Table of Contents

Wiley Series in Probability and Statistics

Title Page





List of Abbreviations

Chapter 1: Introduction

1.1 Basic Notation and Two Simple Examples

1.2 An Offshore Electrical Power Generation System

1.3 Basic Definitions from Binary Theory

1.4 Early Attempts to Define Multistate Coherent Systems

1.5 Exercises

Chapter 2: Basics

2.1 Multistate Monotone and Coherent Systems

2.2 Binary Type Multistate Systems

2.3 Multistate Minimal Path and Cut Vectors

2.4 Stochastic Performance of Multistate Monotone and Coherent Systems

2.5 Stochastic Performance of Binary Type Multistate Strongly Coherent Systems

2.6 Exercises

Chapter 3: Bounds for System Availabilities and Unavailabilities

3.1 Performance Processes of the Components and the System

3.2 Basic Bounds in a Time Interval

3.3 Improved Bounds in a Time Interval using Modular Decompositions

3.4 Improved Bounds at a Fixed Point of Time Using Modular Decompositions

3.5 Strict and Exactly Correct Bounds

3.6 Availabilities and Unavailabilities of the Components

3.7 The Simple Network System Revisited

3.8 The Offshore Electrical Power Generation System Revisited

Chapter 4: An Offshore Gas Pipeline Network

4.1 Description of the System

4.2 Bounds for System Availabilities and Unavailabilities

Chapter 5: Bayesian Assessment of System Availabilities

5.1 Basic Ideas

5.2 Moments for Posterior Component Availabilities and Unavailabilities

5.3 Bounds for Moments for System Availabilities and Unavailabilities

5.4 A Simulation Approach and an Application to the Simple Network System

Chapter 6: Measures of Importance of System Components

6.1 Introduction

6.2 Measures of Component Importance in Nonrepairable Systems

6.3 The Birnbaum and Barlow–Proschan Measures of Component Importance in Repairable Systems and the Latter's Dual Extension

6.4 The Natvig Measure of Component Importance in Repairable Systems and its Dual Extension

6.5 Concluding Remarks

Chapter 7: Measures of Component Importance—A Numerical Study

7.1 Introduction

7.2 Component Importance in Two Three-Component Systems

7.3 Component Importance in the Bridge System

7.4 Application to an Offshore Oil and Gas Production System

7.5 Concluding Remarks

Chapter 8: Probabilistic Modeling of Monitoring and Maintenance

8.1 Introduction and Basic Marked Point Process

8.2 Partial Monitoring of Components and the Corresponding Likelihood Formula

8.3 Incorporation of Information from the Observed System History Process

8.4 Cause Control and Transition Rate Control

8.5 Maintenance, Repair and Aggregation of Operational Periods

8.6 The Offshore Electrical Power Generation System

8.7 The Data Augmentation Approach

Appendix A: Remaining Proofs of Bounds given in Chapter 3

A.1 Proof of the Inequalities 14, 7 and 8 of Theorem 3.12

A.2 Proof of inequality 14 of Theorem 3.13

A.3 Proof of Inequality 10 of Theorem 3.17

Appendix B: Remaining Intensity Matrices in Chapter 4



Wiley Series in Probability and Statistics




David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Harvey Goldstein,Geert Molenberghs, David W. Scott, Adrian F.M. Smith, Ruey S. Tsay, Sanford Weisberg

Editors Emeriti

Vic Barnett, Ralph A. Bradley, J. Stuart Hunter, J.B. Kadane, David G. Kendall, Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume.

Title Page

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.


In the magazine Nature (1986) there was an article on a near catastrophe on the night of 14 April 1984 in a French pressurized water reactor (PWR) at Le Bugey on the Rhône river, not far from Geneva. This incident provides ample motivation for multistate reliability theory.

‘The event began with the failure of the rectifier supplying electricity to one of the two separate 48 V direct-current control circuits of the 900 MW reactor which was on full power at the time. Instantly, a battery pack switched on to maintain the 48 V supply and a warning light began to flash at the operators in the control room. Unfortunately, the operators ignored the light (if they had not they could simply have switched on an auxiliary rectifier).

What then happened was something that had been completely ignored in the engineering risk analysis for the PWR. The emergency battery now operating the control system began to run down. Instead of falling precipitously to zero, as assumed in the “all or nothing” risk analysis, the voltage in the control circuit steadily slipped down from its nominal 48 V to 30 V over a period of three hours. In response a number of circuit breakers began to trip out in an unpredictable fashion until finally the system, with the reactor still at full power, disconnected itself from the grid.

The reactor was now at full power with no external energy being drawn from the system to cool it. An automatic “scram” system then correctly threw in the control rods, which absorbed neutrons and shut off the nuclear reaction. However, a reactor in this condition is still producing a great deal of heat −300 MW in this case. An emergency system is then supposed to switch in a diesel generator to provide emergency core cooling (otherwise the primary coolant would boil and vent within a few hours). But the first generator failed to switch on because of the loss of the first control circuit. Luckily the only back-up generator in the system then switched on, averting a serious accident.’

Furthermore, Nature writes: ‘The Le Bugey incident shows that a whole new class of possible events had been ignored—those where electrical systems fail gradually. It shows that risk analysis must not only take into account a yes or no, working or not working, for each item in the reactor, but the possibility of working with a slightly degraded system.’

This book is partly a textbook and partly a research monograph, also covering research on the way to being published in international journals. The first two chapters, giving an introduction to the area and the basics, are accompanied by exercises. In a course, these chapters should at least be combined with the two first and the three last sections of Chapter 3. This will cover basic bounds in a time interval for system availabilities and unavailabilities given the corresponding component availabilities and unavailabilities, and how the latter can be arrived at. In addition come applications to a simple network system and an offshore electrical power generation system. This should be followed up by Chapter 4, giving a more in-depth application to an offshore gas pipeline network.

The rest of Chapter 3 first gives improved bounds both in a time interval and at a fixed point of time for system availabilities and unavailabilities, using modular decompositions, leaving some proofs to Appendix 1. Some of the results here are new. Strict and exactly correct bounds are also covered in the rest of this chapter.

In Chapter 5 component availabilities and unavailabilities in a time interval are considered uncertain and a new theory for Bayesian assessment of system availabilities and unavailabilities in a time interval is developed including a simulation approach. This is applied to the simple network system. Chapter 6 gives a new theory for measures of importance of system components covering generalizations of the Birnbaum, Barlow–Proschan and Natvig measures from the binary to the multistate case both for unrepairable and repairable systems. In Chapter 7 a corresponding numerical study is given based on an advanced discrete event simulation program. The numerical study covers two three-component systems, a bridge system and an offshore oil and gas production system.

Chapter 8 is concerned with probabilistic modeling of monitoring and maintenance based on a marked point process approach taking the dynamics into account. The theory is illustrated using the offshore electrical power generation system. We also describe how a standard simulation procedure, the data augmentation method, can be implemented and used to obtain a sample from the posterior distribution for the parameter vector without calculating the likelihood explicitly.

Which of the material in the rest of Chapter 3 and in Chapters 5–8 should be part of a course is a matter of interest and taste.

A reader of this book will soon discover that it is very much inspired by the Barlow and Proschan (1975a) book, which gives a wonderful tour into the world of binary reliability theory. I am very much indebted to Professor Richard Barlow for, in every way, being the perfect host during a series of visits, also by some of my students, to the University of California at Berkeley during the 1980s. I am also very thankful to Professor Arnljot Høyland, at what was then named The Norwegian Institute of Technology, for letting me, in the autumn of 1976, give the first course in moderny reliability theory in Norway based on the Barlow and Proschan (1975a) book.

Although I am the single author of the book, parts of it are based on important contributions from colleagues and students to whom I am very grateful. Government grant holder Jørund Gåsemyr is, for instance, mainly responsible for the new results in Chapter 3, for the simulation approach in Chapter 5 and is the first author of the paper on which Chapter 8 is based. Applying the data augmentation approach was his idea. Associate Professor Arne Bang Huseby, my colleague since the mid 1980s, has developed the advanced discrete event simulation program necessary for the numerical study in Chapter 7. The extensive calculations have been carried through by my master student Mads Opstad Reistadbakk. Furthermore, the challenging calculations in Chapter 5 were done by post. doc Trond Reitan. The offshore electrical power generation system case study in Chapter 3 is based on a suggestion by Professor Arne T. Holen at The Norwegian Institute of Technology and developed as part of master theses by my students Skule Sørmo and Gutorm Høgåsen. Finally, Chapter 4 is based on the master thesis of my student Hans Wilhelm Mørch, heavily leaning on the computer package MUSTAFA (MUltiSTAte Fault-tree Analysis) developed by Gutorm Høgåsen as part of his PhD thesis.

Finally I am thankful to the staff at John Wiley & Sons, especially Commissioning Editor, Statistics and Mathematics Dr Ilaria Meliconi for pushing me to write this book. I must admit that I have enjoyed it all the way.

Bent Natvig

Oslo, Norway


We acknowledge Applied Probability for giving permission to include Tables 2.1, 2.2, 2.4 and 2.15 from Natvig (1982a), Figure 1.1 and Tables 1.1 and 3.1 from Funnemark and Natvig (1985) and Figure 1.2 and Tables 2.7–2.10, 3.6 and 3.7 from Natvig et al. (1986). In addition we acknowledge World Scientific Publishing for giving permission to include Figure 4.1 and Tables 4.1–4.15 from Natvig and Mørch (2003).

List of Abbreviations

BCS Binary coherent system
BMS Binary monotone system
BTMMS Binary type multistate monotone system
BTMSCS Binary type multistate strongly coherent system
MCS Multistate coherent system
MMS Multistate monotone system
MSCS Multistate strongly coherent system
MUSTAFA MUltiSTAte Fault-tree Analysis
MWCS Multistate weakly coherent system