Math. Biosci., 245:1 (2013), 96–102  crossref  mathscinet  zmath  isi  elib  scopus
    11. A. I. Zeifman, V. Yu. Korolev, A. V. Korotysheva, S. Ya. Shorgin, “Obschie otsenki ustoichivosti dlya nestatsionarnykh markovskikh tsepei s nepreryvnym vremenem”, Inform. i ee primen., 8:1 (2014), 106–117  mathnet  crossref  elib
    12. A. Zeifman, V. Korolev, Ya. Satin, A. Korotysheva, V. Bening, “Perturbation bounds and truncations for a class of Markovian queues”, Queueing Syst., 76:2 (2014), 205–221  crossref  mathscinet  zmath  isi  scopus
    13. Zeifman A., Satin Ya., Korolev V., Shorgin S., “On Truncations For Weakly Ergodic Inhomogeneous Birth and Death Processes”, Int. J. Appl. Math. Comput. Sci., 24:3 (2014), 503–518  crossref  mathscinet  zmath  isi  scopus
    14. Zeifman A., Korotysheva A., Satin Ya., Korolev V., Shorgin S., Razumchik R., “Ergodicity and Perturbation Bounds For Inhomogeneous Birth and Death Processes With Additional Transitions From and To the Origin”, Int. J. Appl. Math. Comput. Sci., 25:4 (2015), 787–802  crossref  mathscinet  zmath  isi  scopus
    15. Zeifman A.I., Korolev V.Yu., “Two-Sided Bounds on the Rate of Convergence For Continuous-Time Finite Inhomogeneous Markov Chains”, Stat. Probab. Lett., 103 (2015), 30–36  crossref  mathscinet  zmath  isi  elib  scopus
    16. Lorek P., Szekli R., “Computable Bounds on the Spectral Gap For Unreliable Jackson Networks”, Adv. Appl. Probab., 47:2 (2015), 402–424  crossref  mathscinet  zmath  isi  elib
    17. A. I. Zeifman, A. V. Korotysheva, Ya. A. Satin, K. M. Kiseleva, R. V. Razumchik, V. Yu. Korolev, S. Ya. Shorgin, “Otsenki pogreshnosti approksimatsii dlya markovskikh sistem obsluzhivaniya, opisyvaemykh protsessami rozhdeniya i gibeli s dopolnitelnymi perekhodami”, Sistemy i sredstva inform., 27:3 (2017), 37–51  mathnet  crossref  elib
    18. Zeifman A., Korotysheva A., Satin Ya., Razumchik R., Korolev V., Shorgin S., “Ergodicity and Truncation Bounds For Inhomogeneous Birth and Death Processes With Additional Transitions From and to Origin”, Stoch. Models, 33:4, SI (2017), 598–616  crossref  mathscinet  zmath  isi  scopus
    19. Zeifman A., Korotysheva A., Satin Ya., Kiseleva K., Korolev V., Shorgin S., “Bounds For Markovian Queues With Possible Catastrophes”, Proceedings of the 31st European Conference on Modelling and Simulation (ECMS 2017), eds. Paprika Z., Horak P., Varadi K., Zwierczyk P., VidovicsDancs A., Radics J., European Council Modelling & Simulation, 2017, 628–634  isi
    20. Satin Ya., Korotysheva A., Shilova G., Sipin A., Fokicheva E., Kiseleva K., Zeifman A., Korolev V., Shorgin S., “Two-Sided Truncations For the M-T\M-T§Queueing Model”, Proceedings of the 31st European Conference on Modelling and Simulation (ECMS 2017), eds. Paprika Z., Horak P., Varadi K., Zwierczyk P., VidovicsDancs A., Radics J., European Council Modelling & Simulation, 2017, 635–641  isi
    21. Zeifman A. Razumchik R. Satin Ya. Kiseleva K. Korotysheva A. Korolev V., “Bounds on the Rate of Convergence For One Class of Inhomogeneous Markovian Queueing Models With Possible Batch Arrivals and Services”, Int. J. Appl. Math. Comput. Sci., 28:1 (2018), 141–154  crossref  mathscinet  isi  scopus
    22. Zeifman A.I. Korolev V.Yu. Satin Ya.A. Kiseleva K.M., “Lower Bounds For the Rate of Convergence For Continuous-Time Inhomogeneous Markov Chains With a Finite State Space”, Stat. Probab. Lett., 137 (2018), 84–90  crossref  mathscinet  zmath  isi  scopus
    23. Satin Ya., Morozov E., Nekrasova R., Zeifman A., Kiseleva K., Sinitcina A., Sipin A., Shilova G., Gudkova I., “Upper Bounds on the Rate of Convergence For Constant Retrial Rate Queueing Model With Two Servers”, Stat. Pap., 59:4, SI (2018), 1271–1282  crossref  mathscinet  zmath  isi  scopus
    24. Zeifman A. Kiseleva K. Satin Ya. Kryukova A. Korolev V., 2018 10Th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (Icumt 2018): Emerging Technologies For Connected Society, International Conference on Ultra Modern Telecommunications and Control Systems & Workshops, IEEE, 2018  isi
    25. Zeifman A., Satin Y., Kiseleva K., Korolev V., Panfilova T., “On Limiting Characteristics For a Non-Stationary Two-Processor Heterogeneous System”, Appl. Math. Comput., 351 (2019), 48–65  crossref  mathscinet  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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