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Multi-state System

g i , p i. 1. Element with five different performance levels. x. g j 1. g j 2. g j 3. g*. g j 4. g j 0 =0. Multi-state System. Element. Pr{ G  x }. Element with total failure. Multi-state System. Combination of Element s. G. G. Gn. G 2. G 1. 1. 1. 2. 3. 2. 3. 1.

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Multi-state System

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  1. gi , pi 1 Element with five different performance levels x gj1 gj2 gj3 g* gj4 gj0=0 Multi-state System Element Pr{Gx} Element with total failure

  2. Multi-state System Combination of Elements G

  3. G Gn G2 G1 1 1 2 3 2 3 1 1 1 2 3 3 2 2 Multi-state System Structure function G=f(G1,G2,…,Gn) Transmission capacity Transmission time G=f(G1, G2, G3)=min{G1+G2, G3}

  4. G2 G1 gi , pi Gn Multi-state System Generic Model g, p i=1,2,…,n G=f(G1,G2,…,Gn) Acceptability Function F(G,W) (0,1) Pr{G>x} R(W) x W System reliability: R(W)=E(F(G,W))

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

  6. G2 Gn G1 SYSTEM PERFORMANCE MEASURES g, p Reliability Average (expected) performance Expected unsupplied demand

  7. pmf of F(G,W): uG(z)=0.6641z4+0.0657z1.5+0.2459z2.5+0.0243z0 W=2 uW(z)=1z2 F(G,W)=1(GW) U(z)= 0.6641z1(42)+0.0657z1(1.52)+0.2459z1(2.52)+0.0243z1(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

  8. Types of Multi-state Systems Series systems Processing speed Transmission capacity

  9. Functions in composition operators … Series systems Processing speed Identical elements:

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

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

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

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

  14. D E(max(w-G,0)) Flow transmission parallel systems Performance measures w R E

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

  16. D E(max(w-G,0)) Task processing parallel systems Performance measures No work sharing w R E

  17. fser fser fser fpar fser fpar Types of Multi-state Systems Series-parallel systems Generalized RBD method Usystem(z)

  18. Types of Multi-state Systems Bridge systems 1 3 5 2 4 Component ... ... ... Element Transmission capacity Flow dispersion No flow dispersion br(G1, G2, G3, G4, G5) = min{G1, G3}+min{G2, G4} + min{|G1 G3|,|G2 G4|, G5}1((G1 G3)(G2 G4)<0) br(G1, G2, G3, G4, G5) =max{min{G1,G3} min{G2,G4}, min{G1,G5,G4}, min{G2,G5,G3}}

  19. Types of Multi-state Systems Bridge systems Task processing speed No work sharing Work sharing T = min{t1+t3, t2+t4, t1+t5+t4, t2+t5+t3} br(G1, G2, G3, G4, G5) = 1/T =max{ser(G1,G3),ser(G2,G4), ser(G1,G4,G5),ser(G2,G3,G5))} br(G1, G2, G3, G4, G5) =/[(f+G5)+(e+G5)] f = G4, e = G2 if (G2G1) (G3G4) f = G3, e = G1 if (G2G1) > (G3G4)  = G1G2+G1G5+G2G5  = G3G4+G3G5+G4G5

  20. Component m Component 1 Component M ... Types of Multi-state Systems MSS with two failure modes ... ... ... ... R=1-0.5(Q0+Qc) Flow transmission (valves) Element Open Task processing (switches) Close

  21. t1 t1 t2 Open t2 Types of Multi-state Systems MSS with two failure modes Close T=max(t1,t2) T=min(t1,t2) Open Close T=min(t1,t2) T=max(t1,t2)

  22. D(I) t w01 w02 w03 w04 w05 w06 w11 w12 w13 w14 w15 w16 unit 3 unit 4 unit 1 unit 2 unit 5 I Types of Multi-state Systems Weighted voting systems - system output (0,1,x) - threshold - rejection weights - acceptance weights d1(I) d2(I) d3(I) d4(I) d5(I) d6(I) - voting units outputs (0,1,x) unit 6 - system input (0,1)

  23. Example of Weighted Voting System Undersea target detection system ?

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

  25. r3=3, w3 r2=6, w2 r1=2, w1 g1g2 g3g4 { r2=5 Types of Multi-state Systems Multiple sliding window systems { r1=3 …Gn G1 …

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

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

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

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

  30. Types of Multi-state Systems Fault-Tolerant Programming Effect of Versions Sequencing 3-out-of-5 system 5 3 1 3 4 2 2 4 5 1 t2+t4+t5 t1+t3 5 3 3 4 4 2 1 2 5 1 t1+t2+t5 t3+t4

  31. … Types of Multi-state Systems Multiprocessor systems Processing speed S Processing time T=G/S Computational burden G

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

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