# Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme

Christiane Cocozza-Thivent; Robert Eymard

- Volume: 38, Issue: 5, page 853-875
- ISSN: 0764-583X

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topCocozza-Thivent, Christiane, and Eymard, Robert. "Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.5 (2004): 853-875. <http://eudml.org/doc/245218>.

@article{Cocozza2004,

abstract = {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.},

author = {Cocozza-Thivent, Christiane, Eymard, Robert},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {renewal equation; semi-Markov process; convergence of a finite volume scheme},

language = {eng},

number = {5},

pages = {853-875},

publisher = {EDP-Sciences},

title = {Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme},

url = {http://eudml.org/doc/245218},

volume = {38},

year = {2004},

}

TY - JOUR

AU - Cocozza-Thivent, Christiane

AU - Eymard, Robert

TI - Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 5

SP - 853

EP - 875

AB - 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.

LA - eng

KW - renewal equation; semi-Markov process; convergence of a finite volume scheme

UR - http://eudml.org/doc/245218

ER -

## References

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- [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).
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- [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, Proceedings of the Conference $\lambda /\mu $ 14, Bourges, France (October 2004).
- [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. Zbl0981.65095
- [12] W. Feller, An Introduction to Probability Theory and its Applications. Volume II, Wiley (1966). Zbl0138.10207MR210154
- [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. Zbl1020.92025

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