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This article is cited in 31 scientific papers (total in 31 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
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English version:
Theory of Probability and its Applications, 2010, 54:1, 97–113
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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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http://mi.mathnet.ru/eng/tvp2497https://doi.org/10.4213/tvp2497 http://mi.mathnet.ru/eng/tvp/v54/i1/p18
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E. B. Yarovaya, “Models of branching walks and their use in the reliability theory”, Autom. Remote Control, 71:7 (2010), 1308–1324
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van Doorn E.A., “Rate of convergence to stationarity of the system system $M/M/N/N+R$”, TOP, 19:2 (2011), 336–350
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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
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Zeifman A. Korotysheva A., “Perturbation bounds for $M_t/M_t/N$ queue with catastrophes”, Stoch. Models, 28:1 (2012), 49–62
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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
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Zeifman A., Satin Ya., Panfilova T., “Limiting Characteristics for Finite Birth-Death-Catastrophe Processes”, Math. Biosci., 245:1 (2013), 96–102
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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
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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
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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
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Lorek P., Szekli R., “Computable Bounds on the Spectral Gap For Unreliable Jackson Networks”, Adv. Appl. Probab., 47:2 (2015), 402–424
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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
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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
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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
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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
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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
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