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**Stochastic models in reliability.**
*(English)*
Zbl 0926.60006

Applications of Mathematics. 41. New York, NY: Springer. xii, 270 p. (1999).

This monograph deals with the “key research areas of reliability theory, summarizing and extending results obtained in recent years” (from the preface). The new dynamical approach comes from the fact that information about the system grows as the time goes on. In estimating the future states we may use not only the information of the actual state but also of the previous states. Moreover the failure rate does not stay constant but may evolve in deterministic or even stochastic way, depending on the status of some other elements or systems. This book provides a rigorous introduction for those who do not have a good framework in stochastic processes. The problems under consideration are mainly based on the stochastic methods, including martingales, semimartingales, counting processes, renewal processes and optimal stopping times.

The text centered around the topics which form the essentials of reliability comprises five chapters and two appendices. Chapter 1 has introductory character. It deals with the following problems: lifetime models, complex systems, point processes and availability analysis. Chapter 2 is more formal and presents an overview of the basic (traditional) theory of reliability, among others the binary monotone systems and methods of computing reliability. Chapter 3 is mainly devoted to the new concept of hazard function (failure rate), which is a stochastic process determined on a given filtered probability space. Here the smooth semimartingale representation is employed. It makes possible to construct a general lifetime model and to find a lot of reliability models described by point processes (among others a shock model with state-dependent failure probability and minimal repaire models).

In Chapter 4 various methods of computing performance measures of monotone systems with repairable elements are given. The point and interval availability is studied too (multistate systems including). The downtime distribution of the failed system is also considered. Chapter 5 is devoted to the maintenance optimization, the lifetime modeling and optimization criteria for various replacement models. Numerical illustrations for many special cases are presented. A semimartingale approach for solving the optimal stopping time is used in burn-in models. The appendices comprise the background in probability and stochastic processes with filtered probability space and martingale theory and basic theory of renewal processes. Bibliographic notes follow each chapter. The number of references is 178, mainly from the last twenty years. Most of examples focuse on the situation that the systems or/and their components have exponential lifetime distributions.

Resuming we can conclude that the book gives a successful presentation of some classical areas of reliability based on the modern theory of stochastic processes.

The text centered around the topics which form the essentials of reliability comprises five chapters and two appendices. Chapter 1 has introductory character. It deals with the following problems: lifetime models, complex systems, point processes and availability analysis. Chapter 2 is more formal and presents an overview of the basic (traditional) theory of reliability, among others the binary monotone systems and methods of computing reliability. Chapter 3 is mainly devoted to the new concept of hazard function (failure rate), which is a stochastic process determined on a given filtered probability space. Here the smooth semimartingale representation is employed. It makes possible to construct a general lifetime model and to find a lot of reliability models described by point processes (among others a shock model with state-dependent failure probability and minimal repaire models).

In Chapter 4 various methods of computing performance measures of monotone systems with repairable elements are given. The point and interval availability is studied too (multistate systems including). The downtime distribution of the failed system is also considered. Chapter 5 is devoted to the maintenance optimization, the lifetime modeling and optimization criteria for various replacement models. Numerical illustrations for many special cases are presented. A semimartingale approach for solving the optimal stopping time is used in burn-in models. The appendices comprise the background in probability and stochastic processes with filtered probability space and martingale theory and basic theory of renewal processes. Bibliographic notes follow each chapter. The number of references is 178, mainly from the last twenty years. Most of examples focuse on the situation that the systems or/and their components have exponential lifetime distributions.

Resuming we can conclude that the book gives a successful presentation of some classical areas of reliability based on the modern theory of stochastic processes.

Reviewer: D.Bobrowski (Poznań)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

62N05 | Reliability and life testing |