Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 5, pp. 853-875.

In the reliability theory, the availability of a component, characterized by non constant failure and repair rates, is obtained, at a given time, thanks to the computation of the marginal distributions of a semi-Markov process. These measures are shown to satisfy classical transport equations, the approximation of which can be done thanks to a finite volume method. Within a uniqueness result for the continuous solution, the convergence of the numerical scheme is then proven in the weak measure sense, and some numerical applications, which show the efficiency and the accuracy of the method, are given.

DOI : 10.1051/m2an:2004043
Classification : 60K15, 60K20, 65C20, 65M60
Mots-clés : renewal equation, semi-Markov process, convergence of a finite volume scheme 
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     title = {Approximation of the marginal distributions of a {semi-Markov} process using a finite volume scheme},
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Cocozza-Thivent, Christiane; Eymard, Robert. Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 5, pp. 853-875. doi : 10.1051/m2an:2004043. http://archive.numdam.org/articles/10.1051/m2an:2004043/

[1] S. Asmussen, Fitting phase-type pistributions via the EM algorithm. Scand. J. Statist. 23 (1996) 419-441. | Zbl

[2] E. Cinlar, Introduction to Stochastic processes. Prentice-Hall (1975). | MR | Zbl

[3] C. Cocozza-Thivent, Processus stochastiques et fiabilité des systèmes. Springer, Paris, Collection Mathématiques & Applications 28 (1997). | MR | Zbl

[4] C. Cocozza-Thivent and R. Eymard, Marginal distributions of a semi-Markov process and their computations, Ninth ISSAT International Conference on Reliability and Quality in Design, International Society of Science and Applied Technologies, H. Pham and S. Yamada Eds. (2003).

[5] C. Cocozza-Thivent and M. Roussignol, Semi-Markov process for reliability studies. ESAIM: PS 1 (1997) 207-223. | Numdam | Zbl

[6] C. Cocozza-Thivent and M. Roussignol, A general framework for some asymptotic reliability formulas. Adv. Appl. Prob. 32 (2000) 446-467. | Zbl

[7] C. Cocozza-Thivent, R. Eymard, S. Mercier and M. Roussignol, On the marginal distributions of Markov processes used in dynamic reliability, Prépublications du Laboratoire d'Analyse et de Mathématiques Appliquées UMR CNRS 8050, 2/2003 (January 2003).

[8] C. Cocozza-Thivent, R. Eymard and S. Mercier, A numerical scheme to solve integro-differential equations in the dynamic reliability field, PSAM7-ESREL'04, Berlin (June 2004).

[9] C. Cocozza-Thivent, R. Eymard and S. Mercier, Méthodologie et algorithmes pour la quantification de petits systèmes redondants

[10] D.R. Cox, Renewal Theory. Chapman and Hall, London (1982).

[11] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., VII (2000) 723-1020. | Zbl

[12] W. Feller, An Introduction to Probability Theory and its Applications. Volume II, Wiley (1966). | MR | Zbl

[13] A. Fritz, P. Pozsgai and B. Bertsche, Notes on the Analytic Description and Numerical Calculation of the Time Dependent Availability, MMR'2000: Second International Conference on Mathematical Methods in Reliability, Bordeaux, France, July 4-7 (2000) 413-416.

[14] S. Mischler, B. Perthame and L. Ryzhik, Stability in a nonlinear population maturation model. Math. Models Met. App. Sci. 12 (2002) 1-22. | Zbl

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