# 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

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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 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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- D.R. Cox, Renewal Theory. Chapman and Hall, London (1982).
- 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
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- 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.
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