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Systems Dependability Assessment
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PREFACE ix INTRODUCTION xiii PART 1. PREDICTED RELIABILITY OF STATIC SYSTEMS; A GRAPH-THEORY BASED APPROACH 1 CHAPTER 1. STATIC AND TIME INVARIANT SYSTEMS WITH BOOLEAN REPRESENTATION 3 1.1. Notations 3 1.2. Order relation on U 4 1.3. Structure of a system 6 1.3.1. State diagram of a system 6 1.3.2. Monotony of an SF, coherence of a system 7 1.4. Cut-set and tie-set of a system 9 1.4.1. Tie-set 9 1.4.2. Cut-set 10 CHAPTER 2. RELIABILITY OF A COHERENT SYSTEM 13 2.1. Demonstrating example 15 2.2. The reliability block diagram (RBD) 18 2.3. The fault tree (FT) 21 2.4. The event tree 26 2.5. The structure function as a minimal union of disjoint monomials 28 2.5.1. Ordered graph of a monotone structure function 29 2.5.2. Maxima and minima of the ordered graph 31 2.5.3. Ordered subgraphs of the structure function 32 2.5.4. Introductory example 33 2.5.5. Construction of the minimal Boolean form 37 2.5.6. Complexity 43 2.5.7. Comparison with the BDD approach 45 2.6. Obtaining the reliability equation from the Boolean equation 49 2.6.1. The traditional approach 49 2.6.2. Comparison with the structure function by Kaufmann 50 2.7. Obtain directly the reliability from the ordered graph 52 2.7.1. Ordered weighted graph 53 2.7.2. Algorithm 56 2.7.3. Performances of the algorithm 59 CHAPTER 3. WHAT ABOUT NON-COHERENT SYSTEMS? 61 3.1. Example of a non-coherent supposed system 61 3.2. How to characterize the non-coherence of a system? 63 3.3. Extension of the ordered graph method 66 3.3.1. Decomposition algorithm 67 3.4. Generalization of the weighted graph algorithm 68 CONCLUSION TO PART 1 73 PART 2. PREDICTED DEPENDABILITY OF SYSTEMS IN A DYNAMIC CONTEXT 75 INTRODUCTION TO PART 2 77 CHAPTER 4. FINITE STATE AUTOMATON 83 4.1. The context of discrete event system 83 4.2. The basic model 84 CHAPTER 5. STOCHASTIC FSA 89 5.1. Basic definition 89 5.2. Particular case: Markov and semi-Markov processes 90 5.3. Interest of the FSA model 91 5.4. Example of stochastic FSA 92 5.5. Probability of a sequence 93 5.6. Simulation with Scilab 94 5.7. State/event duality 95 5.8. Construction of a stochastic SFA 96 CHAPTER 6. GENERALIZED STOCHASTIC FSA 101 CHAPTER 7. STOCHASTIC HYBRID AUTOMATON 105 7.1. Motivation 105 7.2. Formal definition of the model 105 7.3. Implementation 107 7.4. Example 109 7.5. Other examples 116 7.5.1. Control temperature of an oven 116 7.5.2. Steam generator of a nuclear power plant 118 7.6. Conclusion 120 CHAPTER 8. OTHER MODELS/TOOLS FOR DYNAMIC DEPENDABILITY VERSUS SHA 121 8.1. The dynamic fault trees 121 8.1.1. Principle 121 8.1.2. Equivalence with the FSA approach 124 8.1.3. Covered criteria 126 8.2. The Boolean logic-driven Markov processes 126 8.2.1. Principle 126 8.2.2. Equivalence with the FSA approach 127 8.2.3. Covered criteria 127 8.3. The dynamic event trees (DETs) 128 8.3.1. Principle 128 8.3.2. Equivalence with the FSA approach 129 8.3.3. Covered criteria 130 8.4. The piecewise deterministic Markov processes 131 8.4.1. Principle 131 8.4.2. Equivalence with the FSA approach 131 8.4.3. Covered criteria 132 8.5. Other approaches 132 CONCLUSION AND PERSPECTIVES 135 APPENDIX 137 BIBLIOGRAPHY 173 INDEX 181

About the Author

Pr. Jean-Fran ois AUBRY Professor Emeritus, University of Lorraine, France.Dr. Nicolae BRINZEI, Associate Professor, University of Lorraine, France.

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