#### Overview of this book

This book focuses on hot issues of dynamic system reliability, systematically introducing the reliability modeling and analysis methods for systems with imperfect fault coverage, systems with function dependence, systems subject to deterministic or probabilistic common-cause failures, systems subject to deterministic or probabilistic competing failures, and dynamic standby sparing systems. It presents recent developments of such extensions involving reliability modeling theory, reliability evaluation methods, and features numerous case studies based on real-world examples. The presented dynamic reliability theory can enable a more accurate representation of actual complex system behavior, thus more effectively guiding the reliable design of real-world critical systems. The book begins by describing the evolution from the traditional static reliability theory to the dynamic system reliability theory and provides a detailed investigation of dynamic and dependent behaviors in subsequent chapters. Although written for those with a background in basic probability theory and stochastic processes, the book includes a chapter reviewing the fundamentals that readers need to know in order to understand the contents of other chapters that cover advanced topics in reliability theory and case studies.
Preface
Free Chapter
Nomenclature
1 Introduction
2 Fundamental Reliability Theory
3 Imperfect Fault Coverage
4 Modular Imperfect Coverage
5 Functional Dependence
6 Deterministic Common‐Cause Failure
7 Probabilistic Common‐Cause Failure
9 Probabilistic Competing Failure
10 Dynamic Standby Sparing
Index

# 2.2 Reliability Measures

Several quantitative reliability measures for a nonrepairable unit are presented, including the failure function F(t), reliability function R(t), failure rate function z(t), mean time to failure (MTTF), and mean residual life (MRL). Their definitions are based on a continuous r.v. called time to failure (ttf), which is defined in Section 2.2.1.

## 2.2.1 Time to Failure

The time to failure T is a continuous r.v. defined as the time elapsing from when the unit is first put into function until its first failure. It models the lifetime of a nonrepairable unit.

## 2.2.2 Failure Function

The failure function of the unit is given as the cdf of r.v T, that is

(2.16) F t = P T t = 0 t f x dx , --

where f(x) is the pdf of T. F(t) gives the probability that the unit fails within time interval (0, t].