A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3....
-
Upload
trent-palmerton -
Category
Documents
-
view
226 -
download
1
Transcript of A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3....
![Page 1: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/1.jpg)
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
![Page 2: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/2.jpg)
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, ….
![Page 3: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/3.jpg)
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.
![Page 4: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/4.jpg)
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.
![Page 5: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/5.jpg)
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.
![Page 6: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/6.jpg)
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)
![Page 7: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/7.jpg)
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:
![Page 8: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/8.jpg)
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
![Page 9: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/9.jpg)
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 ,
![Page 10: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/10.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 10
Series system: an example
![Page 11: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/11.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 11
Series system: an example
![Page 12: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/12.jpg)
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.
![Page 13: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/13.jpg)
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.
![Page 14: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/14.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 14
The part-count method
![Page 15: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/15.jpg)
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.
![Page 16: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/16.jpg)
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
...
...
![Page 17: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/17.jpg)
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
![Page 18: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/18.jpg)
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
![Page 19: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/19.jpg)
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
![Page 20: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/20.jpg)
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
![Page 21: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/21.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 21
Parallel system: an example
![Page 22: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/22.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 22
Partial redundancy:
an example
![Page 23: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/23.jpg)
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
![Page 24: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/24.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 24
Series-parallel systems
![Page 25: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/25.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 25
System vs component redundancy
![Page 26: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/26.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 26
Component redundant system: an example
![Page 27: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/27.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 27
Is redundancy always useful ?
![Page 28: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/28.jpg)
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
![Page 29: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/29.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 29
Stand-by redundancyA
B
x0 tA B
![Page 30: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/30.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 30
A
B
Stand-by redundancy (exponential
components)
![Page 31: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/31.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 31
Majority voting redundancy
A1
A2
A3
Voter
![Page 32: A. BobbioReggio Emilia, June 17-18, 20031 Dependability & Maintainability Theory and Methods 3. Reliability Block Diagrams Andrea Bobbio Dipartimento di.](https://reader030.fdocuments.in/reader030/viewer/2022032516/56649c6e5503460f949211ea/html5/thumbnails/32.jpg)
A. Bobbio Reggio Emilia, June 17-18, 2003 32
2:3 majority voting redundancy
A1
A2
A3
Voter