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A. BobbioBertinoro, March 10-14, 20031 Dependability Theory and Methods 3. State Enumeration Andrea Bobbio Dipartimento di Informatica Università del Piemonte Orientale, A. Avogadro 15100 Alessandria (Italy) bobbio@unipmn.itbobbio@unipmn.it - http://www.mfn.unipmn.it/~bobbio Bertinoro, March 10-14, 2003

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A. BobbioBertinoro, March 10-14, 20032 State space Consider a system with n binary components. 1 component i up 0 component i down We introduce an indicator variable x i : x i = The state of the system can be identified as a vector x = (x 1, x 2,.... x n). The state space (of cardinality 2 n ) is the set of all the possible values of x.

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A. BobbioBertinoro, March 10-14, 20033 2-component system

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A. BobbioBertinoro, March 10-14, 20034 3-component system

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Characterization of system states The system has a binary behavior. 1 system up 0 system down We introduce an indicator variable for the system y : y = For each state s corresponding to a single value of the vector x = (x 1, x 2,.... x n). 1 system up 0 system down y = (x) = y = (x) is the structure function

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Characterization of system states The state space can be partitioned in 2 subsets: The structure function y = (x) depends on the system configuration

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A. BobbioBertinoro, March 10-14, 20037 2-component system A1A1 A2 A1A1

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A. BobbioBertinoro, March 10-14, 20038 3-component system A1A1 A2 A3 A1A1A2 A3 a) b)

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A. BobbioBertinoro, March 10-14, 20039 State probability Define: Pr{x i (t) = 1} = R i (t) Pr{x i (t) = 0} = 1 - R i (t) Suppose components are statistically independent; The probability of the system to be in a given state x = (x 1, x 2,...., x n ) at time t is given by the product of the probability of each individual component of being up or down. P {x(t)} = Pr{x 1 (t)} · Pr{x 2 (t)} · … ·Pr{x n (t)}

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A. BobbioBertinoro, March 10-14, 200310 2-component system A1A1 A2 A1A1

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A. BobbioBertinoro, March 10-14, 200311 3-component system

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A. BobbioBertinoro, March 10-14, 200312 Dependability measures

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A. BobbioBertinoro, March 10-14, 200313 Dependability measures

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A. BobbioBertinoro, March 10-14, 200314 2-component series system A1A1A2

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A. BobbioBertinoro, March 10-14, 200315 2-component parallel system A1A1 A2

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A. BobbioBertinoro, March 10-14, 200316 3-component system A1A1 A2 A3 A1A1A2 A3 a) b)

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17 3-component system 2:3 majority voting A1A1 A2 A3 Voter

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18 5 component systems

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A. BobbioBertinoro, March 10-14, 200319 Non series-parallel systems with 5 components A1A1 A2 A3 A4 A5 Independent identically distributed components

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