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Several internal degradation states are considered which are observed when a random inspection occurs. This unit is subject to internal repairable failure, external shocks and preventive maintenance.

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If one internal repairable failure occurs, the unit goes to the repair facility for corrective repair, if a major degradation level is observed by inspection, the unit goes to preventive maintenance and when one external shock happens, this one may produce an aggravation of the internal degradation level, cumulative external damage or external extreme failure non-repairable failure.

Preventive maintenance and corrective repair times follow different distributions.

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The system is modelled in transient regime and relevant performance measures are obtained. A numerical example shows the versatility of the model. In this work we are focused on multi-state systems modeled by means of a special type of semi-Markov processes. The sojourn times are seen to be independent not necessarily identically distributed random variables and assumed to belong to a general class of distributions closed under extrema that includes, in addition to some discrete distributions, several typical reliability distributions like the exponential, Weibull, and Pareto.

A special parametrization is proposed for the parameters describing the system, taking thus into account various types of dependencies of the parameters on the the states of the system. We obtain maximum likelihood estimators of the parameters and plug-in type estimators are furnished for the basic quantities describing the semi-Markov system under study. The most of the contemporary large scale technological systems are functioning under multiple stages of degradation, from their perfect state to their total failure.

Under a proper inspection and maintenance policy, it is feasible the operation of the system to be improved significantly. Our main goal is to model multi-state systems with redundancy and to identify the optimal maintenance policies. The system is inspected periodically. Depending on the condition of the system, either no action takes place or maintenance is carried out, either minimal or major.

The proposed model takes also into account the scenario of imperfect and failed maintenance. The asymptotic behaviour of the system is studied and optimization problems for the asymptotic availability, the downtime cost and the expected cost due to maintenance and unavailability, with respect to inspection intervals, are formulated and solved. We present an extension of the phase-type methodology for modeling of lifetime distributions to include the case of competing risks. This is done by considering finite state Markov chains in continuous time with more than one absorbing state, letting each absorbing state correspond to a particular risk.

The special structure of Coxian phase-type models is considered in particular. The chapter emphasizes the use of phase-type models in statistical modeling and inference for survival and competing risks data. Consider a repairable series system consisting of n units and each of them follows a Markov process with finite state and continuous time.

Under independence assumption among units, the repairable series system has been widely studied by using the Markov process method and Lz -transform method. However, both methods have faced the problem of state exploration although some approximation methods have been used. Thus, it is still an interesting and significant problem to be explored. In this chapter, we investigate repairable series systems by using matrix method which has been widely used in aggregated stochastic processes especially in ion channel modeling and aggregated repairable systems.

The formulas for reliability, instantaneous and interval availabilities are given in matrix form for four kinds of repairable series systems, general repairable series system, general repairable series system with neglected failures, phased-mission repairable series system and phased-mission repairable series system with neglected failures, respectively. Numerical examples are shown to illustrate the results for the four kinds of systems and present how matrix method is used to solve the problem of state exploration in the Lz -transform method.

Finally, the conclusions and some future possible applications are given. This chapter presents a method for evaluating dynamic performance of multi-state systems with a general series parallel structure. The system components can be either repairable binary elements with given time-to-failure and repair time distributions, or 1-out-of- N warm standby configurations of heterogeneous binary elements characterized by different performances and time-to-failure distributions.

The entire system needs to satisfy a random demand defined by a time-dependent distribution. Iterative algorithms are presented for determining performance stochastic processes of individual components. A universal generating function technique is implemented for evaluating the dynamic system performance indices.

Examples are provided to demonstrate applications of the proposed methodology.

Mod-01 Lec-38 Application to reliability theory failure law

In this research, we are concerned with the modeling of optimal maintenance actions in multi-state systems. Most of the imperfect maintenance models that have been investigated in literature use either imperfect preventive maintenance actions or imperfect corrective maintenance actions. It will greatly benefit scientists and researchers working in reliability, as well as practitioners and managers with an interest in reliability and performability analysis.

It can also be used as a textbook or as a supporting text for postgraduate courses in Industrial Engineering, Electrical Engineering, Mechanical Engineering, Applied Mathematics, and Operations Research. Help Centre. My Wishlist Sign In Join. Frenkel Editor , Alex Karagrigoriou Editor. Be the first to write a review.

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