Jorge Navarro - Introduction to System Reliability Theory

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The purpose of this book is to provide the basic tools for a modern post-graduate introductory course on System Reliability Theory. As the excellent classic book by Barlow and Proschan (1975), the present one is just devoted to probabilistic aspects of that theory, including recent results based on signatures, stochastic orders, aging classes, copulas and distortion (or aggregation) functions.

The only requirements for the readers are basic knowledge on Probability Theory and on the Mathematical tools needed in that theory (basic Algebra and Calculus), so that it can serve both for graduate students in Mathematics and for different Engineering students. Some aspects can also be applied to Survival Analysis, Network Reliability or Simple Game Theory. So it could be of interest for other students/researchers in these fields as well.

For that reason, the book includes short introductions to the basic aspects of lifetime modelling, stochastic comparisons, aging classes, mixtures and copula theory needed for the present course. For the interested readers, we provide appropriate references for more advanced results on these topics. Some basic codes written in the statistical (free) program R are also included.

The purpose is to provide the tools for a short basic course (3060 hours) for different graduate students. So, unfortunately, we have to exclude some relevant aspects on Reliability Theory. For example, the book does not include results based on stochastic processes (since they need more advances courses on probability). Fortunately, there are several recent books on this topic available for the interested readers (see, e.g., Aven and Jensen, 1999; Parzen, 1999; Nakagawa, 2008; Cha and Finkelstein, 2018). Analogously, we do not study here statistical aspects related with reliability data. They are left for a possible second volume.

The book is divided into 5 chapters.

In Chap. , we study the basic properties of coherent and semi-coherent binary system structures, obtaining several representations for the structure Boolean function of the system. We do not study non-coherent systems, but we provide some results for multi-state systems with binary components. Relationships with simple games, connectivity properties of networks, mixed systems and fuzzy measures are provided as well.

Chapter is devoted to studying the (random) system lifetime. First, we study its relationship with the component lifetimes. So we introduce the basic tools needed to study random lifetimes as the reliability function, the Mean Time to Failure (MTTF), the hazard (failure) rate function, or the mean residual life function. The different representations based on signatures and distortion functions are studied.

The basic tools to stochastically compare coherent systems are provided in Chap.. We study distribution-free comparisons, that is, orderings that do not depend on the components distribution functions. We consider five cases: systems with Independent and Identically Distributed (IID) components, with Exchangeable (EXC) components, with Identically Distributed (ID) components, with Independent (IND) components and the general case of arbitrary (dependent or independent) components. We use a copula approach to model the dependence structure between the component lifetimes.

In Chap. , we study the process of growing old for both the system and the components. To this end, we use the main aging classes (IFR, NBU, DMRL, ILR and their respective dual classes). In particular, we state conditions for the preservation of some of these aging classes under the formation of coherent systems. We also consider different system residual and inactivity times. The limiting behavior (when the time increases) of some system aging functions is studied as well.

In Chap. , we study several mechanisms that are used to improve the systems performance. One option is to include some redundant units at some positions in the system. Another popular redundancy option is to add standby components in the system to replace the failed components (when they fail). Another one is to repair these failed components (with perfect or minimal repairs). The main questions analyzed in this chapter are: What is the reliability of the (new) redundant system? What are the best positions to add the redundant components? We also study some component importance indices which can be used to determine the best replacement positions.

I want to thank all the people that helped me in writing this book: my department colleagues, my students and my research collaborators. I do not include names here because I do not know if I will be able to cite all of them.