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March 8, 20071 Dynamic Fault Tree analysis using Input/Output Interactive Markov Chains Hichem Boudali, Pepijn Crouzen, and Mariëlle Stoelinga. Formal Methods and Tools group CS, University of Twente, NL.

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March 8, 20072 Motivation (and setting) Systems do fail Example methodology: Dynamic Fault Trees (DFT) -- Reliability Engineering -- Goal: Reduce system failure probability. Methodology: Identify/analyze failure modes and their effects. But: DFTs have drawbacks

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March 8, 20073 Outline Dynamic fault trees (DFT). Definition, Example, Solution, Drawbacks. Input/Output interactive Markov chains (I/O-IMC). DFT semantics in terms of I/O-IMCs. DFT compositional analysis. Translation, || Composition, Abstraction, Aggregation. Case studies. Summary.

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March 8, 20074 Outline Dynamic fault trees (DFT). Definition, Example, Solution, Drawbacks. Input/Output interactive Markov chains (I/O-IMC). DFT semantics in terms of I/O-IMCs. DFT compositional analysis. Translation, || Composition, Abstraction, Aggregation. Case studies. Summary.

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March 8, 20075 Dynamic Fault Trees (DFT) Extend standard fault trees with dynamic gates. Enable modelling complex behaviours and interactions between components. combination & order of failures matter. Unreliability = Prob[System fails within T time units]

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March 8, 20076 (dynamic) Fault trees Upside-down tree (graph) Leaves: Basic events (BE) Nodes: Gates (complex events) BEs + Gates: Elements Arrows: Causal relations One top-node: the root node The top-node models system failure Failure propagation: From leaves to root

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March 8, 20077 DFTs: Static gates (combination)

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March 8, 20078 DFTs: Dynamic gates (order)

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March 8, 20079 DFTs: Basic events (BE) Temperature of a BE: Relevant when used as a spare BE maps to a Basic Physical component

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March 8, 200710 C AB 0.2 0.4 Failure rate: 0.2 f/h Failure rate: 0.4 f/h AND-gate Starting state: A is operational B is operational A has failed B is operational Pr(A fails in T hours) = 1 – e -0.2T As Mean time to failure = 1/0.2 = 5 hours A is operational B has failed A has failed B has failed Convert the DFT into a Continuous-time Markov chain. Analyze CTMC using standard solution techniques. For (partially) static DFT, binary decision diagrams can be used! DFT solution Unreliability = Prob[Being in state ]

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March 8, 200711 DFT example Road trip fails if mobile phone fails BEFORE the car fails Spare tire is cold: It cannot fail when not in use State-Space Explosion! One of the drawbacks Although distinct modules, CTMC generation in One shot

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March 8, 200712 DFT drawbacks State-space explosion. No formal syntax and semantics. Lack of modularity: Dynamic modules (e.g. Tires subsystem in the example) can not be reused. Restrictions on certain inputs to gates (e.g. spare gate). DFT-to-MC* conversion algorithm is hard to extend and/or modify. Compositional Aggregation DAG Compositionality Lift restrictions Extension: At the element level I/O-IMC *: DIFTree algorithm

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March 8, 200713 Outline Dynamic fault trees (DFT). Definition, Example, Solution, Drawbacks. Input/Output interactive Markov chains (I/O-IMC). DFT semantics in terms of I/O-IMCs. DFT compositional analysis. Translation, || Composition, Abstraction, Aggregation. Case studies. Summary.

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March 8, 200714 Input/Output Interactive Markov Chains (I/O-IMC) Combination of I/O automata and CTMC Discrete state space Markovian transitions Interactive transitions Action signature ? - Input actions ! - Output actions ; - Internal actions Input-enabled λ failed! Immediate

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March 8, 200715 Outline Dynamic fault trees (DFT). Definition, Example, Solution, Drawbacks. Input/Output interactive Markov chains (I/O-IMC). DFT semantics in terms of I/O-IMCs. DFT compositional analysis. Translation, || Composition, Abstraction, Aggregation. Case studies. Summary.

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March 8, 200716 f(C)! f(A)? f(B)? f(A)? f(C)! f(A)? f(B)? f(A)? f(B)? f(A)? DFT semantics (DFT element to I/O-IMC) f(A)? f(B)?

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March 8, 200717 DFT semantics (DFT element to I/O-IMC)

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March 8, 200718 Outline Dynamic fault trees (DFT). Definition, Example, Solution, Drawbacks. Input/Output interactive Markov chains (I/O-IMC). DFT semantics in terms of I/O-IMCs. DFT compositional analysis. Translation, || Composition, Abstraction, Aggregation. Case studies. Summary.

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March 8, 200719 Compositional Analysis Translation C AB 0.2 f(A)! 0.4 f(B)! f(A)? f(B)? f(C)!

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March 8, 200720 Compositional Analysis Parallel Composition f(A)? f(B)? f(C)! 0.2 f(A)!

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March 8, 200721 Compositional Analysis Parallel Composition 1 2 3 1 2 3 4 5 1||1 0.2 f(A)! f(A)? f(B)? f(C)! 0.2 f(B)? f(A)! f(C)! 1||2 2||3 3||1 f(B)? 0.2 f(A)! 3||2 4||35||3 Inputs: f(A)? and f(B)? Outputs: f(C)! Inputs: none Outputs: f(A)! C A C || A Synchronize on f(A)

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March 8, 200722 f(A); f(A)! Compositional Analysis Abstraction (hiding) 1||1 0.2 f(B)? 0.2 f(C)! 1||2 2||3 3||1 3||2 4||35||3 C AB Abstraction (hiding): Makes signal internal

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March 8, 200723 f(A); Compositional Analysis Aggregation (weak bisimulation) 1||1 0.2 f(B)? 0.2 f(C)! 1||2 2||3 3||1 3||2 4||35||3 Weak bisimulation: Disregard internal steps Aggregation: Finding a smaller model equivalent (behaviorally) to the original

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March 8, 200724 Compositional-Aggregation Overview Translation Composition + Hiding Aggregation (minimization) Repeat Aggregated system CTMC Result: System failure probability

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March 8, 200725 Outline Dynamic fault trees (DFT). Definition, Example, Solution, Drawbacks. Input/Output interactive Markov chains (I/O-IMC). DFT semantics in terms of I/O-IMCs. DFT compositional analysis. Translation, || Composition, Abstraction, Aggregation. Case studies. Summary.

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March 8, 200726 Case studies Case study Analysis method Max number of states Max number of transitions Unreliability (T=1) (a) DIFTree Comp-Agg 4113 132 24608 426 0.00135668 (b) DIFTree Comp-Agg 8 36 10 119 0.657900 (c) DIFTree Comp-Agg 253 157 1383 756 2.00025 10 -9

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March 8, 200727 Outline Dynamic fault trees (DFT). Definition, Example, Solution, Drawbacks. Input/Output interactive Markov chains (I/O-IMC). DFT semantics in terms of I/O-IMCs. DFT compositional analysis. Translation, || Composition, Abstraction, Aggregation. Case studies. Summary.

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March 8, 200728 Summary Alleviate state-space explosion problem. Formal syntax & semantics. Enhanced DFT modularity: Dynamic module reuse. Lifting restrictions on allowed inputs. Readily extensible framework (extensions at the element level); e.g. repair. Works well for highly-modular dynamic FTs. Compositional semantics for DFTs Gain at the modeling & analysis levels

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March 8, 200729 References H. Boudali, P. Crouzen, M. Stoelinga. Dynamic Fault Tree analysis using Input/Output Interactive Markov Chains, to appear, DSN 2007 proceedings. H. Boudali, P. Crouzen, M. Stoelinga. A compositional semantics for Dynamic Fault Trees in terms of Interactive Markov Chains, Technical report, to appear. More info: hboudali@cs.utwente.nlhboudali@cs.utwente.nl The END!

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March 8, 200730 Extra slides

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March 8, 200731 Future work Weaker bisimulation relation (i.e. more aggressive state reduction) Extension to non-exponential distributions (e.g. use of phase-type distributions) Further extensions to DFT modeling capabilities (i.e. definition of new gates and corresponding I/O-IMC) Fully automated tool (at this point, the tool is only partially automated)

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March 8, 200732 Parallel Composition and Hiding

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March 8, 200733 Aggregation (Weak Bisimulation)

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March 8, 200734 Preservation Theorem (WB is a congruence)

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March 8, 200735 CTMC Compositional-Aggregation Overview Step 1: Translation Step 2a: Parallel Composition Step 2b: Abstraction Step 3: Aggregation Step 4: Repetition Step 2a: (C||A) || B Step 2b: Hide f(B) Step 3: Aggregate (C||A)||B Step 5: CTMC Analysis C AB C A B f(A) f(B) f(C) DFT IOIMC C||A f(C) f(B) f(A) f(B) f(C) C||A||B 0.2 0.4 f(C)! f(C) IOIMC model can be reused! Steps 2–4: Compositional Aggregation

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