Fuzzy based evaluation of dependable systems Mehran Garmehi

Fuzzy based evaluation of dependable systems Mehran Garmehi Spring 1384

Overview l l l l 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.

Dependability (RMS) l l Reliability Maintainability Safety …

Evaluation techniques l l l Fault tree Markov model …

Fault tree model l l Pz = 1 - (1 -Pa)(1 -Pb) P z = P a* P b

Fault tree model l Repair ? Complexity! Scalability!

Markov model l l Some states like FSM Memory less

TMR (Triple Modular Redundancy)

Parameters l l Failure rate λ Coverage c Corrective repair rate μc Preventive repair rate μp

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

Why fuzzy theory is useful? l l 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

Example

Example l P ( t + Δt ) = A. P(t)

Example P ( nΔt ) = A . P (0)

Example Fuzzy parameters

Example : Fuzzy equations l P ( nΔt ) =A . P (0)

Example : Fuzzy matrix

Example result matrix

Example: Normalization

Example: α cut l R ( nΔt ) = P 10 ( nΔt )

Example : Result

References l Barry W. Johnson, ”Design and analysis of fault– l P. S. Cugnasca, M. T. de Andrada, J. B. Camargo, “A fuzzy based approach for the design and l toletant digital systems”, Addison-Wesley publishing, 1989 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