Systems Dependability Assessment Benefits of Petri Net Models

Petri Nets were defined for the study of discrete events systems and later extended for many purposes including dependability assessment. In our knowledge, no book deals specifically with the use of different type of PN to dependability. We propose in addition to bring a focus on the adequacy of Petri net types to the study of various problems related to dependability such as risk analysis and probabilistic assessment.

In the first part, the basic models of PN and some useful extensions are briefly recalled. In the second part, the PN are used as a formal model to describe the evolution process of critical system in the frame of an ontological approach. The third part focuses on the stochastic Petri Nets (SPN) and their use in dependability assessment. Different formal models of SPN are formally presented (semantics, evolution rules…) and their equivalence with the corresponding class of Markov processes to get an analytical assessment of dependability. Simplification methods are proposed in order to reduce the size of analytical model and to make it more calculable. The introduction of some concepts specific to high level PN allows too the consideration of complex systems. Few applications in the field of the instrumentation and control (l&C) systems, safety integrated systems (SIS) emphasize the benefits of SPN for dependability assessment.

Dr. Nicolae BRINZEI, Associate University of Lorraine.

Dr. Mohammed-Habib MAZOUNI, Engineer.

Part 1. Short Review of Petri Net Modeling 1

Introduction to Part 1 3

Chapter 1. Autonomous Petri Nets 5

1.1. Unmarked Petri nets 5

1.1.1. Definitions 5

1.1.2. Drawing 6

1.1.3. Other definitions 7

1.2. Marking of a PN 7

1.2.1. Order relation on markings 8

1.2.2. Enabled transition 9

1.3. Dynamics of autonomous PNs 9

1.3.1. Firing of a transition 9

1.3.2. Transition matrix 11

1.3.3. Firing sequence 11

1.3.4. Reachable marking 12

1.3.5. Fundamental equation 12

1.3.6. Properties of PN 14

1.3.7. Other properties 14

1.3.8. Invariants in a PN 15

1.3.9. Reachability graph 16

Chapter 2. Petri Nets and Event Languages 19

2.1. Labeled PNs 19

2.1.1. Formal definition 19

2.1.2. Generated and marked languages 20

2.2. Example 21

Chapter 3. Comparison Petri Nets – Finite State Automaton 25

3.1. Language expression  26

3.2. Building of the models 27

3.2.1. Synchronization of submodels 28

3.2.2. Resource sharing 29

3.2.3. Construction by refinement 30

3.3. Compactness of the model 32

Chapter 4. Some Extensions of Petri Nets 35

4.1. PN with inhibitor arcs 35

4.2. Timed PN 36

4.2.1. P-timed Petri nets 37

4.2.2. T-timed Petri nets 37

4.3. Synchronized PN 38

4.4. Timed synchronized PN 40

4.5. Interpreted PN 41

4.6. Colored PN 42

4.6.1. Introduction example 42

4.6.2. Formal definition 45

4.6.3. A dedicated software CPN Tools 46

Conclusion to Part 1 51

Part 2. A Formal Approach to Risk Assessment 53

Introduction to Part 2 51

Chapter 5. Ontology-based Accidental Process 61

5.1. Preliminary definitions 61

5.2. Elementary entities: HSE and VTE 63

5.2.1. Hazard supplier entity (HSE) 63

5.2.2. Vulnerable target entity (VTE) 63

5.3. Elementary situations and elementary events 64

5.3.1. State versus situation  64

5.3.2. Initial situation (IS) 64

5.3.3. Initiating event (IEv) 64

5.3.4. Hazard situation (HS) 65

5.3.5. Exposure event (EEv) 65

5.3.6. Exposure situation (ES) 65

5.3.7. Accident situation 65

5.3.8. Hazardous (feared) event (HEv) 65

5.4. Conclusion 66

Chapter 6. Petri Net Modeling of the Accidental Process 67

6.1. Elementary process 68

6.2. Sequence of elementary processes 71

6.3. Modeling the action of a safety barrier 71

6.4. Modeling of a cumulative process 73

6.5. PN as a support for risk assessment 75

6.5.1. Modeling of the damage 75

6.5.2. Modeling of the event frequencies 75

6.5.3. CPN Tools implementation  77

6.5.4. Evaluation rule of the risk 83

6.6. Conclusion  86

Chapter 7. Illustrative Example 87

7.1. Functional description 87

7.2. Building of an accidental process 88

7.2.1. First elementary process 88

7.2.2. Second elementary process 91

7.2.3. Parallel process 92

7.2.4. The whole model 92

7.3. Conclusion 94

Chapter 8. Design and Safety Assessment Cycle 95

8.1. Five essential steps 95

8.2. Ontological interest 98

Conclusion to Part 2 101

Part 3. Stochastic Petri Nets 103

Introduction to Part 3 105

Chapter 9. Basic Concept 107

9.1. Introductory example 107

9.2. Formal definition 108

Chapter 10. Semantics, Properties and Evolution Rules of an SPN 111

10.1. Conservatism properties 112

10.1.1. Conservatism of the mean marking in steady state 112

10.1.2. Conservatism of the flow in steady state 113

10.2. Mean sojourn time in a place of a SPN 113

10.3. Equivalent Markov process 114

10.4. Example of SPN for systems dependability modeling and assessment 116

Chapter 11. Simplification of Complex Models 121

11.1. Introduction 121

11.2. System modeling 122

11.3. Presentation of the quantitative analysis method 124

11.3.1. Steps to obtain an aggregated Markov graph 124

11.3.2. Toward a direct establishment of a reduced Markov graph 137

11.4. Example 137

11.4.1. Failure modeling  138

11.4.2. Study of the different functional and hardware solutions 139

11.4.3. Evaluation of the weighting coefficients from the Petri nets 144

11.4.4. Conclusion 147

Chapter 12. Extensions of SPN  149

12.1. Introduction 149

12.2. Relationship between stochastic Petri nets and stochastic processes  150

12.3. The transition firing policy 151

12.4. Associated stochastic processes  151

12.4.1. Temporal memory based on resampling 152

12.4.2. Temporal memory based on age memory or on enabling memory  153

12.4.3. Stochastic process underlying a stochastic PN  154

12.4.4. Embedded Markov chain of the stochastic process 157

12.4.5. Application to a case study 159

12.5. Synchronization problem in generalized stochastic Petri nets  162

12.5.1. GSPN with internal synchronization 162

12.5.2. SPN with predicates and assertions  164

12.6. Conclusion  168

Part 4. Applications of Stochastic Petri Nets to Assessment Problems in Industrial Systems 169

Introduction to Part 4 171

Chapter 13. Application in Dynamic Reliability  175

13.1. Presentation of the system and hypothesis  175

13.2. System modeling with Petri net  177

13.3. Methodology application  179

13.4. Construction of an aggregated Markov graph 180

13.5. Conclusion  185

Chapter 14. Classical Dependability Assessment  187

14.1. Availability study of a nuclear power plant subsystem 187

14.1.1. CPN modeling 188

14.1.2. Reliability and dependability assessment 192

14.1.3. Conclusion 196

14.2. Common causes failures in nuclear plants (safety oriented)  197

14.2.1. The Atwood model 197

14.2.2. Case study 199

14.2.3. Probabilistic dependability assessment 208

14.2.4. Conclusion 212

Chapter 15. Impact of Failures on System Performances  213

15.1. Reliability evaluation of networked control system 213

15.1.1. Statement of the problem 213

15.1.2. Reliability criteria of an NCS 215

15.1.3. Elements of modeling 216

15.1.4. Simulation and results 225

15.1.5. Evaluation of reliability  230

15.1.6. Conclusion 230

15.2. Railway signaling  231

15.2.1. Introduction 231

15.2.2. Interest  233

15.2.3. Signaling system specifications  234

15.2.4. Elements to be modeled 235

15.2.5. Architecture of the model 236

15.2.6. Example of an elementary model 237

15.2.7. Incident generation 239

15.2.8. Results  239

15.2.9. Conclusion 242

Conclusion 245

Appendix 247

Bibliography  251

Index 261

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