Dependability Theory and Methods 3' State Enumeration

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Dependability Theory and Methods3. State
Enumeration

• Andrea Bobbio
• Dipartimento di Informatica
• Università del Piemonte Orientale, A. Avogadro
• 15100 Alessandria (Italy)
• bobbio_at_unipmn.it - http//www.mfn.unipmn.it/bob
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Bertinoro, March 10-14, 2003
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State space
Consider a system with n binary components.
We introduce an indicator variable x i
1 component i up 0 component i down
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 system
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3-component system
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Characterization of system states
The system has a binary behavior.
We introduce an indicator variable for the system
y
1 system up 0 system down
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 structure function y ? (x) depends on the
system configuration
The state space ? can be partitioned in 2 subsets
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2-component system
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a)
3-component system
b)
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State probability
Prx i(t) 1 R i (t) Prx i(t) 0 1
- R i (t)
Define
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) Prx 1(t) Prx 2(t) Prx
n(t)
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2-component system
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3-component system
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Dependability measures
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Dependability measures
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2-component series system
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2-component parallel system
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3-component system
a)
b)
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3-component system 23 majority voting
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5 component systems
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Non series-parallel systems with 5 components
Independent identically distributed components