Sophie Mercier

Bivariate Gamma wear processes for track geometry modelling, with application to intervention scheduling

This article discusses the intervention scheduling of a railway track, based on the observation of two dependent randomly increasing deterioration indicators. These two indicators are modelled through a bivariate Gamma process constructed by trivariate reduction. Empirical and maximum likelihood estimators are given for the process parameters and tested on simulated data. An expectation-maximisation (EM) algorithm is used to compute the maximum likelihood estimators. A bivariate Gamma process is then fitted to real data of railway track deterioration. Intervention scheduling is defined, ensuring that the railway track remains of good quality with a high probability. The results arecompared to those based on both indicators taken separately, and also on one single indicator. The policy based onthe joint information is proved to be safer than the other ones, which shows the potential of the bivariate model.

A preventive maintenance policy for a continuously monitored system with correlated wear indicators

A continuously monitored system is considered, that gradually and stochastically deteriorates according to a bivariate non-decreasing Levy process. The system is considered as failed as soon as its bivariate deterioration level enters a failure zone, assumed to be an upper set. A preventive maintenance policy is proposed, which involves a delayed replacement, triggered by the reaching of some preventive zone for the system deterioration level. The preventive maintenance policy is assessed through a cost function on an infinite horizon time. The cost function is provided in full form, and tools are provided for its numerical computation. The influence of different parameters on the cost function is studied, both from a theoretical and/or numerical point of view.

Comparison of numerical methods for the assessment of production availability of a hybrid system

Abstract A finite volume (FV) scheme is proposed in order to compute different probabilistic measures for systems from dynamic reliability field. The FV scheme is tested on a small but realistic benchmark case stemmed from gas industry [Labeau PE, Dutuit Y. Fiabilite dynamique et disponibilite de production: un cas illustratif. Proceedings of λ μ 14, Bourges, France, vol. 2. 2004. p. 431–6 [in French]]. The point is to compute the production availability and the annual frequency of loss of nominal production (among other quantities) for a system of gas production. The results of the FV method are compared to those obtained by Monte Carlo simulation, showing the accuracy of the method.

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.

Discrete random bounds for general random variables and applications to reliability

Abstract We here propose some new algorithms to compute bounds for (1) cumulative density functions of sums of i.i.d. nonnegative random variables, (2) renewal functions and (3) cumulative density functions of geometric sums of i.i.d. nonnegative random variables. The idea is very basic and consists in bounding any general nonnegative random variable X by two discrete random variables with range in h N , which both converge to X as h goes to 0. Numerical experiments are lead on and the results given by the different algorithms are compared to theoretical results in case of i.i.d. exponentially distributed random variables and to other numerical methods in other cases.

Monte Carlo Optimization of the Replacement Strategy of Components subject to Technological Obsolescence

Components are technologically obsolescent when challenger units with higher performances become available. An optimal strategy mixing corrective and preventive replacements must be defined in order to minimize the expected total cost induced by this change of component generation. Realistic operational assumptions can be accounted for, as costs are estimated by a Monte Carlo simulation.

A Random Shock Model with Mixed Effect, Including Competing Soft and Sudden Failures, and Dependence

A system is considered, which is subject to external and possibly fatal shocks, with dependence between the fatality of a shock and the system age. Apart from these shocks, the system suffers from competing soft and sudden failures, where soft failures refer to the reaching of a given threshold for the degradation level, and sudden failures to accidental failures, characterized by a failure rate. A non-fatal shock increases both degradation level and failure rate of a random amount, with possible dependence between the two increments. The system reliability is calculated by four different methods. Conditions under which the system lifetime is New Better than Used are proposed. The influence of various parameters of the shocks environment on the system lifetime is studied.

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.

A bivariate failure time model with random shocks and mixed effects

Two components are considered, which are subject to common external and possibly fatal shocks. The lifetimes of both components are characterized by their hazard rates. Each shock can cause the immediate failure of either one or both components. Otherwise, the hazard rate of each component is increased by a non fatal shock of a random amount, with possible dependence between the simultaneous increments of the two failure rates. An explicit formula is provided for the joint distribution of the bivariate lifetime. Aging and positive dependence properties are described, thereby showing the adequacy of the model as a bivariate failure time model. The influence of the shock model parameters on the bivariate lifetime is also studied. Numerical experiments illustrate and complete the study. Moreover, an estimation procedure is suggested in a parametric framework, under a specific observation scheme.

Modeling and quantification of aging systems for maintenance optimization

To conclude, this method quickly gives us reliability quantities that allow us to find an optimal maintenance. The methodology can be used for many systems. However the limitations are the number of aging components and the system complexity. Indeed, the computational time increases with those two parameters. In the future, we will try to apply this method with more complex systems and keep the computational time low. Focus will also be put on the determination of importance and sensibility indicators, in order to be able to identify the essential components of the systems, from the maintenance optimization point of view.