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Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method

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  • Alexander Zeifman

    (Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilova 44-2, 119333 Moscow, Russia
    Department of Applied Mathematics, Vologda State University, Lenina 15, 160000 Vologda, Russia
    Moscow Center for Fundamental and Applied Mathematics, Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia; vkorolev@cs.msu.ru)

  • Yacov Satin

    (Department of Applied Mathematics, Vologda State University, Lenina 15, 160000 Vologda, Russia)

  • Ivan Kovalev

    (Department of Applied Mathematics, Vologda State University, Lenina 15, 160000 Vologda, Russia)

  • Rostislav Razumchik

    (Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilova 44-2, 119333 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia; vkorolev@cs.msu.ru)

  • Victor Korolev

    (Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilova 44-2, 119333 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia; vkorolev@cs.msu.ru
    Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia)

Abstract

The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can benefit from methods providing ergodicity bounds because the latter can indicate how to choose the position and the length of the “distant time interval” (in the periodic case) on which the solution has to be computed. They can also be helpful whenever the state space truncation is required. In this paper one such analytic method—the logarithmic norm method—is being reviewed. Its applicability is shown within the queueing theory context with three examples: the classical time-varying M / M / 2 queue; the time-varying single-server Markovian system with bulk arrivals, queue skipping policy and catastrophes; and the time-varying Markovian bulk-arrival and bulk-service system with state-dependent control. In each case it is shown whether and how the bounds on the rate of convergence can be obtained. Numerical examples are provided.

Suggested Citation

  • Alexander Zeifman & Yacov Satin & Ivan Kovalev & Rostislav Razumchik & Victor Korolev, 2020. "Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method," Mathematics, MDPI, vol. 9(1), pages 1-20, December.
  • Handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:42-:d:469068
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    References listed on IDEAS

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    1. Linda Green & Peter Kolesar, 1991. "The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals," Management Science, INFORMS, vol. 37(1), pages 84-97, January.
    2. Schwarz, Justus Arne & Selinka, Gregor & Stolletz, Raik, 2016. "Performance analysis of time-dependent queueing systems: Survey and classification," Omega, Elsevier, vol. 63(C), pages 170-189.
    3. Anyue Chen & Xiaohan Wu & Jing Zhang, 2020. "Markovian bulk-arrival and bulk-service queues with general state-dependent control," Queueing Systems: Theory and Applications, Springer, vol. 95(3), pages 331-378, August.
    4. Soongeol Kwon & Natarajan Gautam, 2016. "Guaranteeing performance based on time-stability for energy-efficient data centers," IISE Transactions, Taylor & Francis Journals, vol. 48(9), pages 812-825, September.
    5. Liu, Yunan & Whitt, Ward, 2017. "Stabilizing performance in a service system with time-varying arrivals and customer feedback," European Journal of Operational Research, Elsevier, vol. 256(2), pages 473-486.
    6. Armann Ingolfsson & Elvira Akhmetshina & Susan Budge & Yongyue Li & Xudong Wu, 2007. "A Survey and Experimental Comparison of Service-Level-Approximation Methods for Nonstationary M(t)/M/s(t) Queueing Systems with Exhaustive Discipline," INFORMS Journal on Computing, INFORMS, vol. 19(2), pages 201-214, May.
    7. Ward Whitt & Wei You, 2019. "Time-Varying Robust Queueing," Operations Research, INFORMS, vol. 67(6), pages 1766-1782, November.
    8. M. Arns & P. Buchholz & A. Panchenko, 2010. "On the Numerical Analysis of Inhomogeneous Continuous-Time Markov Chains," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 416-432, August.
    9. Peter J. Kolesar & Kenneth L. Rider & Thomas B. Crabill & Warren E. Walker, 1975. "A Queuing-Linear Programming Approach to Scheduling Police Patrol Cars," Operations Research, INFORMS, vol. 23(6), pages 1045-1062, December.
    10. Pier Conti, 1997. "An asymptotic test for a geometric process against a lattice distribution with monotone hazard," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 6(3), pages 213-231, December.
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