Fault Trees

Nikolaos Limnios is a Professor at the University of Technology, Compiègne, France. His research includes reliability, applied stochastic processes and statistics. He has written and edited many books in the reliability field.

• Introduction 11Chapter 1 Single-Component Systems 171.1 Distribution of failure and reliability 171.1.1 Function of distribution and density of failure 171.1.2 Survival function: reliability 181.1.3 Hazard rate 191.1.4 Maintainability 191.1.5 Mean times 201.1.6 Mean residual lifetime 211.1.7 Fundamental relationships 211.1.8 Some probability distributions 221.2 Availability of the repairable systems 251.2.1 Instantaneous availability 251.2.2 Asymptotic availability 261.2.3 Mean availability 261.2.4 Asymptotic mean availability 271.3 Reliability in discrete time 271.3.1 Discrete distributions 281.3.2 Reliability 281.4 Reliability and maintenance 291.4.1 Periodic test: repair time is negligible 291.4.2 Periodic test: repair time is not negligible 301.4.3 Mean duration of a hidden failure 301.5 Reliability data 31Chapter 2 Multi-Component Systems 332.1 Structure function 332.2 Modules andmodular decomposition 362.3 Elementary structure systems 372.3.1 Series system 372.3.2 Parallel system 382.3.3 System k-out-of-n 382.3.4 Parallel-series system 392.3.5 Series-parallel system 392.4 Systems with complex structure 402.5 Probabilistic study of the systems 422.5.1 Introduction 422.5.2 Inclusion-exclusion method 432.5.3 Disjoint products 442.5.4 Factorization 462.5.5 Reliability bounds 46Chapter 3 Construction of Fault Trees 493.1 Basic ideas and definitions 493.1.1 Graphic symbols 523.1.2 Use of the operators 533.2 Formal definition and graphs 563.3 Stages of construction 573.3.1 Preliminary analysis 583.3.2 Specifications 593.3.3 Construction 593.4 Example of construction 603.4.1 Preliminary analysis 603.4.2 Specifications 623.4.3 Construction 623.5 Automatic construction 63Chapter 4 Minimal Sets 674.1 Introduction 674.2 Methods of study 684.2.1 Direct methods 684.2.2 Descending methods 714.2.3 Ascending methods 734.3 Reduction 744.4 Other algorithms for searching the cut sets 754.5 Inversion of minimal cut sets 764.6 Complexity of the search for minimal cut sets 78Chapter 5 Probabilistic Assessment 795.1 The problem of assessment 795.2 Direct methods 805.2.1 AND operator 815.2.2 OR operator 815.2.3 Exclusive OR operator 825.2.4 k-out-of-n operator 835.2.5 Priority-AND operator 835.2.6 IF operator 835.3 Methods of minimal sets 845.3.1 Inclusion-exclusion development 845.3.2 Disjoint products 855.3.3 Kitt method 865.4 Method of factorization 885.5 Direct recursive methods 905.5.1 Recursive inclusion-exclusion method 905.5.2 Method of recursive disjoint products 915.6 Other methods for calculating the fault trees 925.7 Large fault trees 935.7.1 Method of Modarres and Dezfuli [MOD 84] 935.7.2 Method of Hughes [HUG 87] 945.7.3 Schneeweiss method [SCH 87] 955.7.4 Brown method [BRO 90] 95Chapter 6 Influence Assessment 976.1 Uncertainty 976.1.1 Introduction 976.1.2 Methods for evaluating the uncertainty 986.1.3 Evaluation of the moments 996.2 Importance 1036.2.1 Introduction 1036.2.2 Structural importance factors 1056.2.3 Probabilistic importance factors 1066.2.4 Importance factors over the uncertainty 109Chapter 7 Modules – Phases – Common Modes 1117.1 Introduction 1117.2 Modular decomposition of an FT 1117.2.1 Module and better modular representation 1117.2.2 Modularization of a fault tree 1147.3 Multiphase fault trees 1167.3.1 Example 1177.3.2 Transformation of a multiphase system 1187.3.3 Method of eliminating the minimal cut sets 1187.4 Common mode failures 119Chapter 8 Extensions: Non-Coherent, Delay and Multistate Fault Trees 1238.1 Non-coherent fault trees 1238.1.1 Introduction 1238.1.2 An example of a non-coherent FT 1268.1.3 Prime implicants and implicates 1268.1.4 Probabilistic study 1288.2 Delay fault trees 1298.2.1 Introduction 1298.2.2 Treatment 1298.3 FTs and multistate systems 1318.3.1 Multistate systems 1318.3.2 Structure function 1328.3.3 Stochastic description and function of reliability 1358.3.4 Fault trees with restrictions 1368.3.5 Multistate fault trees 138Chapter 9 Binary Decision Diagrams 1439.1 Introduction 1439.2 Reduction of the Shannon tree 1439.2.1 Graphical representation of a BDD 1439.2.2 Formal BDD 1459.2.3 Probabilistic calculation 1479.3 Probabilistic assessment of the FTs based on the BDD 1489.4 Research about the prime implicants 1519.5 Algorithmic complexity 153Chapter 10 Stochastic Simulation of Fault Trees 15510.1 Introduction 15510.2 Generation of random variables 15510.2.1 Generation of a uniform variable 15510.2.2 Generation of discrete random variables 15710.2.3 Generation of real random variables 15810.3 Implementation and evaluation of the method 15910.3.1 The Monte Carlo method 15910.3.2 Estimating the probability of the top event 16010.3.3 Precision of the estimation 16110.3.4 Acceleration of the convergence 16410.3.5 Rare events 165Exercises 167Appendices 177A BDD Algorithms in FT Analysis 179A1 Introduction 179A2 Obtaining the BDD 180A3 Algorithm of probabilistic assessment 182A4 Importance factors 183A5 Prime implicants 184B European Benchmark Fault Trees 187B1 Description of the data 187B2 Fault tree: Europe-1 188B2.1 Structure of the fault tree (structural data) 188B2.2 Probabilistic data 190B2.3 Results 190B3 Fault tree: Europe-2 191B3.1 Structure of the fault tree 191B3.2 Probabilistic data 192B3.3 Results 192B4 Fault tree: Europe-3 193B4.1 Structure of the FT 193B4.2 Probabilistic data 195B4.3 Results 195C Some Results of Probabilities 197Main Notations 201Bibliography 205Index 221