1. Multi-state System Element Pr{G  x} Element with total failure Element with five different performance levels g*gj4gj4 g j3 gj2gj2 gj1gj1 g j0 =0 x 1 g i, p i

2. Multi-state System Combination of Element s G

3. Multi-state System Structure function Gn G2G2 G1G1 G G=  (G 1,G 2,…,G n  1 2 3 1 2 3 1 2 3 1 2 3 2 1 G=  (G 1, G 2, G 3  min{G 1 +G 2, G 3 }

4. Multi-state System Generic Model i=1,2,…,n g i, p i G=  (G 1,G 2,…,G n  Gn G2G2 G1G1 g, p Acceptability Function F(G,WF(G,W (0,1  System reliability: R(W)=E(F(G,W  Pr{G>x} W x R(W)R(W)

5. SYSTEM PERFORMANCE MEASURES Average (expected) performance Reliability G x Pr{G>x} Demand Expected unsupplied demand

6. SYSTEM PERFORMANCE MEASURES Average (expected) performance Reliability Expected unsupplied demand Gn G2G2 G1G1 g, p

7. u G (z)=0.6641z 4 +0.0657z 1.5 +0.2459z 2.5 +0.0243z 0 W=2 u W (z)=1z 2 F(G,W)=1(G  W) U(z)= 0.6641z 1(4  2) +0.0657z 1(1.5  2) +0.2459z 1(2.5  2) +0.0243z 1(0  2) R = U’(1) = 0.6641  1(4  2)+0.0657  1(1.5  2)+0.2459  1(2.5  2) +0.0243  1(0  2) = 0.6641+0.2459 = 0.91 pmf of F(G,W):

8. Types of Multi-state Systems Series systems

9. Functions in composition operators … Identical elements:

10. Series systems Functions in composition operators … Identical elements:

11. Series systems Performance measures E(max(w-G,0)) 0 Processing speed: Transmission capacity:

12. Types of Multi-state Systems Parallel systems Flow dispersion No flow dispersion Work sharing No work sharing

13. Parallel systems Flow dispersion Functions in composition operators n identical elements: Flow dispersion

14. Flow transmission parallel systems Performance measures w-p 11 g 11 -p 21 g 21 0w>g 11 +g 21 g 11 p 11 (p 21 -1)+g 21 p 21 (p 11 -1)+w(1-p 11 p 21 ) p 11 p 21 g 21 <w<g 11 +g 21 p 11 g 11 +p 21 g 21 (1-p 21 )(w-g 11 p 11 ) p 21 g 11 <w < g 21 (1-p 11 )(1-p 21 )w p 11 +p 21 -p 11 p 21 0<w < g 11 E(max(w-G,0))

15. Parallel systems No work sharing Functions in composition operators n identical elements: No work sharing

16. Task processing parallel systems Performance measures w-p 11 g 11 -p 21 g 21 +p 11 p 21 g 11 0w>g 21 p 11 (1- p 21 )g 11 +p 21 g 21 (1-p 21 )(w-g 11 p 11 )p 21 g 11 <w  g 21 (1-p 11 )(1-p 21 )wp 11 +p 21 -p 11 p 21 0<w  g 11 No work sharing E(max(w-G,0))

17. Types of Multi-state Systems Series-parallel systems Generalized RBD method  ser  par  ser  par U system (z)

18. Types of Multi-state Systems Bridge systems... Element Component 1 2 3 4 5...  br (G 1, G 2, G 3, G 4, G 5 ) = min{G 1, G 3 }+min{G 2, G 4 } + min{|G 1  G 3 |, |G 2  G 4 |, G 5 }1((G 1  G 3 )(G 2  G 4 )<0) Flow dispersionNo flow dispersion  br (G 1, G 2, G 3, G 4, G 5 ) = max{min{G 1,G 3 } min{G 2,G 4 }, min{G 1,G 5,G 4 }, min{G 2,G 5,G 3 }}

19. Types of Multi-state Systems Bridge systems T = min{t 1 +t 3, t 2 +t 4, t 1 +t 5 +t 4, t 2 +t 5 +t 3 }  br (G 1, G 2, G 3, G 4, G 5 ) = 1/T = max{  ser (G 1,G 3 ),  ser (G 2,G 4 ),  ser (G 1,G 4,G 5 ),  ser (G 2,G 3,G 5 ))} No work sharingWork sharing  br (G 1, G 2, G 3, G 4, G 5 ) =  /[( f+G 5 )  +( e+G 5 )  ] f = G 4, e = G 2 if (G 2  G 1 )   (G 3  G 4 )  f = G 3, e = G 1 if (G 2  G 1 )  > (G 3  G 4 )   = G 1 G 2 +G 1 G 5 +G 2 G 5  = G 3 G 4 +G 3 G 5 +G 4 G 5

20. Types of Multi-state Systems MSS with two failure modes... Component m Component 1 Component M Element Close Open Flow transmission (valves) Task processing (switches) R=1-0.5(Q 0 +Q c )

21. t1t1 t2t2 t1t1 t2t2 T=max(t 1,t 2 ) Types of Multi-state Systems MSS with two failure modes Open Close T=min(t 1,t 2 ) Open Close T=max(t 1,t 2 )

22. d 1 (I) d 2 (I) d 3 (I) d 4 (I) d 5 (I) d 6 (I) w 1 1 w 1 2 w 1 3 w 1 4 w 1 5 w 1 6  I D(I)D(I) w 0 1 w 0 2 w 0 3 w 0 4 w 0 5 w 0 6 unit 1unit 2 unit 3unit 4 unit 5 unit 6 - system output (0,1,x) - threshold - voting units outputs (0,1,x) - acceptance weights - system input (0,1) - rejection weights Types of Multi-state Systems Weighted voting systems

23. Example of Weighted Voting System Undersea target detection system

24. r k-out-of-r-from-n: n r Types of Multi-state Systems Sliding window systems

25. r 1 =2, w 1 r 2 =6, w 2 r 3 =3, w 3 g 1 g 2 g 3 g 4 Types of Multi-state Systems Multiple sliding window systems r 1 =3 r 2 =5 G 1 … …G n

26. Linear Circular Types of Multi-state Systems Consecutively connected systems

27. Single terminal Types of Multi-state Systems Multi-state networks Multiple terminals Tree structure Node states

28. Hardware Software Success Failure Input Output Types of Multi-state Systems Software systems

29. N-Version Programming Fault-Tolerant Programming Version 1 Version 2 Version N … Voter M Identical Outputs Correct Result Failure Recovery Blocks Scheme Version 1 AT Correct Result + - Version 2 AT Correct Result + - … Version N AT Correct Result + - Failure Types of Multi-state Systems

30. t1+t3t1+t3 13 245 1 2 3 4 5 3 215 4 3 1 2 5 4 Effect of Versions Sequencing t2+t4+t5t2+t4+t5 t3+t4t3+t4 t1+t2+t5t1+t2+t5 3-out-of-5 system Fault-Tolerant Programming Types of Multi-state Systems

31. Multiprocessor systems … S G

32. Types of Multi-state Systems Grid computing services Resource RMS Request for service Resource