Fault tree analysis is an important technique in determining the
safety and dependability of complex systems. Fault trees are used
as a major tool in the study of system safety as well as in
reliability and availability studies.

The basic methods - construction, logical analysis,
probability evaluation and influence study - are described in
this book. The following extensions of fault trees, non-coherent
fault trees, fault trees with delay and multi-performance fault
trees, are also explained. Traditional algorithms for fault tree
analysis are presented, as well as more recent algorithms based on
binary decision diagrams (BDD).

Introduction 11

Chapter 1 Single-Component Systems 17

1.1 Distribution of failure and reliability 17

1.1.1 Function of distribution and density of failure 17

1.1.2 Survival function: reliability 18

1.1.3 Hazard rate 19

1.1.4 Maintainability 19

1.1.5 Mean times 20

1.1.6 Mean residual lifetime 21

1.1.7 Fundamental relationships 21

1.1.8 Some probability distributions 22

1.2 Availability of the repairable systems 25

1.2.1 Instantaneous availability 25

1.2.2 Asymptotic availability 26

1.2.3 Mean availability 26

1.2.4 Asymptotic mean availability 27

1.3 Reliability in discrete time 27

1.3.1 Discrete distributions 28

1.3.2 Reliability 28

1.4 Reliability and maintenance 29

1.4.1 Periodic test: repair time is negligible 29

1.4.2 Periodic test: repair time is not negligible 30

1.4.3 Mean duration of a hidden failure 30

1.5 Reliability data 31

Chapter 2 Multi-Component Systems 33

2.1 Structure function 33

2.2 Modules andmodular decomposition 36

2.3 Elementary structure systems 37

2.3.1 Series system 37

2.3.2 Parallel system 38

2.3.3 System k-out-of-n 38

2.3.4 Parallel-series system 39

2.3.5 Series-parallel system 39

2.4 Systems with complex structure 40

2.5 Probabilistic study of the systems 42

2.5.1 Introduction 42

2.5.2 Inclusion-exclusion method 43

2.5.3 Disjoint products 44

2.5.4 Factorization 46

2.5.5 Reliability bounds 46

Chapter 3 Construction of Fault Trees 49

3.1 Basic ideas and definitions 49

3.1.1 Graphic symbols 52

3.1.2 Use of the operators 53

3.2 Formal definition and graphs 56

3.3 Stages of construction 57

3.3.1 Preliminary analysis 58

3.3.2 Specifications 59

3.3.3 Construction 59

3.4 Example of construction 60

3.4.1 Preliminary analysis 60

3.4.2 Specifications 62

3.4.3 Construction 62

3.5 Automatic construction 63

Chapter 4 Minimal Sets 67

4.1 Introduction 67

4.2 Methods of study 68

4.2.1 Direct methods 68

4.2.2 Descending methods 71

4.2.3 Ascending methods 73

4.3 Reduction 74

4.4 Other algorithms for searching the cut sets 75

4.5 Inversion of minimal cut sets 76

4.6 Complexity of the search for minimal cut sets 78

Chapter 5 Probabilistic Assessment 79

5.1 The problem of assessment 79

5.2 Direct methods 80

5.2.1 AND operator 81

5.2.2 OR operator 81

5.2.3 Exclusive OR operator 82

5.2.4 k-out-of-n operator 83

5.2.5 Priority-AND operator 83

5.2.6 IF operator 83

5.3 Methods of minimal sets 84

5.3.1 Inclusion-exclusion development 84

5.3.2 Disjoint products 85

5.3.3 Kitt method 86

5.4 Method of factorization 88

5.5 Direct recursive methods 90

5.5.1 Recursive inclusion-exclusion method 90

5.5.2 Method of recursive disjoint products 91

5.6 Other methods for calculating the fault trees 92

5.7 Large fault trees 93

5.7.1 Method of Modarres and Dezfuli [MOD 84] 93

5.7.2 Method of Hughes [HUG 87] 94

5.7.3 Schneeweiss method [SCH 87] 95

5.7.4 Brown method [BRO 90] 95

Chapter 6 Influence Assessment 97

6.1 Uncertainty 97

6.1.1 Introduction 97

6.1.2 Methods for evaluating the uncertainty 98

6.1.3 Evaluation of the moments 99

6.2 Importance 103

6.2.1 Introduction 103

6.2.2 Structural importance factors 105