On the Decreasing Failure Rate property for general counting process. Results based on conditional interarrival times

In the present paper we consider general counting processes stopped at a random time $T$, independent of the process. Provided that $T$ has the decreasing failure rate (DFR) property, we give sufficient conditions on the arrival times so that the number of events occurring before $T$ preserves the DFR property of $T$. These conditions involve the study of the conditional interarrival times. As a main application, we prove the DFR property in a context of maintenance models in reliability, by the consideration of Kijima type I virtual age models under quite general assumptions

Shock models governed by an inverse gamma mixed Poisson process

We study three classes of shock models governed by an inverse gamma mixed Poisson process (IGMP), namely a mixed Poisson process with an inverse gamma mixing distribution. In particular, we analyze (1) the extreme shock model, (2) the delta-shock model, and the (3) cumulative shock model. For the latter, we assume a constant and an exponentially distributed random threshold and consider different choices for the distribution of the amount of damage caused by a single shock. For all the treated cases, we obtain the survival function, together with the expected value and the variance of the failure time. Some properties of the inverse gamma mixed Poisson process are also disclosed

Optimal replacement policy under a general failure and repair model: Minimal versus worse than old repair

We analyze the optimal replacement policy for a system subject to a general failure and repair model. Failures can be of one of two types: catastrophic or minor. The former leads to the replacement of the system, whereas minor failures are followed by repairs. The novelty of the proposed model is that, after repair, the system recovers the operational state but its condition is worse than that just prior to failure (worse than old). Undertrained operators or low quality spare parts explain this deficient maintenance. The corresponding failure process is based on the Generalized Pólya Process which presents both the minimal repair and the perfect repair as special cases. The system is replaced by a new one after the first catastrophic failure, and also undergoes two sorts of preventive maintenance based on age and after a predetermined number of minor failures whichever comes first. We derive the long-run average cost rate and study the optimal replacement policy. Some numerical examples illustrate the comparison between the as bad-as-old and the worse than old conditions

(Batch) Markovian arrival processes: the identifiability issue and other applied aspects

Mención Internacional en el título de doctorThis dissertation is mainly motivated by the problem of statistical modeling via a specific point process, namely, the Batch Markovian arrival processes. Point processes arise in a wide range of situations of our daily activities, such as people arriving to a bank, claims of an insurance company or failures in a system. They are defined by the occurrence of an event at a specific time, where the event occurrences may be understood from different perspectives, either by the arrival of a person or group of people in a waiting line, the different claims to the insurance companies or failures occurring in a system. Point processes are defined in terms of one or several stochastic processes which implies more versatility than mere single random variables, for modeling purposes. A traditional assumption when dealing with the analysis of point processes is that the occurrence of events are independent and identically distributed, which considerably simplifies the theoretical calculations and computational complexity, and again because of simplicity, the Poisson process has been widely considered in stochastic modelling. However, the independence and exponentiability assumptions become unrealistic and restrictive in practice. For example, in teletraffic or insurance contexts it is usual to encounter dependence amongst observations, high variability, arrivals occurring in batches, and therefore, there is a need of more realistic models to fit the data. In particular, in this dissertation we investigate new theoretical and applied properties concerning the (batch) Markovian arrival processes, or (B)MAP, which is well known to be a versatile class of point process that allows for dependent and non-exponentially distributed inter-event times as well as correlated batches. They inherit the tractability of the Poisson processes, and turn out suitable models to fit data with statistical features that differ from the classical Poisson assumptions. In addition, in spite of the large amount of works considering the BMAP, still there are a number of open problems which are of interest and which shall be considered in this dissertation. This dissertation is organized as follows. In Chapter 1, we present a brief theoretical background that introduces the most important concepts and properties that are needed to carry out our analyses. We give a theoretical background of point processes and describe them from a probabilistic point of view. We introduce the Markovian point processes and its main properties, and also provide some point process estimation backdrop with a review of recent works. An important problem to consider when the statistical inference for any model is to be developed is the uniqueness of its representation, the identifiability problem. In Chapter 2 we analyze the identifiability of the non-stationary two-state MAP. We prove that, when the sample information is given by the inter-event times, then, the usual parametrization of the process is redundant, that is, the process is nonidentifiable. We present a methodology to build an equivalent non-stationary two-state MAPs from any fixed one. Also, we provide a canonical and unique parametrization of the process so that the redundant versions of the same process can be reduced to its canonical version. In Chapter 3 we study an estimation approach for the parameters of the non-stationary version of the MAP under a specific observed information. The framework to be considered is the modelling of the failures of N electrical components that are identically distributed, but for which it is not reasonable to assume that the operational times related to each component are independent and identically distributed. We propose a moments matching estimation approach to fit the data to the non-stationary two-state MAP. A simulated and a real data set provided by the Spanish electrical group Iberdrola are used to illustrate the approach. Unlike Chapters 2 and 3, which are devoted to the Markovian arrival process, Chapters 4 and 5 focus on its arrivals-in-batches counterpart, the BMAP. The capability of modeling non-exponentially distributed and dependent inter-event times as well as correlated batches makes the BMAP suitable in different real-life settings as teletraffic, queueing theory or actuarial contexts, to name a few. In Chapter 4 we analyze the identifiability issue of the BMAP. Specifically, we explore the identifiability of the stationary two-state BMAP noted as BMAP2(k), where k is the maximum batch arrival size, under the assumptions that both the inter-event times and batches sizes are observed. It is proven that for k ≥ 2 the process cannot be identified. The proof is based on the construction of an equivalent BMAP2(k) to a given one, and on the decomposition of a BMAP2(k) into k BMAP2(2)s. In Chapter 5 we study the auto-correlation functions of the inter-event times and batch sizes of the BMAP. This chapter examines the characterization of both auto-correlation functions for the stationary BMAP2(k), for k ≥ 2, where four behavior patterns are identified for both functions for the BMAP2(2). It is proven that both auto-correlation functions decrease geometrically as the time lag increases. Also, the characterization of the autocorrelation functions has been extended for the general BMAPm(k) case, m ≥ 3. To conclude, Chapter 6 summarizes the most significant contributions of this dissertation, and also give a short description of possible research lines.Esta tesis está motivada por el problema de modelización estadística mediante un tipo específico de procesos puntuales, los procesos de llegada Markovianos en tandas. Los procesos puntuales surgen en una gran variedad de situaciones de la vida real, como las personas que llegan a un banco, reclamaciones en compañías de seguro o fallos en un sistema. Los procesos puntuales se definen como la ocurrencia de eventos en diferentes instantes temporales, donde las ocurrencias de eventos se pueden entender desde diferentes perspectivas, llegadas de personas o un grupo de personas a una cola, las distintas reclamaciones en una compañía de seguros o los fallos que ocurren en un sistema. Los procesos puntuales se definen en términos de uno o varios procesos estocásticos lo que implica más versatilidad, en términos de modelización, que la que se obtiene mediante variables aleatorias que no consideren la dimensión temporal. Una suposición tradicional en la literatura al estudiar y analizar procesos puntuales es que los tiempos entre la ocurrencia de eventos son independientes e idénticamente distribuidos, lo que simplifica considerablemente los cálculos teóricos y la complejidad computacional. Adicionalmente, por simplicidad, el proceso de Poisson ha sido ampliamente considerado en modelización estocástica. Sin embargo, las suposiciones de independencia y exponenciabilidad son poco realistas en la práctica. Por ejemplo, en el contexto teletráfico o de seguros es usual encontrar dependencia entre las observaciones, alta variabilidad, llegadas que ocurren en tandas, por lo que hay una necesidad de ajustar los datos a modelos más reales. En particular, en esta tesis investigamos nuevas propiedades teóricas y aplicadas sobre los procesos de llegada Markovianos (en tanda), denotados (B)MAP, que son conocidos por ser procesos puntuales versátiles que permiten la dependencia y no-exponenciabilidad de los tiempos entre eventos, así como la correlación entre las tandas. Ya que heredan la manejabilidad de los procesos de Poisson, son procesos adecuados para ajustar datos con características estadísticas que difieren de los supuestos clásicos de Poisson. Además, a pesar de la gran cantidad de trabajos que consideran los BMAP, todavía hay una serie de problemas abiertos que son de interés y que serán considerados en esta tesis. La estructura de esta tesis es la siguiente. En el Capítulo 1, se presenta una breve revisión teórica que introduce las definiciones y propiedades más importantes necesarias para el desarrollo de nuestros análisis. Se definen los procesos puntuales y se describen desde un punto de vista probabilístico. Se introducen los procesos puntuales Markovianos y sus propiedades principales, además se proporciona una revisión de la literatura sobre la estimación de los procesos puntuales. Un problema importante a considerar cuando se quieren desarrollar métodos de inferencia sobre cualquier modelo es la unicidad de su parametrización, o alternativamente, el problema de identificabilidad. En el Capítulo 2 estudiamos el problema de identificabilidad del MAP no estacionario con dos estados. Se demuestra que, cuando la información muestral está dada por los tiempos entre eventos, entonces, la parametrización usual del proceso es redundante, esto es, el proceso es no-identificable. Se presenta un procedimiento para construir un MAP no estacionario con dos estados equivalente a uno fijo. Además, se proporciona una parametrización canónica y única del proceso, de manera que las versiones redundantes o equivalentes de un mismo proceso se pueden reducir a su versión canónica. En el Capítulo 3 se estudia un método de estimación para los parámetros del MAP no estacionario con dos estados. El esquema que se considerará es la modelización de los fallos de N componentes eléctricos que son idénticamente distribuidos, pero que no es razonable considerar que los tiempos operacionales asociados a cada componente son independientes ni idénticamente distribuidos. Se propone un método de igualdad de momentos para ajustar datos a un MAP no estacionarios con dos estados. Se presenta un ejemplo simulado y un ejemplo con datos reales proporcionados por la compañía eléctrica Iberdrola para ilustrar la metodología propuesta. A diferencia de los capítulos 2 y 3, que están dedicados a los procesos de llegada Markovianos, los capítulos 4 y 5 se centran en su generalización para considerar llegadas en tandas, el BMAP. La capacidad de modelar tiempos entre eventos dependientes y no-exponenciales, así como llegadas en tandas correladas, hace que los BMAP sean modelos apropiados en problemas de la vida real, como en contextos teletráficos, de teoría de colas o actuariales, entre otros. En el Capítulo 4 se explora la identificabilidad para el BMAP estacionario de 2 estados, BMAP2(k), donde k es el tamaño máximo de las tandas, bajo la suposición de que los tiempos entre eventos y los tamaños de las tandas son los datos observados. Se demuestra que para k ≥ 2 el proceso no es único. La demostración se basa en la construcción de un BMAP2(k) equivalente a uno fijo, y en la descomposición de un BMAP2(k) en k BMAP2(2)s. En el Capítulo 5 se estudia las funciones de autocorrelación para los tiempos entre-eventos y las llegadas en tanda del BMAP. Además, también se examina la caracterización de ambas funciones de autocorrelación para el BMAP2(k), k ≥ 2, estacionario, donde se identifican cuatro patrones para el BMAP2(2). Se demuestra que ambas funciones de autocorrelación decrecen geométricamente. Finalmente, se extiende la caracterización de las funciones de autocorrelación para el caso general BMAPm(k), m ≥ 3. Finalmente, en el Capítulo 6 se resumen las contribuciones más importantes de esta tesis y futuras líneas de investigación.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Rafael Pérez Ocón.- Secretario: D Auria , Bernardo.- Vocal: Mogens Blad

Modèles de fiabilité et de maintenance prédictive de systèmes sujets à des défaillances interactives

RÉSUMÉ: L’interaction des défaillances est une thématique qui prend une ampleur considérable dans le monde de la recherche industrielle moderne. Les systèmes sont de plus en plus complexes et leurs fonctionnements et défaillances sur le long terme sont sujets à diverses sources d’influence internes et externes. Les actifs physiques en particulier sont soumis à l’impact du temps, de l’environnement et du rythme de leur utilisation. Connaître ces sources d’influence n’est pas suffisant car il importe de comprendre quelles sont les relations qui les lient afin de planifier de façon efficiente la maintenance des actifs. En effet, cette dernière peut s’avérer très couteuse et sa mauvaise planification peut conduire à l’utilisation de systèmes dangereux pouvant engendrer des évènements catastrophiques. La fiabilité est un vaste domaine. Elle propose une large panoplie de modèles mathématiques qui permettent de prédire le fonctionnement et les défaillances des actifs physiques. Ceci dit, les concepts des modèles les plus appliqués à ce jour se basent sur des hypothèses parfois simplistes et occultent bien souvent certaines relations de dépendances qui régissent un système. L’interaction des défaillances dans le cadre des dépendances stochastiques est abordée par de nombreux travaux de recherches. Par contre, la compréhension et l’implémentation de ces travaux demeurent un défi pour les spécialistes en maintenance qui ont besoin de modèles réalistes pour une maintenance préventive efficace. Cette thèse traite de la fiabilité et la maintenance prédictive des actifs physiques en exploitation et sujets à divers modes de défaillance interactifs. Elle établit avant tout l’importance d’accorder une attention particulière à l’interaction des défaillances dans le domaine de la fiabilité et de la maintenance. Dans une revue de littérature, les concepts et les méthodes de modélisation et d’optimisation en fiabilité et en maintenance préventive sont présentés. Les divers types de dépendances dans un système sont discutés. Un cas d’application, à savoir celui des ponceaux en béton, est proposé. Les travaux entrepris par la suite fournissent avant tout un cadre pour la modélisation de la fiabilité incluant l’interaction des défaillances. A cette fin, une étude comparative des modèles existants les plus pertinents est effectuée de points de vue conceptuel, méthodologique et applicatif. Le cadre étant défini, un modèle basé sur les chocs extrêmes et les chaînes de Markov est construit afin de valoriser le caractère séquentiel des défaillances interactives. Cette proposition est améliorée pour prendre en compte la dégradation du système. Une stratégie de maintenance prédictive est conséquemment développée. Toutes ces approches sont appliquées à un ensemble de ponceaux en béton observés sur plusieurs années. Cela permet d’expliquer les dépendances entre l’occurrence de déplacements et l’occurrence de fissures dans une structure. Tous ces concepts et résultats sont finalement discutés afin de déterminer des perspectives réalistes pour une étude approfondie de l’interactivité d’un point de vue fiabiliste et dans un but stratégique pour la planification de la maintenance.----------ABSTRACT: Failure interaction is a subject gaining growing attention in the world of modern industrial research. Systems are becoming increasingly complex. Their life cycles are subject to various internal and external influences. Physical assets in particular are impacted by time, environment and usage. Knowing these sources of influence is not enough. Indeed, it is important to understand the relationships between them in order to plan effectively for the maintenance of assets. Maintenance can be quite expensive. Thus, poor planning can lead to dangerous systems that could cause catastrophic events. Reliability engineering offers a wide range of mathematical models to predict failures. That being said, the concepts of the most widely applied models in the industry are often based on simplistic assumptions and tend to overlook certain dependencies within a system. Failure interaction in the context of stochastic dependencies is largely addressed in the literature. However, understanding and implementing the proposed approaches remains a challenge for maintenance specialists that need realistic models for efficient maintenance planning. This thesis focuses on the reliability and predictive maintenance of physical assets subject to interactive failure modes. First of all, it emphasizes the importance of paying particular attention to failure interaction. In a literature review, the concepts and methods for modeling and optimizing reliability and preventive maintenance are presented. The diverse dependencies in a system are discussed. A case study is proposed, namely concrete culverts. Subsequently, the research provides a framework for modeling reliability that integrates the interaction of failures. To this end, the most relevant models in the literature are comparatively studied from a conceptual, methodological and applicative point of view. In the defined framework, a model based on extreme shocks and Markov processes is built in order to represent the sequential nature of interactive failures. This approach is extended to take into account the natural degradation of a system. A predictive maintenance strategy is consequently developed. All these models are applied to a set of concrete culverts observed over several years. The dependences between the occurrence of displacements and the occurrence of cracks in a structure are explained through these approaches. Finally, these concepts and results are discussed in order to determine realistic perspectives for in-depth studies of the impact of failure interaction on reliability and for strategic maintenance plannin

Optimal Periodic Inspection of a Stochastically Degrading System

This thesis develops and analyzes a procedure to determine the optimal inspection interval that maximizes the limiting average availability of a stochastically degrading component operating in a randomly evolving environment. The component is inspected periodically, and if the total observed cumulative degradation exceeds a fixed threshold value, the component is instantly replaced with a new, statistically identical component. Degradation is due to a combination of continuous wear caused by the component\u27s random operating environment, as well as damage due to randomly occurring shocks of random magnitude. In order to compute an optimal inspection interval and corresponding limiting average availability, a nonlinear program is formulated and solved using a direct search algorithm in conjunction with numerical Laplace transform inversion. Techniques are developed to significantly decrease the time required to compute the approximate optimal solutions. The mathematical programming formulation and solution techniques are illustrated through a series of increasingly complex example problems

Unreliable Retrial Queues in a Random Environment

This dissertation investigates stability conditions and approximate steady-state performance measures for unreliable, single-server retrial queues operating in a randomly evolving environment. In such systems, arriving customers that find the server busy or failed join a retrial queue from which they attempt to regain access to the server at random intervals. Such models are useful for the performance evaluation of communications and computer networks which are characterized by time-varying arrival, service and failure rates. To model this time-varying behavior, we study systems whose parameters are modulated by a finite Markov process. Two distinct cases are analyzed. The first considers systems with Markov-modulated arrival, service, retrial, failure and repair rates assuming all interevent and service times are exponentially distributed. The joint process of the orbit size, environment state, and server status is shown to be a tri-layered, level-dependent quasi-birth-and-death (LDQBD) process, and we provide a necessary and sufficient condition for the positive recurrence of LDQBDs using classical techniques. Moreover, we apply efficient numerical algorithms, designed to exploit the matrix-geometric structure of the model, to compute the approximate steady-state orbit size distribution and mean congestion and delay measures. The second case assumes that customers bring generally distributed service requirements while all other processes are identical to the first case. We show that the joint process of orbit size, environment state and server status is a level-dependent, M/G/1-type stochastic process. By employing regenerative theory, and exploiting the M/G/1-type structure, we derive a necessary and sufficient condition for stability of the system. Finally, for the exponential model, we illustrate how the main results may be used to simultaneously select mean time customers spend in orbit, subject to bound and stability constraints