Bounds and approximations for continuous-time Markovian transition probabilities and large systems

We propose new bounds and approximations for the transition probabilities of a continuous-time Markov process with finite but large state-space. The bounding and approximating procedures have been exposed in another paper [S. Mercier, Numerical bounds for semi-Markovian quantities and applications t...

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Published in:	European journal of operational research Vol. 185; no. 1; pp. 216 - 234
Main Author:	Mercier, Sophie
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier B.V 16.02.2008 Elsevier Elsevier Sequoia S.A
Series:	European Journal of Operational Research
Subjects:	Algorithms Applied sciences Approximation Eulers equations Exact sciences and technology Finite volume method Markov analysis Markov processes Mathematics Numerical bounds and approximations Operational research and scientific management Operational research. Management science Probability Reliability Reliability theory. Replacement problems Studies Time-dependent quantities Markov processes Numerical bounds and approximations Reliability Time-dependent quantities Markov process Euler equation Probabilistic approach Finite volume method Continuous time State space State space method Numerical approximation Continuous process Time dependence Marginal distribution Transition probability
ISSN:	0377-2217, 1872-6860
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Abstract	We propose new bounds and approximations for the transition probabilities of a continuous-time Markov process with finite but large state-space. The bounding and approximating procedures have been exposed in another paper [S. Mercier, Numerical bounds for semi-Markovian quantities and applications to reliability, in revision for Methodology and Computing in Applied Probability] in the more general context of a continuous-time semi-Markov process with countable state-space. Such procedures are here specialized to the Markovian finite case, leading to much simpler algorithms. The aim of this paper is to test such algorithms versus other algorithms from the literature near from ours, such as forward Euler approximation, external uniformization and a finite volume method from [C. Cocozza-Thivent, R. Eymard, Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme, ESAIM: M2AN 38(5) (2004) 853–875].
AbstractList	We propose new bounds and approximations for the transition probabilities of a continuous-time Markov process with finite but large state-space. The bounding and approximating procedures have been exposed in another paper [S. Mercier, Numerical bounds for semi-Markovian quantities and applications to reliability, in revision for Methodology and Computing in Applied Probability] in the more general context of a continuous-time semi-Markov process with countable state-space. Such procedures are here specialized to the Markovian finite case, leading to much simpler algorithms. The aim of this paper is to test such algorithms versus other algorithms from the literature near from ours, such as forward Euler approximation, external uniformization and a finite volume method from [C. Cocozza-Thivent, R. Eymard, Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme, ESAIM: M2AN 38(5) (2004) 853–875]. We propose new bounds and approximations for the transition probabilities of a continuous-time Markov process with finite but large state-space. The bounding and approximating procedures have been exposed in another paper [S. Mercier, Numerical bounds for semi-Markovian quantities and applications to reliability, in revision for Methodology and Computing in Applied Probability] in the more general context of a continuous-time semi-Markov process with countable state-space. Such procedures are here specialized to the Markovian finite case, leading to much simpler algorithms. The aim of this paper is to test such algorithms versus other algorithms from the literature near from ours, such as forward Euler approximation, external uniformization and a finite volume method from [C. Cocozza-Thivent, R. Eymard, Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme, ESAIM: M2AN 38(5) (2004) 853-875]. [PUBLICATION ABSTRACT] We propose new bounds and approximations for the transition probabilities of a continuous-time Markov process with finite but large state-space. The bounding and approximating procedures have been exposed in another paper [S. Mercier, Numerical bounds for semi-Markovian quantities and applications to reliability, in revision for Methodology and Computing in Applied Probability] in the more general context of a continuous-time semi-Markov process with countable state-space. Such procedures are here specialized to the Markovian finite case, leading to much simpler algorithms. The aim of this paper is to test such algorithms versus other algorithms from the literature near from ours, such as forward Euler approximation, external uniformization and a finite volume method from [C. Cocozza-Thivent, R. Eymard, Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme, ESAIM: M2AN 38(5) (2004) 853-875]. (c) 2007 Elsevier B.V. All rights reserved.
Author	Mercier, Sophie
Author_xml	– sequence: 1 givenname: Sophie surname: Mercier fullname: Mercier, Sophie email: Sophie.Mercier@univ-mlv.fr organization: Laboratoire d’Analyse et de Mathématiques Appliquées (CNRS - UMR 8050), Université de Marne-la-Vallée, Cité Descartes, 5 boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée Cedex 2, France
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References_xml	– year: 1989 ident: bib1 article-title: Introduction to Numerical Linear Algebra and Optimisation – volume: 20 start-page: 801 year: 1979 end-page: 836 ident: bib10 article-title: Nineteen dubious ways to compute the exponential of a matrix publication-title: SIAM Review – year: 1975 ident: bib2 article-title: Introduction to Stochastic Processes – year: 2003 ident: bib14 article-title: Introduction to Probability Models – volume: 177 start-page: 378 year: 2007 end-page: 405 ident: bib8 article-title: Discrete random bounds for general random variables and applications to reliability publication-title: European Journal of Operational Research – volume: 86 start-page: 105 year: 2004 end-page: 115 ident: bib12 article-title: An experimental study on iterative methods to compute transient solutions of large Markov models publication-title: Reliability Engineering & System Safety – year: 1997 ident: bib6 article-title: Markov Processes for Stochastic modelling – volume: 45 start-page: 3 year: 2003 end-page: 49 ident: bib11 article-title: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later publication-title: SIAM Review – volume: vol. 28 year: 1997 ident: bib3 article-title: Processus stochastiques et fiabilité des systèmes publication-title: Mathématiques et Applications – reference: C. 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Mercier, Encadrement de quantités fiabilistes pour un gros système markovien, Actes du congrès Lambda-Mu 15 (15e Congrès de Maıˆtrise des Risques et de Sûreté de Fonctionnement), Lille, octobre 2006 (in French). – volume: 20 start-page: 801 year: 1979 ident: 10.1016/j.ejor.2006.12.036_bib10 article-title: Nineteen dubious ways to compute the exponential of a matrix publication-title: SIAM Review doi: 10.1137/1020098 – volume: 86 start-page: 105 issue: 1 year: 2004 ident: 10.1016/j.ejor.2006.12.036_bib12 article-title: An experimental study on iterative methods to compute transient solutions of large Markov models publication-title: Reliability Engineering & System Safety doi: 10.1016/j.ress.2004.01.007 – year: 1975 ident: 10.1016/j.ejor.2006.12.036_bib2 – volume: 29 start-page: 345 issue: 3 year: 1999 ident: 10.1016/j.ejor.2006.12.036_bib15 article-title: A numerical study of large sparse matrix exponentials arising in Markov chains publication-title: Computational Statistics & Data Analysis doi: 10.1016/S0167-9473(98)00062-0 – volume: 177 start-page: 378 issue: 1 year: 2007 ident: 10.1016/j.ejor.2006.12.036_bib8 article-title: Discrete random bounds for general random variables and applications to reliability publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2005.09.043 – year: 1989 ident: 10.1016/j.ejor.2006.12.036_bib1 – volume: 40 start-page: 257 year: 1989 ident: 10.1016/j.ejor.2006.12.036_bib13 article-title: Markov and Markov reward models: A survey of numerical approaches publication-title: European Journal of Operations Research doi: 10.1016/0377-2217(89)90335-4 – volume: 38 start-page: 853 issue: 5 year: 2004 ident: 10.1016/j.ejor.2006.12.036_bib4 article-title: Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme publication-title: ESAIM: M2AN doi: 10.1051/m2an:2004043 – ident: 10.1016/j.ejor.2006.12.036_bib7 – ident: 10.1016/j.ejor.2006.12.036_bib5 – ident: 10.1016/j.ejor.2006.12.036_bib9 – volume: 45 start-page: 3 year: 2003 ident: 10.1016/j.ejor.2006.12.036_bib11 article-title: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later publication-title: SIAM Review doi: 10.1137/S00361445024180 – year: 2003 ident: 10.1016/j.ejor.2006.12.036_bib14 – volume: vol. 28 year: 1997 ident: 10.1016/j.ejor.2006.12.036_bib3 article-title: Processus stochastiques et fiabilité des systèmes – year: 1994 ident: 10.1016/j.ejor.2006.12.036_bib16 – year: 1997 ident: 10.1016/j.ejor.2006.12.036_bib6
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SubjectTerms	Algorithms Applied sciences Approximation Eulers equations Exact sciences and technology Finite volume method Markov analysis Markov processes Mathematics Numerical bounds and approximations Operational research and scientific management Operational research. Management science Probability Reliability Reliability theory. Replacement problems Studies Time-dependent quantities
Title	Bounds and approximations for continuous-time Markovian transition probabilities and large systems
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