Nonparametric estimation of renewal processes from count data.

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

International audienceIn 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

A finite-volume scheme for dynamic reliability models

International audienceIn a model arising in the dynamic reliability study of a system, the probability of the state of the system is completely described by the Chapman-Kolmogorov equations, which are scalar linear hyperbolic partial differential equations coupled by their right-hand side, the solution of which are probability measures. We propose in this paper a finite-volume scheme to approximate these measures. We show, thanks to the proof of the tightness of the approximate solution, that the conservation of the probability mass leads to a compactness property. The convergence of the scheme is then obtained in the space of continuous functions with respect to the time variable, valued in the set of probability measures on R-d. We finally show on a numerical example the accuracy and efficiency of the approximation method

MUT of a one out of two system with preventive maintenance

International audienceWe are interested in the MUT (Mean Up Time) of a one out of two system in cold standby with preventive maintenance: a preventive maintenance occurs when the working unit reaches a given age. We study in details the stationary distribution of the Markov chain describing the state of the system at the beginning of its working periods. We give exact analytical formulas from which we derive a way to compute the MUT and we compare the results with those of Smith and Decker [M.A.J. Smith, R. Decker, Preventive maintenance in a 1 out of n system: The uptime, downtime and costs, European Journal of Operational Research 99 (1997) 565-583] which are based on approximations. We also investigate some discontinuities problems. (c) 2006 Elsevier B.V. All rights reserved

Switching Lévy Process

This paper introduces Switching Processes, called SP. Their constructions are inspired by the PDMP's ones (PDMP stands for Piecewise Deterministic Markov Process). A Markov process, called the intrinsic process, replaces the PDMP's flow. Jumps are added ; they occur randomly as their locations ; their distributions depend on the process's trajectory between them. When the intrinsic process is a Levy process, thanks to its Lévy-Itô decomposition as a semi-martingale, we obtain the expected Kolmogorov equations for the SP. The results are extended to Itô-Lévy processes, in particular to diffusion processes.Dans cet article, nous introduisons les processus avec changements de rythmes. Leur construction est inspirée par celle des PDMP (Piecewise Deterministic Markov Process). Ces processus, notés SP pour Switching Processes, sont construits à partir d'un processus dit intrinsèque qui remplace le flot déterministe de la construction des PDMP. Des sauts sont ajoutés à ce processus intrinsèque. Ils se produisent à des instants aléatoires, les lois de ces instants et leurs localisations dépendent de la trajectoire du processus entre ceux-ci. Lorsque le processus intrinsèque est un processus de Lévy, son écriture comme semi-martingale (décomposition de Lévy-Itô) nous permet d'obtenir les équations de Kolmogorov auxquelles on s'attend pour le SP. Les résultats s'étendent aux processus d'Itô-Lévy et en particulier aux diffusions

Switching Lévy Process

This paper introduces Switching Processes, called SP. Their constructions are inspired by the PDMP's ones (PDMP stands for Piecewise Deterministic Markov Process). A Markov process, called the intrinsic process, replaces the PDMP's flow. Jumps are added ; they occur randomly as their locations ; their distributions depend on the process's trajectory between them. When the intrinsic process is a Levy process, thanks to its Lévy-Itô decomposition as a semi-martingale, we obtain the expected Kolmogorov equations for the SP. The results are extended to Itô-Lévy processes, in particular to diffusion processes.Dans cet article, nous introduisons les processus avec changements de rythmes. Leur construction est inspirée par celle des PDMP (Piecewise Deterministic Markov Process). Ces processus, notés SP pour Switching Processes, sont construits à partir d'un processus dit intrinsèque qui remplace le flot déterministe de la construction des PDMP. Des sauts sont ajoutés à ce processus intrinsèque. Ils se produisent à des instants aléatoires, les lois de ces instants et leurs localisations dépendent de la trajectoire du processus entre ceux-ci. Lorsque le processus intrinsèque est un processus de Lévy, son écriture comme semi-martingale (décomposition de Lévy-Itô) nous permet d'obtenir les équations de Kolmogorov auxquelles on s'attend pour le SP. Les résultats s'étendent aux processus d'Itô-Lévy et en particulier aux diffusions

Processus de renouvellement markovien Processus de Markov déterministes par morceaux

A PDMP (Piecewise Deterministic Markov Processes) is a Markov process with jumps and the following distinctive feature : a deterministic trajectory between jumps. The jumps distributions (times and locations) depend on the deterministic evolution between them. The PDMP have been introduced by M.H.A. Davis, ours are slightly more general. PDMP's are used to model various situations in the field of insurance, biology, supply management, dependability management and so on. Some examples are provided in this book.This book's originality is to strongly link the PDMP study to the underlying Markov renewal process.It is equivalent to get the Markov renewal process (Y_n, T_n)_{n ≥ 0} or the process (Z_t, A_t)_{t ≥ 0} defined by Z_t=Y_n, A_t=t-T_n for T_n ≤ t < T_{n+1}. The process (Z_t, A_t)_{t ≥ 0}, called the CSMP (Completed Semi-Markov Process), is a PDMP with a very simple deterministic behavior since it is linear ; the deterministic flow $\phi$, involved in the PDMP's definition, only appears in the Markov renewal kernel of the process (Y_n, T_n)_{n ≥ 0}. The marked point processes are widely used in our approach. The PDMP can be written as Phi_t=phi(Z_t, A_t), therefore many results on this process can easily be deduced from those on the associated CSMP.In the last chapter, we consider what we call Switching Processes or SP. They are obtained by replacing the deterministic flow of the PDMP by a Markov process. With the approach used for the PDMP study, we display some results for these new processes.Un PDMP (Piecewise Deterministic Markov Processes) est un processus de Markov qui évolue de manière déterministe entre des sauts aléatoires. La loi de ces sauts (instants et lieux) est fonction de l'évolution déterministe entre ceux-ci. Les PDMP, introduits par M.H.A. Davis, servent à modéliser de nombreux phénomènes qui vont de l'assurance à la biologie, en passant par le transfert de données informatiques, la gestion de stocks, la sûreté de fonctionnement, etc. Des exemples sont donnés dans cet ouvrage.Dans ce livre nous étudions des PDMP, un peu plus généraux que ceux définis par M.H.A. Davis, en nous appuyant fortement sur le processus de renouvellement markovien sous-jacent. Se donner ce processus de renouvellement markovien (Y_n, T_n)_{n ≥ 0} équivaut à se donner le processus (Z_t, A_t)_{t ≥ 0} défini par Z_t=Y_n, A_t=t-T_n si T_n ≤ < T_{n+1}. Le processus (Z_t, A_t)_{t ≥ 0}, que nous appelons le CSMP (Completed Semi-Markov Process) est un PDMP particulièrement simple puisque son évolution déterministe est linéaire ; le flot déterministe markovien phi entrant dans la définition du PDMP initial n'intervient que dans la loi des sauts, c'est-à-dire dans le noyau de renouvellement markovien du processus (Y_n, T_n)_{n ≥ 0}. Les processus ponctuels marqués ont une place de choix dans notre approche. Le PDMP peut s'écrire Phi_t=phi(Z_t, A_t) si bien que de nombreux résultats sur celui-ci découlent immédiatement de l'étude du CSMP associé.Dans le dernier chapitre, nous regardons ce que donne notre approche si, dans la définition du PDMP, nous remplaçons l'évolution déterministe entre les sauts par un processus de Markov. Les processus obtenus peuvent servir à modéliser des phénomènes avec changements de rythmes, nous les appelons des SP, pour Switching Processes