Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method
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 t...
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MDPI AG
2020-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/1/42 |
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| author | Alexander Zeifman Yacov Satin Ivan Kovalev Rostislav Razumchik Victor Korolev |
| author_facet | Alexander Zeifman Yacov Satin Ivan Kovalev Rostislav Razumchik Victor Korolev |
| author_sort | Alexander Zeifman |
| collection | DOAJ |
| description | 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 <inline-formula><math display="inline"><semantics><mrow><mi>M</mi><mo>/</mo><mi>M</mi><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> 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. |
| first_indexed | 2024-03-10T13:44:39Z |
| format | Article |
| id | doaj.art-8953154d45544fb8923fb18ed69bb1b3 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T13:44:39Z |
| publishDate | 2020-12-01 |
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| series | Mathematics |
| spelling | doaj.art-8953154d45544fb8923fb18ed69bb1b32023-11-21T02:44:59ZengMDPI AGMathematics2227-73902020-12-01914210.3390/math9010042Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm MethodAlexander Zeifman0Yacov Satin1Ivan Kovalev2Rostislav Razumchik3Victor Korolev4Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilova 44-2, 119333 Moscow, RussiaDepartment of Applied Mathematics, Vologda State University, Lenina 15, 160000 Vologda, RussiaDepartment of Applied Mathematics, Vologda State University, Lenina 15, 160000 Vologda, RussiaInstitute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilova 44-2, 119333 Moscow, RussiaInstitute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilova 44-2, 119333 Moscow, RussiaThe 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 <inline-formula><math display="inline"><semantics><mrow><mi>M</mi><mo>/</mo><mi>M</mi><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> 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.https://www.mdpi.com/2227-7390/9/1/42continuous-time Markov chainsergodicity boundsdiscrete state spacerate of convergencelogarithmic norm |
| spellingShingle | Alexander Zeifman Yacov Satin Ivan Kovalev Rostislav Razumchik Victor Korolev Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method Mathematics continuous-time Markov chains ergodicity bounds discrete state space rate of convergence logarithmic norm |
| title | Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method |
| title_full | Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method |
| title_fullStr | Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method |
| title_full_unstemmed | Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method |
| title_short | Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method |
| title_sort | facilitating numerical solutions of inhomogeneous continuous time markov chains using ergodicity bounds obtained with logarithmic norm method |
| topic | continuous-time Markov chains ergodicity bounds discrete state space rate of convergence logarithmic norm |
| url | https://www.mdpi.com/2227-7390/9/1/42 |
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