Mathematical Models for Systems Reliability
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Summary
Evolved from the lectures of a recognized pioneer in developing the theory of reliability, Mathematical Models for Systems Reliability provides a rigorous treatment of the required probability background for understanding reliability theory.
This classroom-tested text begins by discussing the Poisson process and its associated probability laws. It then uses a number of stochastic models to provide a framework for life length distributions and presents formal rules for computing the reliability of nonrepairable systems that possess commonly occurring structures. The next two chapters explore the stochastic behavior over time of one- and two-unit repairable systems. After covering general continuous-time Markov chains, pure birth and death processes, and transitions and rates diagrams, the authors consider first passage-time problems in the context of systems reliability. The final chapters show how certain techniques can be applied to a variety of reliability problems.
Illustrating the models and methods with a host of examples, this book offers a sound introduction to mathematical probabilistic models and lucidly explores how they are used in systems reliability problems.
Table of Contents
Preliminaries
The Poisson process and distribution
Waiting time distributions for a Poisson process
Statistical estimation theory
Generating a Poisson process
Nonhomogeneous Poisson process
Binomial, geometric, and negative binomial distributions
Statistical Life Length Distributions
Stochastic life length models
Models based on the hazard rate
General remarks on large systems
Reliability of Various Arrangements of Units
Series and parallel arrangements
Series-parallel and parallel-series systems
Various arrangements of switches
Standby redundancy
Reliability of a One-Unit Repairable System
Exponential times to failure and repair
Generalizations
Reliability of a Two-Unit Repairable System
Steady-state analysis
Time-dependent analysis via Laplace transform
On model 2(c)
Continuous-Time Markov Chains
The general case
Reliability of three-unit repairable systems
Steady-state results for the n-unit repairable system
Pure birth and death processes
Some statistical considerations
First Passage Time for Systems Reliability
Two-unit repairable systems
Repairable systems with three (or more) units
Repair time follows a general distribution
Embedded Markov Chains and Systems Reliability
Computations of steady-state probabilities
Mean first passage times
Integral Equations in Reliability Theory
Introduction
Example 1: Renewal process with a general distribution
Example 2: One-unit repairable system
Example 3: Effect of preventive replacements or maintenance
Example 4: Two-unit repairable system
Example 5: One out of n repairable systems
Example 6: Section 7.3 revisited
Example 7: First passage time distribution
References
Index
A Problems and Comments section appears at the end of each chapter.
Editorial Reviews
… The material presented in the book is classic material, but is also timeless because the basic theory, probability and statistical rigor and applications to system reliability are still relevant today and important for any student or practitioner of reliability theory. … This book has a very good set of problems, exercises and comments on how to solve these problems for every chapter. These problems are excellent if one wants to use this book as a textbook. … The rigor and mathematical development in the book is excellent. This book is also a good reference book in the field of reliability theory for researchers … .
—Journal of Quality Technology, Vol. 41, No. 2, April 2009
