Multistate systems reliability : theory with applications

263 Pages·2011·1.533 MB·English

by  Bent Natvig

Multistate Systems Reliability
Theory with Applications
WILEY SERIES IN PROBABILITY AND STATISTICS
Established by WALTERA. SHEWHART and SAMUEL S. WILKS
Editors
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Jozef L. Teugels
A complete list of the titles in this series appears at the end of this volume.
Multistate Systems Reliability
Theory with Applications
Bent Natvig
University of Oslo, Norway
A John Wiley and Sons, Ltd., Publication
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Natvig,Bent,1946-
Multistatesystemsreliabilitytheorywithapplications/BentNatvig.
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ISBN978-0-470-69750-4(cloth)
1.Reliability(Engineering)2.Stochasticsystems.I.Title.
TA169.N382011
620(cid:1).00452 – dc22
2010034245
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Typesetin11/13Times-RomanbyLaserwordsPrivateLimited,Chennai,India
To my Mother, Kirsten,
for giving me jigsaws instead of cars,
and to Helga
for letting me (at least for the last 40 years)
run in the sun
and work when normal people relax.

Contents
Preface xi
Acknowledgements xv
List of abbreviations xvii
1 Introduction 1
1.1 Basic notation and two simple examples 2
1.2 An offshore electrical power generation system 4
1.3 Basic definitions from binary theory 6
1.4 Early attempts to define multistate coherent systems 9
1.5 Exercises 10
2 Basics 11
2.1 Multistate monotone and coherent systems 11
2.2 Binary type multistate systems 16
2.3 Multistate minimal path and cut vectors 22
2.4 Stochastic performance of multistate monotone and
coherent systems 30
2.5 Stochastic performance of binary type multistate strongly
coherent systems 43
2.6 Exercises 46
3 Bounds for system availabilities and unavailabilities 49
3.1 Performance processes of the components
and the system 50
3.2 Basic bounds in a time interval 53
3.3 Improved bounds in a time interval using modular
decompositions 61
3.4 Improved bounds at a fixed point of time using modular
decompositions 67
3.5 Strict and exactly correct bounds 75
viii CONTENTS
3.6 Availabilities and unavailabilities of the components 80
3.7 The simple network system revisited 84
3.8 The offshore electrical power generation system
revisited 91
4 An offshore gas pipeline network 99
4.1 Description of the system 99
4.2 Bounds for system availabilities and unavailabilities 107
5 Bayesian assessment of system availabilities 113
5.1 Basic ideas 113
5.2 Moments for posterior component availabilities and
unavailabilities 117
5.3 Bounds for moments for system availabilities and
unavailabilities 119
5.4 A simulation approach and an application to the simple
network system 127
6 Measures of importance of system components 133
6.1 Introduction 133
6.2 Measures of component importance in nonrepairable
systems 134
6.2.1 The Birnbaum measure 135
6.2.2 The Barlow–Proschan measure 137
6.2.3 The Natvig measure 138
6.3 The Birnbaum and Barlow–Proschan measures
of component importance in repairable systems and
the latter’s dual extension 142
6.3.1 The Birnbaum measure 142
6.3.2 The Barlow–Proschan measure 143
6.3.3 The dual extension of the Barlow–Proschan
measure 147
6.4 The Natvig measure of component importance in
repairable systems and its dual extension 149
6.4.1 The Natvig measure 149
6.4.2 The dual extension of the Natvig measure 154
6.5 Concluding remarks 160
7 Measures of component importance – a numerical study 161
7.1 Introduction 161
7.2 Component importance in two three-component systems 162
7.3 Component importance in the bridge system 173