Multi-state System (MSS) Basic Concepts MSS is able to perform its task with partial performance...
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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
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
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
)(,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
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
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
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
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
...
Differential Equations forPerformance Distribution
)()()()()(
)()()(
332223211122
2211121
tptptpdt
tdp
tptpdt
tdp
… = …
)()()(
1,1,1 tptpdt
tdpkkkkkk
k
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
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
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
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
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
Polynomials “Multiplication”
},min{2213
},min{2113
},min{2212
},min{2112
},min{2211
},min{2111
22132113
22122112
22112111)(
gggg
gggg
gggg
zppzpp
zppzpp
zppzppzU
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
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
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
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
Individual UGF
0.222
0212 )()( zpztpzu
5.112
0111 )()( zpztpzu
0.433
8.132
0313 )()( zpzpztpzu
UGF for Entire MSS
)}()},(),({{ 321 zuzuzups
)}(),(),({)( 321 zuzuzuzU
UGF for parallel connected elements 1 and 2
)}(),({)( 2112 zuzuzu p
5.32212
22211
5.12112
02111 )()()()()()()()( ztptpztptpztptpztptp
UGF for elements connected in series
)}(),({)()( 312123 zuzuzuzU s
5
1
)()(i
gi
iztpzU
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
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)
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.
