Virtual series-system models of imperfect repair

Novel models of imperfect repair are fitted to classic reliability datasets. The models suppose that a virtual system comprises a component and a remainder in series. On failure of the component, the component is renewed, and on failure of the remainder, the component is renewed and the remainder is minimally repaired. It follows that the repair process is a counting process that is the superposition of a renewal process and a Poisson process. The repair effect, that is, the extent to the system is repaired by renewal of the component, depends on the relative intensities of the superposed processes. The repair effect may be negative, when the intensity of the part that is a renewal process is a decreasing function. Other special cases of the model exist (renewal process, Poisson process, superposed renewal process and homogeneous Poisson process). Model fit is important because the nature of the model and corresponding parameter values determine the effectiveness of maintenance, which we also consider. A cost-minimizing repair policy may be determined provided the cost of preventive-repair is less than the cost of corrective-repair and the repairable part is ageing. If the remainder is ageing, then policy needs to be adapted as it ages

Two methods to approximate the superposition of imperfect failure processes

Suppose a series system is composed of a number of repairable components. If a component fails, it is repaired immediately and the effectiveness of the repair may be imperfect. Then the failure process of the component can be modelled by an imperfect failure process and the failure process of the system is the superposition of the failure processes of all components. In the literature, there is a bulk of research on the superimposed renewal process (SRP) for the case where the repair on each component is assumed perfect. For the case that the component causing the system to fail is unknown and that repair on a failed component is imperfect, however, there is little research on modelling the failure process of the system. Typically, the likelihood functions for the superposition of imperfect failure processes cannot be given explicitly. Approximation-based models have to be sought. This paper proposes two methods to model the failure process of a series system in which the failure process of each component is assumed an arithmetic reduction of intensity and an arithmetic reduction of age model, respectively. The likelihood method of parameter estimation is given. Numerical examples and real-world data are used to illustrate the applicability of the proposed models

Ferramenta Automática para Avaliação da Fiabilidade e Disponibilidade de Sistemas Reparáveis

Nesta dissertação apresentam-se as contribuições de um trabalho de investigação que tem como objetivo o desenvolvimento e implementação de uma ferramenta automática que disponibiliza, quase em tempo real, indicadores de fiabilidade e de disponibilidade para um qualquer sistema reparável. A ferramenta recebe dados recolhidos no equipamento em estudo, modela o processo gerador das avarias por recurso a vários modelos matemáticos/estatísticos, seleciona o modelo mais adequado aos dados recolhidos e, como base nesse modelo, apresenta um conjunto de indicadores úteis. Estes indicadores podem ser usados, tanto para prever o comportamento do sistema, como para avaliar como é que este responde a ações externas, como, por exemplo, à respetiva manutenção. Os trabalhos são iniciados com uma revisão dos conceitos e instrumentos fundamentais que suportam, grosso modo, o trabalho desenvolvido, seguindo-se o desenho, implementação e validação da ferramenta. A ferramenta é desenvolvida em ambiente Python e de forma automática. Este automatismo facilita a obtenção de resultados em tempo quase real e permite que o perito do processo tenha acesso a um conjunto de indicadores do estado do sistema sem que tenha conhecimentos específicos sobre modelação de processo geradores de avarias. O resumo dos dados recolhidos, assim como a informação produzida pela ferramenta desenvolvida são apresentados de forma organizada através de um dashboard, também desenvolvido em ambiente Python, de forma que a interpretação dos resultados seja mais rápida e eficiente.This dissertation presents the contributions of a research work that aims to develop and implement an automatic tool that provides, almost in real time, reliability, and availability indicators for any repairable system. The tool receives data collected from the equipment, models the process that generates the faults using various mathematical/statistical models, selects the most appropriate model for the data collected and based on that model, presents a set of useful indicators. These indicators can be used both to predict the behavior of the system and to assess how it responds to external actions, such as, for example, its maintenance. The work begins with a review of the fundamental concepts and instruments that roughly support the work developed, followed by the design, implementation, and validation of the tool. The tool is developed in a Python environment and automatically. This automatism facilitates obtaining results in near real time and allows the process expert to have access to a set of system status indicators without having specific knowledge about modeling fault-generating processes. The summary of the collected data, as well as the information produced by the developed tool, are presented in an organized way through a dashboard, also developed in a Python environment, so that the interpretation of the results is faster and more efficient

Modèles de renouvellement avec effets de tendance, et application à l'assurance pour fautes des professionnels de la santé

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2018-2019.Dans cette thèse, nous présentons une classe très large de processus de dénombrement, incluant le processus de renouvellement et le processus de Poisson non-homogène, à laquelle s’ajouteront des taux d’escompte stochastiques afin de modéliser les coûts agrégés liés aux assurances pour fautes des professionnels de la santé. Ainsi, dans l’introduction, nous présentons certaines caractéristiques importantes du processus des coûts agrégés liés aux assurances pour fautes des professionnels de la santé. Au chapitre 1, nous présentons des concepts théoriques préalables à l’élaboration et l’application du modèle mathématique qui sera proposé au chapitre 4. Au chapitre 2, nous présentons des résultats liés aux processus de Poisson non-homogène composé et de Cox composé, avec escompte. En particulier, nous y présentons des expressions analytiques pour les fonctions génératrices des moments qui seront inversées numériquement en utilisant la transformée de Fourier afin d’obtenir une approximation de la fonction de répartition. Au chapitre 3, nous considérons une classe de processus qui généralise celle étudiée au chapitre 2 : les processus de renouvellement composés, avec effet de tendance et escompte. Pour cette nouvelle classe, nous obtenons des formules récursives pour le calcul des moments ainsi que des expressions analytiques pour la fonction génératrice des moments, fonction qui peut être inversée analytiquement ou numériquement dans plusieurs cas particuliers afin d’obtenir une expression exacte ou une approximation de la fonction de répartition. Au chapitre 4, nous présentons les hypothèses du modèle stochastique qui servira à évaluer le risque du processus des coûts agrégés liés aux assurances pour fautes des professionnels de la santé, ce dernier généralisant la classe de modèles considérée au chapitre 3. Au chapitre 5, nous calibrons le modèle proposé au chapitre 4 sur la base de données des réclamations « fermées » d’une compagnie d’assurance de la Floride. Finalement, nous concluons cette thèse avec un résumé des nouveaux résultats et une discussion sur les avenues de recherches potentielles liées à la présente thèse.In this thesis, we present a very large class of counting processes including the renewal process and the non-homogeneous Poisson process, to which we add stochastic discount rates, in order to model the aggregate cost related to medical malpractice insurance. In the introduction, we present some important characteristics related to the cost process of medical malpractice insurance. In Chapter 1, we present some theoretical concepts that will be used to build the aggregate cost process related to the medical malpractice insurance model that is proposed in Chapter 4. In Chapter 2, we present some results related to the compound non-homogeneous Poisson and compound Cox processes with a discount factor. In particular, we derive an analytic expression for the moment generating functions that will be inverted numerically using Fourier transforms in order to obtain an approximation of the probability distribution function. In Chapter 3, we study a class of models that generalizes the class of models studied in Chapter 2 : the compound trend renewal process with discount factor. For this new class of processes, we obtain recursive formulas for the moment calculations and an analytic expression for the moment generating function. The moment generating function can be inverted analytically or numerically for many particular cases in order to obtain an exact expression or an approximation of the probability distribution function. In Chapter 4, we present the stochastic model that will be used to measure the risk of an agregate cost related to medical malpractice insurance, which also generalizes the class of models considered in Chapter 3. In Chapter 5, we calibrate the model proposed in Chapter 4 on the closed claims database of Florida. The conclusion follows with a short summary of the results and an outline of some extensions for future research

Nonparametric estimation in trend-renewal processes

This thesis gives an introduction to stochastic modeling of repairable systems with failure and maintenance data, in particular the nonhomogeneous Poisson process and the trend-renewal process. It is studying kernel-based methods for nonparametric estimation of the trend function of trend-renewal processes and presents a method using weighted kernel estimation. These weights are found by maximization of the likelihood function that they are included in. The method is then tested on both real and simulated data sets