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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.

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Presentation on theme: "A. BobbioBertinoro, March 10-14, 20031 Dependability Theory and Methods 3. State Enumeration Andrea Bobbio Dipartimento di Informatica Università del Piemonte."— Presentation transcript:

1 A. BobbioBertinoro, March 10-14, Dependability Theory and Methods 3. State Enumeration Andrea Bobbio Dipartimento di Informatica Università del Piemonte Orientale, A. Avogadro Alessandria (Italy) - Bertinoro, March 10-14, 2003

2 A. BobbioBertinoro, March 10-14, 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.

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

4 A. BobbioBertinoro, March 10-14, component system

5 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

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

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

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

9 A. BobbioBertinoro, March 10-14, 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)}

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

11 A. BobbioBertinoro, March 10-14, component system

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

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

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

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

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

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

18 18 5 component systems

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


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