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 - Modélisation Mathématique et Analyse Numérique (2004)

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

Abstract

top
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

top

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 - 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

top
  1. [1] S. Asmussen, Fitting phase-type pistributions via the EM algorithm. Scand. J. Statist. 23 (1996) 419–441. Zbl0898.62104
  2. [2] E. Cinlar, Introduction to Stochastic processes. Prentice-Hall (1975). Zbl0341.60019MR380912
  3. [3] C. Cocozza-Thivent, Processus stochastiques et fiabilité des systèmes. Springer, Paris, Collection Mathématiques & Applications 28 (1997). Zbl0883.60002MR1619554
  4. [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. [5] C. Cocozza-Thivent and M. Roussignol, Semi-Markov process for reliability studies. ESAIM: PS 1 (1997) 207–223. Zbl1002.60580
  6. [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. [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. [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. [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. [10] D.R. Cox, Renewal Theory. Chapman and Hall, London (1982). 
  11. [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. [12] W. Feller, An Introduction to Probability Theory and its Applications. Volume II, Wiley (1966). Zbl0138.10207MR210154
  13. [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. [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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.