Christiane Cocozza-Thivent

Processus des misanthropes

SummaryWe consider a system of identical interacting particles moving on the lattice ℤd. The rate at which a particle at the site x jumps to the site y is p(y−x)b(η(x), η(y)) where p is an irreducible probability on ℤd and b(η(x), η(y)) is an increasing (resp. decreasing) function of the number η(x) (resp. η(y)) of particles at site x (resp. y). We study the convergence of the system to equilibrium and describe the invariant measures.

The failure rate in reliability: approximations and bounds

We consider models typical to the area of reliability, and a failure rate function for processes describing the dynamics of these models. Various approximations of the failure rate function are proposed and their accuracies are investigated. The basic case studied in the paper is a regenerative model. Some interesting particular cases (Markov, semi-Markov, etc.) are considered. All proposed estimates are stated in a tractable analytic form.

Characterization of the marginal distributions of Markov processes used in dynamic reliability

In dynamic reliability, the evolution of a system is described by a piecewise deterministic Markov process (It,Xt)t≥0 with state-space E×ℝd, where E is finite. The main result of the present paper is the characterization of the marginal distribution of the Markov process (It,Xt)t≥0 at time t, as the unique solution of a set of explicit integro-differential equations, which can be seen as a weak form of the Chapman-Kolmogorov equation. Uniqueness is the difficult part of the result.

The failure rate in reliability: numerical treatment

This paper contains approximations and bounds of failure rate functions of redundant systems. We illustrate properties of these bounds and their accuracy.

A general framework for some asymptotic reliability formulas

The authors prove that certain reliability formulas which link asymptotic availability, mean normal operation time, mean time between failures, mean number of failures over a period of time and asymptotic Vesely rate, and which are well known in the case of modelling using a Markov jump process or an alternating renewal process, are also true in the context of more general modelling.

A Numerical Scheme to Solve Integro-Differential Equations in the Dynamic Reliability Field

A system is considered, which evolves in time. At time t, its state is described by its “physical” state denoted by I t and by some “environmental” conditions, such as temperature, pressure or so on, and denoted by X t. The “physical” state I t is assumed to take its values in a finite state-space E whereas the “environmental” conditions X t take their values in R d. For i j in E, the transition rate at time t from state i to state j depends on the value (say x) of the “environmental” conditions X t and is denoted by a i j x). The evolution of the environmental conditions is described by a set of differential equations, which depends on the state of the item. More precisely, given that the system is in state I t i for all t∈ [a b], then X t fulfils the following equation: for all t∈ [a b], where v is an application from E x Rd to Rd. This is the general context of dynamic reliability.

MUT of a one out of two system with preventive maintenance

Abstract We 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.

Some Models and Mathematical Results for Reliability of Systems of Components

We discuss the modelization of systems of components and the associated reliability formulas which can be proved. We present some mathematical results which can be seen as validations of engineers’ practice. The exhibited tools are the renewal theory and martingale technics.