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

2. 0 g nom 1 r Pr{g>x} x 0 gngn 1 r xg 1 g 2... Binary elementMulti-state element

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

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

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

6. SWS Applications: Manufacturing n r

7. r n SWS Applications: Service System

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

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

10. g i,k g i+1,k g i+2,k g i+r-1,k g i+r,j  +g i+r,k -g i,j... Composition Operator Operator for Determining Group Unreliability

11. gigi g i+1 g i+2 g i+r-1 g i+r,j... Like term collection in the the u-function g i+r-1 g i,1 g i,2 g i,3 g i,N i...

12. Algorithm for SWS Reliability Determination

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

14. Reliability Importance of SWS Elements 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 I j = R/ r j I w

15. Optimal Sequencing of SWS Elements R w 2,1,6,5,4,8,7,10,3,95,1,8,9,6,4,7,3,10,25,9,3,1,4,7,10,8,6,2 SWS Elements Performance distribution SWS Reliability

16. A B R A (3) = p 4 ; R A (4) =0 R B (3) = p 4 +4(1-p)p 3 ; R B (4) = p 4 5—9—3—1—4 — 7—10—8 — 6 — 2 — —6,7,10— —2,5 — 1,4— —3,8,9 — — Uneven allocation of SWS elements

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

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

19. r=3r=5 Group Survivability Importance I j = R/ s j

20. r 1 =2, w 1 r 2 =6, w 2 r 3 =3, w 3 g 1 g 2 g 3 g 4 Multiple sliding window systems r 1 =3 r 2 =5 G 1 … …G n

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