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A. Bobbio Reggio Emilia, June 17-18, 2003 1
Dependability & Maintainability Theory and Methods
3. Reliability Block Diagrams
Andrea BobbioDipartimento di Informatica
Università del Piemonte Orientale, “A. Avogadro”
15100 Alessandria (Italy)
[email protected] - http://www.mfn.unipmn.it/~bobbio/IFOA
IFOA, Reggio Emilia, June 17-18, 2003
A. Bobbio Reggio Emilia, June 17-18, 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 Reggio Emilia, June 17-18, 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.
A. Bobbio Reggio Emilia, June 17-18, 2003 4
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.
A. Bobbio Reggio Emilia, June 17-18, 2003 5
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.
A. Bobbio Reggio Emilia, June 17-18, 2003 6
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)
A. Bobbio Reggio Emilia, June 17-18, 2003 7
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:
A. Bobbio Reggio Emilia, June 17-18, 2003 8
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
A. Bobbio Reggio Emilia, June 17-18, 2003 9
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 ,
A. Bobbio Reggio Emilia, June 17-18, 2003 10
Series system: an example
A. Bobbio Reggio Emilia, June 17-18, 2003 11
Series system: an example
A. Bobbio Reggio Emilia, June 17-18, 2003 12
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.
A. Bobbio Reggio Emilia, June 17-18, 2003 13
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.
A. Bobbio Reggio Emilia, June 17-18, 2003 14
The part-count method
A. Bobbio Reggio Emilia, June 17-18, 2003 15
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.
A. Bobbio Reggio Emilia, June 17-18, 2003 16
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
...
...
A. Bobbio Reggio Emilia, June 17-18, 2003 17
Parallel system
"failedhassystemparallelThe"pE
"failedhavecomponentsnAll"____
2
__
1 ... nEEE
)...()(____
2
__
1
__
np EEEPEP )()...()(____
2
__
1 nEPEPEP
Therefore:
)(1)( pp EPEP
A. Bobbio Reggio Emilia, June 17-18, 2003 18
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
A. Bobbio Reggio Emilia, June 17-18, 2003 19
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
A. Bobbio Reggio Emilia, June 17-18, 2003 20
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
A. Bobbio Reggio Emilia, June 17-18, 2003 21
Parallel system: an example
A. Bobbio Reggio Emilia, June 17-18, 2003 22
Partial redundancy:
an example
A. Bobbio Reggio Emilia, June 17-18, 2003 23
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
A. Bobbio Reggio Emilia, June 17-18, 2003 24
Series-parallel systems
A. Bobbio Reggio Emilia, June 17-18, 2003 25
System vs component redundancy
A. Bobbio Reggio Emilia, June 17-18, 2003 26
Component redundant system: an example
A. Bobbio Reggio Emilia, June 17-18, 2003 27
Is redundancy always useful ?
A. Bobbio Reggio Emilia, June 17-18, 2003 28
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
A. Bobbio Reggio Emilia, June 17-18, 2003 29
Stand-by redundancyA
B
x0 tA B
A. Bobbio Reggio Emilia, June 17-18, 2003 30
A
B
Stand-by redundancy (exponential
components)
A. Bobbio Reggio Emilia, June 17-18, 2003 31
Majority voting redundancy
A1
A2
A3
Voter
A. Bobbio Reggio Emilia, June 17-18, 2003 32
2:3 majority voting redundancy
A1
A2
A3
Voter