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On Sharp Bounds on the Rate of Convergence for Finite Continuous-Time Markovian Queueing Models

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Computer Aided Systems Theory – EUROCAST 2017 (EUROCAST 2017)

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Abstract

Finite inhomogeneous continuous-time Markov chains are studied. For a wide class of such processes an approach is proposed for obtaining sharp bounds on the rate of convergence to the limiting characteristics. Queueing examples are considered.

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References

  1. Di Crescenzo, A., Giorno, V., Nobile, A.G., Ricciardi, L.M.: On the M/M/1 queue with catastrophes and its continuous approximation. Queueing Syst. 43(4), 329–347 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Van Doorn, E.A.: Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Probab. 17, 514–530 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Van Doorn, E.A., Zeifman, A.I., Panfilova, T.L.: Bounds and asymptotics for the rate of convergence of birth-death processes. Theory Probab. Appl. 54, 97–113 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Van Doorn, E.A., Zeifman, A.I.: On the speed of convergence to stationarity of the Erlang loss system. Queueing Syst. 63, 241–252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fricker, C., Robert, P., Tibi, D.: On the rate of convergence of Erlang’s model. J. Appl. Probab. 36, 1167–1184 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Giorno, V., Nobile, A.G., Spina, S.: On some time non-homogeneous queueing systems with catastrophes. Appl. Math. Comput. 245, 220–234 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Granovsky, B.L., Zeifman, A.I.: The N-limit of spectral gap of a class of birthdeath Markov chains. Appl. Stoch. Models Bus. Ind. 16(4), 235–248 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, J., Zhang, L.: Decay property of stopped Markovian bulk-arriving queues with c-servers. Stoch. Models 32, 674–686 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mitrophanov, A.Y.: The spectral gap and perturbation bounds for reversible continuous-time Markov chains. J. Appl. Probab. 41, 1219–1222 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Nelson, R., Towsley, D., Tantawi, A.N.: Performance analysis of parallel processing systems. IEEE Trans. Softw. Eng. 14(4), 532–540 (1988)

    Article  Google Scholar 

  11. Satin, Y.A., Zeifman, A.I., Korotysheva, A.V.: On the rate of convergence and truncations for a class of Markovian queueing systems. Theory Probab. Appl. 57, 529–539 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zeifman, A.I.: Some properties of a system with losses in the case of variable rates. Autom. Remote Control 50(1), 82–87 (1989)

    Google Scholar 

  13. Zeifman, A.I.: Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes. Stoch. Proc. Appl. 59, 157–173 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zeifman, A.I.: On the nonstationary Erlang loss model. Autom. Remote Control 70(12), 2003–2012 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zeifman, A.I., Korotysheva, A., Satin, Y., Shilova, G., Pafilova, T.: On a queueing model with group services. Lect. Notes CCIS. 356, 198–205 (2013)

    MATH  Google Scholar 

  16. Zeifman, A., Satin, Y., Panfilova, T.: Limiting characteristics for finite birthdeath-catastrophe processes. Math. Biosci. 245(1), 96–102 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zeifman, A., Satin, Y., Korolev, V., Shorgin, S.: On truncations for weakly ergodic inhomogeneous birth and death processes. Int. J. Appl. Math. Comput. Sci. 24, 503–518 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zeifman, A., Korotysheva, A., Korolev, V., Satin, Y., Bening, V.: Perturbation bounds and truncations for a class of Markovian queues. Queueing Syst. 76, 205–221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zeifman, A.I., Korolev, V.Y.: On perturbation bounds for continuous-time Markov chains. Stat. Probab. Lett. 88, 66–72 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zeifman, A.I., Korolev, V.Y.: Two-sided bounds on the rate of convergence for continuous-time finite inhomogeneous Markov chains. Stat. Probab. Lett. 103, 30–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zeifman, A.I., Shilova, G.N., Korolev, V.Y., Shorgin, S.Y.: On sharp bounds of the rate of convergence for some queueing models. In: Proceedings 29th European Conference on Modeling and Simulation, ECMS 2015, Varna, Bulgaria, pp. 622–625 (2015)

    Google Scholar 

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Acknowledgments

The work was supported by the Ministry of Education of the Russian Federation (the Agreement number 02.a03.21.0008 of 24 June 2016), by the Russian Foundation for Basic Research, projects no. 15-01-01698, 15-07-05316.

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Authors and Affiliations

  1. Vologda State University, Vologda, Russia

    Alexander Zeifman, Alexander Sipin, Galina Shilova, Ksenia Kiseleva, Anna Korotysheva & Yacov Satin

  2. Institute of Informatics Problems FRC CSC RAS, Moscow, Russia

    Alexander Zeifman & Victor Korolev

  3. ISEDT RAS, Vologda, Russia

    Alexander Zeifman

  4. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia

    Victor Korolev

  5. Hangzhou Dianzi University, Hangzhou, China

    Victor Korolev

  6. RUDN University, Moscow, Russia

    Ksenia Kiseleva

Authors
  1. Alexander Zeifman
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  2. Alexander Sipin
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  3. Victor Korolev
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  4. Galina Shilova
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  5. Ksenia Kiseleva
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  6. Anna Korotysheva
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  7. Yacov Satin
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Corresponding author

Correspondence to Alexander Zeifman .

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Editors and Affiliations

  1. University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Roberto Moreno-Díaz

  2. Johannes Kepler University Linz, Linz, Austria

    Franz Pichler

  3. University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Alexis Quesada-Arencibia

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Zeifman, A. et al. (2018). On Sharp Bounds on the Rate of Convergence for Finite Continuous-Time Markovian Queueing Models. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2017. EUROCAST 2017. Lecture Notes in Computer Science(), vol 10672. Springer, Cham. https://doi.org/10.1007/978-3-319-74727-9_3

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  • DOI: https://doi.org/10.1007/978-3-319-74727-9_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-74726-2

  • Online ISBN: 978-3-319-74727-9

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