A. Bobbio Bertinoro, March 10-14, 2003 1

Dependability Theory and Methods

3. State Enumeration

Andrea BobbioDipartimento di Informatica

Università del Piemonte Orientale, “A. Avogadro”15100 Alessandria (Italy)

bobbio@unipmn.it - http://www.mfn.unipmn.it/~bobbio

Bertinoro, March 10-14, 2003

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State spaceState 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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2-component system2-component system

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3-component system3-component system

Characterization of system statesCharacterization of system statesThe 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 downy = (x)=

y = (x) is the structure function

Characterization of system statesCharacterization 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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2-component system2-component systemA1

A2

A1 A2

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3-component system3-component system A1

A2

A3

A1 A2

A3

a)

b)

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State probabilityState 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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2-component system2-component systemA1

A2

A1 A2

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3-component system3-component system

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Dependability measures

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Dependability measures

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2-component series system2-component series systemA1 A2

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2-component parallel system2-component parallel systemA1

A2

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3-component system3-component system

A1

A2A3

A1 A2

A3

a)

b)

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3-component system3-component system2:3 majority voting2:3 majority voting

A1

A2

A3

Voter

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5 5 component component

systemssystems

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Non series-parallel systemsNon series-parallel systemswith 5 componentswith 5 components

A1

A2

A3

A4

A5

Independent identicallydistributed components