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

Christiane Cocozza-Thivent; Robert Eymard

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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.

How to cite

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Cocozza-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 38.5 (2010): 853-875. <http://eudml.org/doc/194243>.

@article{Cocozza2010,
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},
keywords = {Renewal equation; semi-Markov process; convergence of a finite volume scheme.},
language = {eng},
month = {3},
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/194243},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
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/194243
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).  Zbl1078.60075
  5. C. Cocozza-Thivent and M. Roussignol, Semi-Markov process for reliability studies. ESAIM: PS1 (1997) 207–223.  Zbl1002.60580
  6. C. Cocozza-Thivent and M. Roussignol, A general framework for some asymptotic reliability formulas. Adv. Appl. Prob.32 (2000) 446–467.  Zbl0961.60090
  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).  Zbl1101.60068
  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).  Zbl1109.65011
  9. C. Cocozza-Thivent, R. Eymard and S. Mercier, Méthodologie et algorithmes pour la quantification de petits systèmes redondants, Proceedings of the Conference λ / μ 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).  Zbl0138.10207
  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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