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Hierarchy of the Binary Models
r=n k=r
k-out-of-r-from-n:Fr
n
Consecutive k-out-of-nk
nn
k-out-of-n:F
Multi-state Models
k-out-of-n
Weighted k-out-of-nWu, Chen (1994)
Parallel Multi-state System
Multi-stateconsecutive k-out-of-n
Hwang, Yao (1989),Kossow, Preuss (1995)
Consecutive k-out-of-n
Chiang, Niu (1981),Bollinger (1982)
Sliding WindowSystems
Levitin (2002)
k-out-of-r-from-nGriffith (1986)
r=n k=r
r
k-out-of-r-from-n:
}1,0{
11,1
m
rh
hmm
G
rnhkG
Sliding window system definition
1
111 ),...,(),...,(
rn
hrhhn GGfGGF
Acceptability function
Any function of r variables Any real value
Total number of groups: n-r+1
...
Each element belongs to no more than r groups
...
Sliding window systems
Cyclic Buffer
gi,k
gi+1,kgi+2,k
gi+r-1,k
...
...
i
k,iK
1k
Ok,ii zp)z(
Element State Distribution
r-Group State Distribution
i
j,iN
1j
gj,ii zq)z(u
Representing Multi-state Elements and Groups
gi,k
gi+1,kgi+2,k
gi+r-1,k
gi+r,j+gi+r,k-gi,j
...
...
Composition Operator
rij,rik,i
i N
1j
gOj,rik,i
K
1krii1i zqp)z(u)z()z(
Operator for Determining Group Unreliability
iK
1kk,ik,ii ).w)O((1p))z((
gi
gi+1gi+2
gi+r-1
gi+r,j
...
...
)(1
,,
1
,,
Ni
kki
kiOm
k
kiOki pzzp
Like term collection in the the u-function
g i+r-1 g i+r-1
gi,1 gi,2 gi,3 gi,Ni
...
Algorithm for SWS Reliability Determination
1. Initialization
F=0; 1-r(z) = 0z .
Determine u-functions of the individual MEs uj(z).
2. Main loop
Repeat the following for j=1,…,n:
2.1. Obtain (z)ju(z)rjΨ(z)r1jΨ .
2.2. If jr add value W)(z),r1jδ(Ψ to F
and remove all the terms with <W from (z)r1jΨ
3. Obtain the SWS reliability as R=1-F.
0
0.2
0.4
0.6
0.8
1
0 3 6 9 12 15 18 21 24 27 30W
R
2 3 4 5 6 7 8 9 10r:
0
0.3
0.6
0.9
0 1 2 3 4x
P{G>x)
Element performance distribution
Example of SWS reliability Determination
10 identical elements
Reliability Importance of SWS Elements
0
0.3
0.6
0.9
0 100 200 300 400 500 600
1 2 3 4 5
6 7 8 9 10
No 1 2 3 4 5 6 7 8 9 10
r 0.87 0.90 0.83 0.95 0.92 0.89 0.80 0.85 0.82 0.95
g 200 200 400 300 100 400 100 200 300 200
Irrelevant element
Most important
element
Ij= R/ rj
I
w
Optimal Sequencing of SWS Elements
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16
optimal for w=6 optimal for w=8 optimal for w=10
R
w
1 2 3 4 5p g p g p g p g p g
0.03 0 0.1 0 0.17 0 0.05 0 0.08 00.22 2 0.1 1 0.83 6 0.25 3 0.2 10.75 5 0.4 2 - - 0.4 5 0.15 2
- - 0.4 4 - - 0.3 6 0.45 4- - - - - - - - 0.12 5
6 7 8 9 10p g p g p g p g p g
0.01 0 0.2 0 0.05 0 0.2 0 0.05 00.22 4 0.1 3 0.25 4 0.1 3 0.25 20.77 5 0.1 4 0.7 6 0.15 4 0.7 6
- - 0.6 5 - - 0.55 5 - -
2,1,6,5,4,8,7,10,3,9 5,1,8,9,6,4,7,3,10,2 5,9,3,1,4,7,10,8,6,2
SWS Elements Performance distribution
SWS Reliability
A
B
RA(3) = p4; RA(4) =0
RB(3) = p4+4(1-p)p3; RB(4) = p4
5—9—3—1—4—7—10—8—6—2
— —6,7,10— —2,5—1,4— —3,8,9— —
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
W
R
Uneven allocation of SWS elements
r=3 r=5
0
0.2
0.4
0.6
0.8
1
0 5 10 15
w
R
M=1 M=2 M=3
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25w
R
M=1 M=2 M=3
M=4 M=5
Optimal Grouping Solutions for Different r and M
r=3 r=5
0
0.1
0.2
0.3
0.4
0.5
0 5 10 15w
I
CSG1 CSG2 CSG3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15 20 25w
I
CSG1 CSG2 CSG3
Group Survivability Importance
Ij= R/ sj