# Multistate Systems Reliability Theory with Applications

# Multistate Systems Reliability Theory with Applications

ISBN: 978-0-470-97708-8 December 2010 264 Pages

## Description

Most books in reliability theory are dealing with a description of component and system states as binary: functioning or failed. However, many systems are composed of multi-state components with different performance levels and several failure modes. There is a great need in a series of applications to have a more refined description of these states, for instance, the amount of power generated by an electrical power generation system or the amount of gas that can be delivered through an offshore gas pipeline network.This book provides a descriptive account of various types of multistate system, bound-for multistate systems, probabilistic modeling of monitoring and maintenance of multistate systems with components along with examples of applications.

Key Features:

- Looks at modern multistate reliability theory with applications covering a refined description of components and system states.
- Presents new research, such as Bayesian assessment of system availabilities and measures of component importance.
- Complements the methodological description with two substantial case studies.

Reliability engineers and students involved in the field of reliability, applied mathematics and probability theory will benefit from this book.

**Preface.**

**Acknowledgements.**

**List of abbreviations.**

**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.

**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.

**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.

**4 An offshore gas pipeline network.**

4.1 Description of the system.

4.2 Bounds for system availabilities and unavailabilities.

**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.

**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.

**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.

**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.**

**References.**

**Index.**