Probabilistic Reliability Models E-Book

Igor A. Ushakov

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 201

Veröffentlichungsjahr: 2012

Beschreibung

Practical Approaches to Reliability Theory in Cutting-Edge Applications Probabilistic Reliability Models helps readers understand and properly use statistical methods and optimal resource allocation to solve engineering problems. The author supplies engineers with a deeper understanding of mathematical models while also equipping mathematically oriented readers with a fundamental knowledge of the engineeringrelated applications at the center of model building. The book showcases the use of probability theory and mathematical statistics to solve common, real-world reliability problems. Following an introduction to the topic, subsequent chapters explore key systems and models including: * Unrecoverable objects and recoverable systems * Methods of direct enumeration * Markov models and heuristic models * Performance effectiveness * Time redundancy * System survivability * Aging units and their related systems * Multistate systems Detailed case studies illustrate the relevance of the discussed methods to real-world technical projects including software failure avalanches, gas pipelines with underground storage, and intercontinental ballistic missile (ICBM) control systems. Numerical examples and detailed explanations accompany each topic, and exercises throughout allow readers to test their comprehension of the presented material. Probabilistic Reliability Models is an excellent book for statistics, engineering, and operations research courses on applied probability at the upper-undergraduate and graduate levels. The book is also a valuable reference for professionals and researchers working in industry who would like a mathematical review of reliability models and the relevant applications.

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

Preface

Acronyms and Notation

Acronyms

Notation

Chapter 1: What Is Reliability?

1.1 Reliability as a Property of Technical Objects

1.2 Other “Ilities”

1.3 Hierarchical Levels of Analyzed Objects

1.4 How Can Reliability be Measured?

1.5 Software Reliability

Chapter 2: Unrecoverable Objects

2.1 Unit

2.2 Series Systems

2.3 Parallel System

2.4 Structure of Type “K-out-of-n”

2.5 Realistic Models of Loaded Redundancy

2.6 Reducible Structures

2.7 Standby Redundancy

2.8 Realistic Models of Unloaded Redundancy

Chapter 3: Recoverable Systems: Markov Models

3.1 Unit

3.2 Series System

3.3 Dubbed System

3.4 Parallel Systems

3.5 Structures of Type “M-out-of-n”

Chapter 4: Recoverable Systems: Heuristic Models

4.1 Preliminary Notes

4.2 Poisson Process

4.3 Procedures over Poisson Processes

4.4 Asymptotic Thinning Procedure over Stochastic Point Process

4.5 Asymptotic Superposition of Stochastic Point Processes

4.6 Intersection of Flows of Narrow Impulses

4.7 Heuristic Method for Reliability Analysis of Series Recoverable Systems

4.8 Heuristic Method for Reliability Analysis of Parallel Recoverable Systems

4.9 Brief Historical Overview and Related Sources

Bibliography

Chapter 5: Time Redundancy

5.1 System with Possibility of Restarting Operation

5.2 Systems with “Admissibly Short Failures”

5.3 Systems with Time Accumulation

5.4 Case Study: Gas Pipeline with an Underground Storage

5.5 Brief Historical Overview and Related Sources

Bibliography

Chapter 6: “Aging” Units and Systems of “Aging” Units

6.1 Chebyshev Bound

6.2 “Aging” Unit

6.3 Bounds for Probability of Failure-Free Operations

6.4 Series System Consisting of “Aging” Units

6.5 Series System

6.6 Parallel System

6.7 Bounds for the Coefficient of Operational Availability

6.8 Brief Historical Overview and Related Sources

Bibliography

Chapter 7: Two-Pole Networks

7.1 General Comments

7.2 Method of Boolean Function Decomposition

7.3 Method of Paths and Cuts

7.4 Brief Historical Overview and Related Sources

Bibliography

Chapter 8: Performance Effectiveness

8.1 Effectiveness Concepts

8.2 General Idea of Effectiveness Evaluation

8.3 Additive Type of System Units' Outcomes

8.4 Case Study: ICBM Control System

8.5 Systems with Intersecting Zones of Action

8.6 Practical Recommendation

8.7 Brief Historical Overview and Related Sources

Bibliography

Chapter 9: System Survivability

9.1 Illustrative Example

9.2 Brief Historical Overview and Related Sources

Bibliography

Chapter 10: Multistate Systems

10.1 Preliminary Notes

10.2 Generating Function

10.3 Universal Generating Function

10.4 Multistate Series System

10.5 Multistate Parallel System

10.6 Reducible Systems

10.7 Conclusion

10.8 Brief Historical Overview and Related Sources

Bibliography

Appendix A: Main Distributions Related to Reliability Theory

A.1 Discrete Distributions

A.2 Continuous Distributions

Appendix B: Laplace Transformation

Appendix C: Markov Processes

C.1 General Markov Process

C.2 Birth–Death Process

Appendix D: General Bibliography

Handbooks

Textbooks and Monographs

Index

Cover Image: © Victor Tongdee/iStockphoto

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Ushakov, I. A. (Igor Alekseevich)

Probabilistic reliability models / Igor A. Ushakov.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-34183-4 (hardback)

1. Reliability (Engineering)–Mathematical models. 2. Probabilities. I. Title.

TA169.U84 2012

320'.00452015192—dc23

2012005946

To Boris Gnedenko, a mentor throughout my life

Preface

I dedicate this book to a great man who was more than my mentor. He took the place of my father who passed away relatively early. Boris Gnedenko was an outstanding mathematician and exceptional teacher. In addition, he was a magnetic personality who gathered around him tens of disciples and students. He was a founder of the famous Moscow Reliability School that produced a number of first-class mathematicians (among them Yuri Belyaev, Igor Kovalenko, Jacob Shor, and Alexander Solovyev) and talented reliability engineers (including Ernest Dzirkal, Vadim Gadasin, Boris Kozlov, Igor Pavlov, Allan Perrote, and Anatoly Raykin).

Why did I write this book?

Since the beginning of my career, I have been working at the junction of engineering and mathematics—I was a reliability engineer. As an engineer by education, I never had a proper mathematical background; however, life has forced me to submerge in depth into the area of probability theory and mathematical statistics. And I was lucky to meet at the start of my career the “three pillars on which rested the reliability theory” in Russia, namely, Boris Gnedenko, Alexander Solovyev, and Yuri Belyaev. They helped me understand the nuances and physical sense of many mathematical methods.

Thus, I have decided to share with the readers my experience, as well as many real mathematical insights, that happened when I submerged myself into reliability theory.

Boris Gnedenko once told me: “Mathematical reliability models are engendered by practice, so they have to be adequate to reality and should not be too complex by their nature.”

To get an understanding of “real reliability,” one goes through a series of painful mistakes in solving real problems. Engineering intuition arrives to mathematicians only after years of working in reliability engineering. At the same time, proper mathematical knowledge comes to reliability engineers after multiple practical uses of mathematical methods and having experienced “finger sensation” of formulas and numbers.

I remember my own thorny way in the course of my professional career. In writing this reliability textbook, I have tried to include as much as possible “physical” explanations of mathematical methods applied in solving reliability problems, as well as “physical” explanations of engineering objects laid on the basis of mathematical models.

At the end of the book, the reader can find a wide list of monographs on reliability. I must, however, note a few books that, in my opinion, are basic in this area. They are (in order of publication) the monographs by Igor Bazovsky (1961), David K. Lloyd and Myron Lipow (1962), Richard Barlow and Frank Proschan (1965), and Boris Gnedenko, Yuri Belyaev, and Alexander Solovyev (1965). These books cover the entire area of probabilistic reliability modeling and contain many important theoretical and practical concepts.

Igor Ushakov

San Diego, CaliforniaMarch 31, 2012

Acronyms and Notation

Acronyms

ACAvailability coefficientd.f.Distribution functionDFRDecreasing failure rateFRFailure rateGFGenerating functioni.i.d.Independent and identically distributed (about r.v.)IFRIncreasing failure rateLTLaplace transformMDTMean downtimeMTBFMean time between failuresMTTFMean time to failureOACOperational availability coefficientPEIPerformance effectiveness indexPFFOProbability of failure-free operationr.v.Random variableRBDReliability block diagramTTFRandom time to failureUGFUniversal generating function

Notation

F(t)Distribution functionKSystem stationary availability coefficientK(t)System nonstationary availability coefficientpk(t)Probability of failure-free operation of unit kP(t)Probability of system's failure-free operationqk(t)Probability of failure of unit kQ(t)Probability of system's failureTMean time to/between failuresXRandom variableλFailure rateτDowntimeξRandom time to/between failuresηRandom recovery timeψ(·)System's structural functionSum by k from 1 to nProduct by k from 1 to nLogic sum (“or”)Logic product (“and”)Logic sum by k from 1 to nLogic product by k from 1 to nSet of natural numbers from 1 to n

Chapter 1

What Is Reliability?

1.1 Reliability as a Property of Technical Objects

Reliability of a technical object is its ability to perform required operations successfully. Usually, it is assumed that an object is used in accordance with its technical requirements and is supported by appropriate maintenance.

One of the outstanding Russian specialists in cybernetics, academician Axel Berg, has said: “Reliability is quality expanded in time.”

Reliability is a broad concept. Of course, its main characterization is the failure-free operation while performing required tasks. However, it also includes such features as availability, longevity, recoverability, safety, survivability, and other important properties of technical objects.

Speaking of reliability, one has to introduce a concept of failure. What does it mean—“successful operation?” Where is the limit of “successfulness?”