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Hierarchy of the Binary Models

r = n. k = r. Hierarchy of the Binary Models. {. k-out-of-r-from-n:F. {. r. n. {. {. k-out-of-n:F. Consecutive k-out-of-n. {. k. n. n. Pr{g>x}. 1. r. g nom. 0. x. Binary element. Multi-state element. Pr{g>x}. 1. r. g n. 0. g 1 g 2. x. r = n. k = r.

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Hierarchy of the Binary Models

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  1. r=n k=r Hierarchy of the Binary Models { k-out-of-r-from-n:F { r n { { k-out-of-n:F Consecutive k-out-of-n { k n n

  2. Pr{g>x} 1 r gnom 0 x Binary element Multi-state element Pr{g>x} 1 r gn 0 g1 g2 ... x

  3. r=n k=r Multi-state Models k-out-of-r-from-n Griffith (1986) Sliding Window Systems Levitin (2002) Consecutive k-out-of-n Chiang, Niu (1981), Bollinger (1982) Multi-state consecutive k-out-of-n Hwang, Yao (1989), Kossow, Preuss (1995) Weighted k-out-of-n Wu, Chen (1994) k-out-of-n Parallel Multi-state System

  4. Sliding window system definition k-out-of-r-from-n: Any function of r variables Any real value Acceptability function { r

  5. Sliding window systems { { { ... Total number of groups: n-r+1 { { ... Each element belongs to no more than r groups

  6. { SWS Applications:Manufacturing n { r

  7. { SWS Applications: Service System { n r

  8. { SWS Applications: Quality Control { n r Deviation Levels 3 2 1 0 1 2 3

  9. Representing Multi-state Elements and Groups Element State Distribution r-Group State Distribution gi+2,k gi+1,k ... gi,k S gi+r-1,k Cyclic Buffer ...

  10. Operator for Determining Group Unreliability Composition Operator gi+2,k gi+1,k gi+r,j ... gi,k S +gi+r,k-gi,j gi+r-1,k ...

  11. Like term collection in the the u-function gi+2 gi+1 gi+r,j ... gi gi+r-1 ... gi,1 gi,2 gi,3 gi,Ni gi+r-1 gi+r-1 ...

  12. Algorithm for SWS Reliability Determination

  13. Example of SWS reliability Determination 10 identical elements Element performance distribution P{G>x) x r:

  14. Ij= R/ rj 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 Reliability Importance of SWS Elements Most important element I w

  15. Optimal Sequencing of SWS Elements SWS Elements Performance distribution SWS Reliability R w 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

  16. A B Uneven allocation of SWS elements 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——

  17. Optimal Grouping of SWS Elements in the Presence of Common Cause Failures

  18. Optimal Grouping Solutions for Different r and M r=3 r=5

  19. Ij= R/ sj Group Survivability Importance r=3 r=5

  20. r3=3, w3 r2=6, w2 r1=2, w1 g1g2 g3g4 Multiple sliding window systems { r2=5 { r1=3 …Gn G1 …

  21. { { >w3 { >w2 >w1 Example of SMWS

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