Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance...
-
Upload
morgan-mills -
Category
Documents
-
view
215 -
download
0
Transcript of Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance...
![Page 1: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/1.jpg)
![Page 2: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/2.jpg)
Multi-state System (MSS)Basic Concepts
MSS is able to perform its task with partial performance
“all or nothing” type of failure criterion cannot be formulated
![Page 3: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/3.jpg)
Oil Transportation system
A
C E
1
2 3
1
2
3
D
G1(t) {0,1.5}
G2(t) {0,2}
G3(t) {0, 1.8, 4}
)}(),()(min{)( 321 tGtGtGtG
))(),(),(()( 21 tGtGtGtG n
![Page 4: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/4.jpg)
Generic MSS model
)(tG j },...,{ 1 jjkj gg
Performance stochastic processes for each system element j :
System structure function that produces the stochastic process corresponding to the output performance of the entire MSS
))(),...,(()( 1 tGtGtG n
![Page 5: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/5.jpg)
)(,112)(
,323 )(
,332
)(,121
)(,321
)(,312
4
1.8
1 1.5, 2, 4
3.5
2 0, 2, 4
23
1.5, 0, 41.5
7 1.5, 0, 1.81.5
111.5, 0, 0
0
8
0
6 0, 2, 1.8
1.8
100, 2, 0
0
120, 0, 0
0
5 0, 0, 40
9 0, 0, 1.80
1.5, 2, 1.8
1.5, 2, 0
)(,221
)(,112
)(,121
)(,212
)(,221
)(,212
)(,323
)(,332
)(,212
)(,221
)(,312 )(
,321
)(,121
)(,112
)(,312
)(,212
)(,312
)(,321 )(
,212 )(
,221
)(,112
)(,121
)(,221
)(,332
)(,212
)(,221
)(,112
)(,121 )(
,323
)(,323 )(
,332
)(,332
)(,112 )(
,121
State-space diagram for the flow transmission MSS
![Page 6: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/6.jpg)
Straightforward Reliability Assessmentfor MSS
Stage 1. State-space diagram building or model construction for MSS
Difficult non-formalized process that may cause numerous mistakes even for relatively small MSS
Stage 2. Solving models with hundreds of states
Can challenge the computer resources available
![Page 7: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/7.jpg)
RBD Method: multi-state interpretation
each block of the reliability block diagram represents one multi-state element of the system
each block's j behavior is defined by the corresponding performance stochastic process
logical order of the blocks in the diagram is defined by the system structure function
)(tG j
![Page 8: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/8.jpg)
Combined Universal Generating Function (UGF) and Random Processes
Method
1-st stage: a model of stochastic process should be built for every multi-state element. Based on this model a state probabilities for every MSS's element can be obtained.
2-nd stage: an output performance distribution for the entire MSS at each time instant t should be defined using UGF technique
![Page 9: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/9.jpg)
Multi-state Element Markov Model
k
k-1
2
1
k,k-1
k-1,k-2
3,2
2,1
......k-1,k
k-2,k-1
2,3
1,2
...
![Page 10: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/10.jpg)
Differential Equations forPerformance Distribution
)()()()()(
)()()(
332223211122
2211121
tptptpdt
tdp
tptpdt
tdp
… = …
)()()(
1,1,1 tptpdt
tdpkkkkkk
k
![Page 11: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/11.jpg)
ENTIRE MULTI-STATE SYSTEM RELIABILITY EVALUATION
based on determined states probabilities for all elements, UGF for each individual element should be defined
by using composition operators over UGF of individual elements and their combinations in the entire MSS structure, one can obtain the resulting UGF for the entire MSS
![Page 12: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/12.jpg)
Individual UGF
Individual UGF for element j
jjk
j
jjg
jkg
jg
jj ztpztpztpztu )(...)()(),( 2121
…
…
)(tpjjk
jjkg
)(2 tp j)(1 tp j
2jg1jg
Element j
![Page 13: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/13.jpg)
UGF for Entire MSS
UGF for MSS with n elements and the arbitrary structure function is defined by using composition operator:
1
1
112
2
1
1
11
1
1 1
),...,(
11
111
).z(...
)z,...,z()(
k
i
k
i
ggn
jji
k
i
k
i
gni
k
i
gi
n
n
nnii
j
n
n
nni
n
i
p
ppzU
![Page 14: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/14.jpg)
Example: MSS consists of two elements
1 2
G(t)=min{G1(t),G2(t)}
G1(t) G2(t)
1312111312111 )( ggg zpzpzpzu
222122212 )( gg zpzpzu
![Page 15: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/15.jpg)
UGF for Entire MSS
3
1
2
1
),(2
1
2
12
3
11
1 2
2211
2
22
21
11
1
)z(
)z,z()(
i i
gg
jji
i
gi
i
gi
ii
j
ii
p
ppzU
![Page 16: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/16.jpg)
Polynomials “Multiplication”
},min{2213
},min{2113
},min{2212
},min{2112
},min{2211
},min{2111
22132113
22122112
22112111)(
gggg
gggg
gggg
zppzpp
zppzpp
zppzppzU
![Page 17: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/17.jpg)
Numerical Example
1
2
3
G(t)=min{G1(t)+G2(t), G3(t)}
Entire MSS
}5.1,0{)(1 tG
}2,0{)(2 tG
}4,8.1,0{)(3 tG
![Page 18: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/18.jpg)
State-space diagrams of the system elements.
Element 1 Element 2 Element 3
)(,323
)(,332
)(,312 )(
,321
g22=2.0
g21=0
g33=4.0
g32=1.8
g31=0
)(,112 )(
,121
1
2
(2)2,1λ
(2)1,2μ
1
2
1
3
2
g12=1.5
g11=0
![Page 19: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/19.jpg)
Differential Equations
For element 1:
)()(/)(
)()(/)(
11)1(2,112
)1(1,212
12)1(1,211
)1(2,111
tptpdttdp
tptpdttdp
For element 2:
)()(/)(
)()(/)(
21)2(2,122
)2(1,222
22)2(1,221
)2(2,121
tptpdttdp
tptpdttdp
![Page 20: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/20.jpg)
For element 3:
)()(/)(
)(
)()()(/)(
)()(/)(
32)3(3,233
)3(2,333
31)3(2,1
32)3(3,2
)3(1,233
)3(2,332
32)3(1,231
)3(2,131
tptpdttdp
tp
tptpdttdp
tptpdttdp
![Page 21: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/21.jpg)
Individual UGF
0.222
0212 )()( zpztpzu
5.112
0111 )()( zpztpzu
0.433
8.132
0313 )()( zpzpztpzu
![Page 22: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/22.jpg)
UGF for Entire MSS
)}()},(),({{ 321 zuzuzups
)}(),(),({)( 321 zuzuzuzU
![Page 23: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/23.jpg)
UGF for parallel connected elements 1 and 2
)}(),({)( 2112 zuzuzu p
5.32212
22211
5.12112
02111 )()()()()()()()( ztptpztptpztptpztptp
![Page 24: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/24.jpg)
UGF for elements connected in series
)}(),({)()( 312123 zuzuzuzU s
5
1
)()(i
gi
iztpzU
![Page 25: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/25.jpg)
Resulting Performance Distribution for the Entire MSS
min8.13 tonsg )()()( 22323 tptptp
min0.24 tonsg
,01 g )()()()()()()()( 221131123121111 tptptptptptptptp
min5.12 tonsg )]()()[()()( 333221122 tptptptptp
min5.35 tonsg
)()()()( 2211334 tptptptp
)()()()( 2212335 tptptptp
![Page 26: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/26.jpg)
Probabilities of different performance levels
0
0.25
0.5
0.75
1
0 0.04 0.08 0.12 0.16 0.2
time (years)
p1(t)p3 (t)p4(t)
p2(t)
p5(t)
![Page 27: Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance “all or nothing” type of failure criterion cannot be.](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697bfa51a28abf838c9783a/html5/thumbnails/27.jpg)
CONCLUSIONS The presented method extends classical
reliability block diagram method to repairable multi-state system.
The procedure is well formalized and based on natural decomposition of entire multi-state system.
Instead of building the complex model for the entire multi-state system, one should built n separate relatively simple models for system elements.
Instead of solving one high-order system of differential (for Markov process) or integral (for semi-Markov process) equations one has to solve n low-order systems for each system element.