A. Bobbio Bertinoro, March 10-14, 2003 1
Dependability Theory and Methods
2. Reliability Block Diagrams
Andrea BobbioDipartimento di Informatica
Università del Piemonte Orientale, “A. Avogadro”15100 Alessandria (Italy)
[email protected] - http://www.mfn.unipmn.it/~bobbio
Bertinoro, March 10-14, 2003
A. Bobbio Bertinoro, March 10-14, 2003 2
Model Types in DependabilityModel Types in DependabilityCombinatorial models assume that components are statistically independent: poor modeling power coupled with high analytical tractability.
Reliability Block Diagrams, FT, ….
State-space models rely on the specification of the whole set of possible states of the system and of the possible transitions among them.
CTMC, Petri nets, ….
A. Bobbio Bertinoro, March 10-14, 2003 3
Reliability Block Diagrams
Each component of the system is represented as a block;
System behavior is represented by connecting the blocks;
Failures of individual components are assumed to be independent;
Combinatorial (non-state space) model type.
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Reliability Block Diagrams (RBDs)
Schematic representation or model;Shows reliability structure (logic) of a system;Can be used to determine dependability measures;A block can be viewed as a “switch” that is
“closed” when the block is operating and “open” when the block is failed;
System is operational if a path of “closed switches” is found from the input to the output of the diagram.
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Reliability Block Diagrams (RBDs)Can be used to calculate:
– Non-repairable system reliability given: Individual block reliabilities (or failure rates); Assuming mutually independent failures events.
– Repairable system availability given:Individual block availabilities (or MTTFs and
MTTRs);Assuming mutually independent failure and
restoration events;Availability of each block is modeled as 2-state
Markov chain.
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Series system of n components.
Components are statistically independent
Define event Ei = “component i functions properly.”
Series system in RBD
)()...()( )...(
)""(
2121 nn EPEPEPEEEP
P
properly g functionin is system series
A1 A2 An
P(Ei) is the probability “component i functions properly” the reliability R i(t) (non repairable) the availability A i(t) (repairable)
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Reliability of Series system
Series system of n components.
Components are statistically independent
Define event Ei = "component i functions properly.”
)()...()( )...(
)""(
2121 nn EPEPEPEEEP
P
properly ng functioni is system series
A1 A2 An
n
iis tRtR
1
)()(
Denoting by R i(t) the reliability of component i
Product law of reliabilities:
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Series system with time-independent failure rate
Let i be the time-independent failure rate of component i. Then:
The system reliability Rs(t) becomes:
Rs(t) = e- s t with s = i
i=1
n
Ri (t) = e- i t
1 1MTTF = —— = ———— s i
i=1
n
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Availability for Series System
Assuming independent repair for each component,
where Ai is the (steady state or transient) availability of component i
n
iis
n
i ii
in
iis
tAtA
MTTRMTTF
MTTFAA
1
11
)()(
or ,
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Series system: an example
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Series system: an example
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Improving the Reliability of a Series System
Sensitivity analysis:
R s R s S i = ———— = ———— R i R i
The optimal gain in system reliability is obtained by improving the least reliable component.
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The part-count method
It is usually applied for computing the reliability of electronic equipment composed of boards with a large number of components.
Components are connected in series and with time-independent failure rate.
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The part-count method
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Redundant systems
When the dependability of a system does not reach the desired (or required) level:
Improve the individual components;
Act at the structure level of the system, resorting to redundant configurations.
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Parallel redundancy
A system consisting of n
independent components in parallel.
It will fail to function only if all n
components have failed.
Ei = “The component i is functioning”
Ep = “the parallel system of n component is
functioning properly.”
A1
An
...
...
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Parallel system
"failedhassystemparallelThe"pE
"failedhavecomponentsnAll"____
2
__
1 ... nEEE
)...()(____
2
__
1
__
np EEEPEP )()...()(____
2
__
1 nEPEPEP
Therefore:
)(1)( pp EPEP
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Parallel redundancy
Fi (t) = P (Ei) Probability component i
is not functioning (unreliability)
Ri (t) = 1 - Fi (t) = P (Ei) Probability
component i is functioning
(reliability)
A1
An
...
...
—
Fp (t) = Fi (t) i=1
n
Rp (t) = 1 - Fp (t) = 1 - (1 - Ri (t)) i=1
n
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2-component parallel system
For a 2-component parallel system:
Fp (t) = F1 (t) F2 (t)
Rp (t) = 1 – (1 – R1 (t)) (1 – R2 (t)) =
= R1 (t) + R2 (t) – R1 (t) R2 (t)
A1
A2
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2-component parallel system: constant failure rate
For a 2-component parallel system
with constant failure rate:
Rp (t) =
A1
A2
e- 1 t + e
- 2 t – e- ( 1 + 2 ) t
1 1 1MTTF = —— + —— – ———— 1 2 1 + 2
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Parallel system: an example
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Partial redundancy:
an example
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Availability for parallel system
Assuming independent repair,
where Ai is the (steady state or transient) availability of component i.
n
iip
n
i ii
in
iip
tAtAor
MTTRMTTF
MTTRAA
1
11
))(1(1)(
1)1(1
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Series-parallel systems
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System vs component redundancy
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Component redundant system: an example
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Is redundancy always useful ?
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Stand-by redundancyA
B
The system works continuouslyduring 0 — t if:
a) Component A did not fail between 0 — t
b) Component A failed at x between 0 — t , and component B survived from x to t .
x0 tA B
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Stand-by redundancyA
B
x0 tA B
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A
B
Stand-by redundancy (exponential
components)
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Majority voting redundancy
A1
A2
A3
Voter
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2:3 majority voting redundancy
A1
A2
A3
Voter
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