Aven T., Jensen U. Stochastic Models in Reliability

New York: Springer, 1999. - 270 p. - ISBN: 0-387-98633-2.The aim of Stochastic Models in Reliability is to give a comprehensive up-to-date presentation of some of the classical areas of reliability, based on a more advanced probabilistic framework using the modern theory of stochastic processes. This framework allows the analyst to formulate general failure models, establish formulas for computing various performance measures, as well as to determine how to identify optimal replacement policies in complex situations. A number of special cases analyzed previously can be included in this framework. Our book presents a unifying approach to some of the key research areas of reliability theory, summarizing and extending results obtained in recent years. Having future work in this area in mind, it will be useful to have at hand a general setup where the conditions and assumptions are formulated independently of particular models.This book comprises five chapters in addition to two appendices.
Chapter 1 gives a short introduction to stochastic models of reliability, linking existing theory and the topics treated in this book. It also contains an overview of some questions and problems to be treated in the book. In addition Section 1.1.6 explains why martingale theory is a useful tool for describing and analyzing the structure of complex reliability models. In the final section of the chapter we briefly discuss some important aspects of reliability modeling and analysis, and present two real-life examples. To apply reliability models in practice successfully, there are many challenges related to modeling and analysis that need to be faced. However, it is not within the scope of this book to discuss these challenges in detail. Our text is an introduction to the topic and of motivational character.
Chapter 2 presents an overview of some parts of basic reliability theory: the theory of complex (monotone) systems, both binary and multistate systems, as well as lifetime distributions and nonparametric classes of lifetime distributions. The aim of this chapter has not been to give a complete overview of the existing theory, but to highlight important areas and give a basis for the coming chapters.
Chapter 3 presents a general set-up for analyzing failure-prone systems. A (semi-) martingale approach is adopted. This general approach makes it possible to formulate a unifying theory of both nonrepairable and repairable systems, and it includes point processes, counting processes, and Markov processes as special cases. The time evolution of the system can also be analyzed on different information levels, which is one of the main attractions of the (semi-) martingale approach. Attention is drawn to the failure rate process, which is a key parameter of the model. Several examples of application of the set-up are given, including a monotone (coherent) system of possibly dependent components, and failure time and (minimal) repair
models. A model for analyzing the time to failure based on risk reserves (the difference between total income and accumulated costs of repairs) is also covered.
In the next two chapters we look more closely at types of models for analyzing situations where the system and its components could be repaired or replaced in the case of failures, and where we model the downtime or costs associated with downtimes.
Chapters 3 to 5 are based on stochastic process theory, including theory of martingales and point, counting, and renewal processes. For the sake of completeness and to help the reader who is not familiar with this theory, two appendices have been included summarizing the mathematical basis and some key results.
Appendix A gives a general introduction to probability and stochastic process theory, whereas Appendix B gives a presentation of results from renewal theory. Appendix A also summarizes basic notation and symbols.