Fuzzy based evaluation of dependable systems

1. Fuzzy based evaluation of dependable systems Mehran Garmehi Spring 1384

2. Overview • Why we need evaluation of dependable systems ? • How to do evaluation? • Why Markov model? • A simple example . • Why fuzzy theory is useful? • How fuzzy theory can be used in Markov model ? • An example :Evaluation of a dependable system using Fuzzy Markov Model .

3. Dependability (RMS) • Reliability • Maintainability • Safety • …

4. Evaluation techniques • Fault tree • Markov model • …

5. Pz = Pa* Pb Pz = 1- (1-Pa)(1-Pb) Fault tree model

6. Fault tree model • Repair ? • Complexity! • Scalability!

7. Markov model • Some states like FSM • Memory less

8. TMR (Triple Modular Redundancy)

9. Parameters • Failure rate λ • Coverage c • Corrective repair rateμc • Preventive repair rate μp

11. Markov model probabilities P ( t + Δt ) = A.P(t)

12. Why fuzzy theory is useful? • In practice λ, μp, μc and c are not crisp • Usually these values are given in a triangular manner (min , max and mid ) • These parameters may vary during the evaluation process • The matrix A can be indicated in fuzzy form

13. Example

14. Example • P ( t + Δt ) = A. P(t)

15. Example P ( nΔt ) = A .P (0)

16. Example Fuzzy parameters

17. Example : Fuzzy equations • P̃ ( nΔt ) =Ã . P (0)

18. Example : Fuzzy matrix

19. Example result matrix

20. Example: Normalization

21. Example: α cut • R̃ ( nΔt ) = P̃10 ( nΔt )

22. Example : Result

23. References • Barry W. Johnson,”Design and analysis of fault–toletant digital systems”,Addison-Wesley publishing, 1989 • P.S. Cugnasca, M.T. de Andrada, J.B. Camargo, “A fuzzy based approach for the design and evaluation of dependable systems using the markov model”, Proceedings of the 1999 Pacific Rim International Symposium on Dependable Computing • H. Zimmermann,”Fuzzy set theory”, Kluwer academic publisher, 1996