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2015 | 25 | 4 | 787-802
Tytuł artykułu

Ergodicity and perturbation bounds for inhomogeneous birth and death processes with additional transitions from and to the origin

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, timedependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided.
Rocznik
Tom
25
Numer
4
Strony
787-802
Opis fizyczny
Daty
wydano
2015
otrzymano
2014-11-28
poprawiono
2015-01-20
poprawiono
2015-08-07
Twórcy
  • Department of Applied Mathematics, Vologda State University, S. Orlova, 6, Vologda, Russia
  • Institute of Socio-Economic Development of Territories, Russian Academy of Sciences, Gorkogo Str., 56A, Vologda, Russia
  • Institute of Informatics Problems, FRC CSC, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, 119333, Russia
  • Department of Applied Mathematics, Vologda State University, S. Orlova, 6, Vologda, Russia
  • Institute of Informatics Problems, FRC CSC, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, 119333, Russia
autor
  • Department of Applied Mathematics, Vologda State University, S. Orlova, 6, Vologda, Russia
  • Institute of Informatics Problems, FRC CSC, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, 119333, Russia
  • Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Leninskie Gory, Moscow, Russia
  • Institute of Informatics Problems, FRC CSC, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, 119333, Russia
  • Institute of Informatics Problems, FRC CSC, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, 119333, Russia
  • Institute of Informatics Problems, FRC CSC, Russian Academy of Sciences, Vavilova str., 44-2, Moscow, 119333, Russia
  • Department of Applied Informatics and Probability Theory, Peoples' Friendship University of Russia, Miklukho-Maklaya str., 6, Moscow, 117198, Russia
Bibliografia
  • Chen, A., Pollett, P., Li, J. and Zhang, H. (2010). Markovian bulk-arrival and bulk-service queues with state-dependent control, Queueing Systems 64(3): 267-304.
  • Chen, A. and Renshaw, E. (1997). The m|m|1 queue with mass exodus and mass arrives when empty, Journal of Applied Probability 34(1): 192-207.
  • Chen, A. and Renshaw, E. (2004). Markov bulk-arriving queues with state-dependent control at idle time, Advances in Applied Probability 36(2): 499-524.
  • Daleckij, J. and Krein, M. (1975). Stability of solutions of differential equations in Banach space, Bulletin of the American Mathematical Society 81(6): 1024-102.
  • Gaidamaka, Y., Pechinkin, A., Razumchik, R., Samouylov, K. and Sopin, E. (2014). Analysis of an M/G/1/R queue with batch arrivals and two hysteretic overload control policies, International Journal of Applied Mathematics and Computer Science 24(3): 519-534, DOI: 10.2478/amcs-2014-0038.
  • Granovsky, B. and Zeifman, A. (2004). Nonstationary queues: Estimation of the rate of convergence, Queueing Systems 46(3-4): 363-388.
  • Li, J. and Chen, A. (2013). The decay parameter and invariant measures for Markovian bulk-arrival queues with control at idle time, Methodology and Computing in Applied Probability 15(2): 467-484.
  • Parthasarathy, P. and Kumar, B.K. (1991). Density-dependent birth and death processes with state-dependent immigration, Mathematical and Computer Modelling 15(1): 11-16.
  • Van Doorn, E., Zeifman, A. and Panfilova, T. (2010). Bounds and asymptotics for the rate of convergence of birth-death processes, Theory of Probability and Its Applications 54(1): 97-113.
  • Zeifman, A. (1995). Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes, Stochastic Processes and Their Applications 59(1): 157-173.
  • Zeifman, A., Bening, V. and Sokolov, I. (2008). ContinuousTime Markov Chains and Models, Elex-KM, Moscow.
  • Zeifman, A. and Korolev, V. (2014). On perturbation bounds for continuous-time Markov chains, Statistics & Probability Letters 88(1): 66-72.
  • Zeifman, A. and Korotysheva, A. (2012). Perturbation bounds for Mt /Mt /N queue with catastrophes, Stochastic Models 28(1): 49-62.
  • Zeifman, A., Korotysheva, A., Satin, Y. and Shorgin, S. (2010). On stability for nonstationary queueing systems with catastrophes, Informatics and Its Applications 4(3): 9-15.
  • Zeifman, A., Leorato, S., Orsingher, E., Satin, Y. and Shilova, G. (2006). Some universal limits for nonhomogeneous birth and death processes, Queueing Systems 52(2): 139-151.
  • Zeifman, A., Satin, Y., Korolev, V. and Shorgin, S. (2014). On truncations for weakly ergodic inhomogeneous birth and death processes, International Journal of Applied Mathematics and Computer Science 24(3): 503-518, DOI: 10.2478/amcs-2014-0037.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-amcv25i4p787bwm
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