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.

Multi-state System

),Pr(

},...,{ 0

ijiij

iikii

gGp

ggG

Element

Pr{Gx}

Element with total failure

Element with five different performance levels

g* gj4 gj3gj2gj1gj0=0x

1

gi , pi

Multi-state System

Combination of Elements

G

Multi-state System

Structure function

Gn

G2

G1

G G=(G1,G2,…,Gn

1 2

3

1

23

1

2 3

1 2

3 2

1

G=(G1, G2, G3min{G1+G2, G3}

Multi-state SystemGeneric Model

i=1,2,…,ngi , pi

G=(G1,G2,…,Gn

GnG2

G1

g, p

Acceptability Function

F(G,W (0,1

WG

WGWGWGF

,0

,1)(1),(

System reliability: R(W)=E(F(G,W

Pr{G>x}

W x

R(W)

SYSTEM PERFORMANCE MEASURESSYSTEM PERFORMANCE MEASURES

Average (expected) performance

Reliability

G

x

Pr{G>x}

Dem

and

Expected unsupplied

demand

SYSTEM PERFORMANCE MEASURESSYSTEM PERFORMANCE MEASURES

Average (expected) performance

Reliability

Expected unsupplied demand

GnG2

G1

g, p),Pr(

},...,{ 1

jj

k

gGp

ggG

Wjg

j

k

jjj pWgFpR

0),(

k

jjj gpE

0

)(

),0max(0

jWjg

j

k

jjj

gWp

gWp

)1('),(

)()()(),(

)),((

UWGF

zuzuzUWGF

WGFER

WF

G

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(42)+0.0657z1(1.52)+0.2459z1(2.52)+0.0243z1(02)

R = U’(1) = 0.66411(42)+0.06571(1.52)+0.24591(2.52)+0.02431(02) = 0.6641+0.2459 = 0.91

pmf of F(G,W):

Types of Multi-state SystemsSeries systems

2121

121

21

)/1/1(/1

GGGG

TTTG

TTT

),min( 21 GGG

Series systems

2121

121

21

)/1/1(/1

GGGG

TTTG

TTT

Functions in composition operators

1

1

11 )(

11

01

11 )1()}(),...,({)(

n

jjg

j

n

jj

n

jn zpzpzuzuzU

…

ngnn zpzpzU /0)1()( Identical elements:

Series systems

),min( 21 GGG

Functions in composition operators

…

Identical elements:

},...,min{1

1

01

11

111min

)1())(),...,(()( nggj

n

jj

n

jn zpzpzuzuzU

gnn zpzp 0)1(

11

01)1()( jg

jjj zpzpzu

gw ˆ

gw ˆ0

n

jjp

11

n

jjpgw

n

jjpgw

n

jjpw

11ˆ

11)ˆ()

111(

)1

11(

n

jjpw

n

jjpg

11ˆ

Series systemsPerformance measures

E(max(w-G,0))

0

},...,min{ˆ 111 nggg

n

jjgg

1

11/1ˆ

Processing speed:

Transmissioncapacity:

Types of Multi-state SystemsParallel systems

21 GGG

),max( 21 GGG

Flow dispersion

No flow dispersion

21 GGG

),max( 21 GGG

Work sharing

No work sharing

Parallel systems

21 GGG Flow dispersion

Functions in composition operators

n identical elements:

21112111

211121

11g2111

02111 )-(1)-(1)-)(1-(1)(

gggzppzppzppzppzU

n

k

kgknk zppknk

nzU

0)1(

)!(!

!)(

Flow dispersion 11

01)1()( jg

jjj zpzpzu

Flow transmission parallel systemsPerformance measures

w>g11+g210w-p11g11-p21g21

g21<w<g11+g21p11p21g11p11(p21-1)+g21p21(p11-1)+w(1-p11p21)

g11<w< g21p21(1-p21)(w-g11p11) p11g11+p21g21

0<w<g11p11+p21-p11p21(1-p11)(1-p21)w

11

01)1()( jg

jjj zpzpzu

E(max(w-G,0))

Parallel systems

),max( 21 GGG No work sharing

Functions in composition operators

gnn zpzpzU ))1(1()1()( 0 n identical elements:

.)(1))(1(1

)(1)(1))(1(1)(

21112111

2111

2121110

2111)max(

2111

112121110

2111

ggg,g

gg

zpzp-pzp-p-zpp

zp-pzp-pzp-p-zU

No work sharing 1

10

1)1()( jgjjj zpzpzu

Task processing parallel systemsPerformance measures

w>g21 0w-p11g11-p21g21+p11p21g11

g11<w g21 p21 (1-p21)(w-g11p11) p11(1-p21)g11+p21g21

0<wg11 p11+p21-p11p21 (1-p11)(1-p21)w

No work sharing 11

01)1()( jg

jjj zpzpzu

E(max(w-G,0))

Types of Multi-state SystemsSeries-parallel systems

Generalized RBD methodGeneralized RBD method

ser ser

serpar

ser par Usystem(z)

Types of Multi-state SystemsBridge systems

...Element

Component

1

2

3

45

... ...

br(G1, G2, G3, G4, G5) = min{G1,

G3}+min{G2, G4} + min{|G1 G3|, |G2 G4|, G5}1((G1 G3)(G2 G4)<0)

Flow dispersion No flow dispersion

br(G1, G2, G3, G4, G5) =max{min{G1,G3}

min{G2,G4}, min{G1,G5,G4}, min{G2,G5,G3}}

Types of Multi-state SystemsBridge systems

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))}

No work sharing Work sharing

br(G1, G2, G3, G4, G5) =/[(f+G5)+

(e+G5)]f = G4, e = G2 if (G2G1) (G3G4)f = G3, e = G1 if (G2G1) > (G3G4)

= G1G2+G1G5+G2G5

= G3G4+G3G5+G4G5

Types of Multi-state SystemsMSS with two failure modes

... ... ...... ...

Component mComponent 1 Component M

Element

Close

Open

Flow transmission (valves)

Task processing (switches)

R=1-0.5(Q0+Qc)

t1

t2

t1 t2

T=max(t1,t2)

Types of Multi-state SystemsMSS with two failure modes

Open Close

T=min(t1,t2)

T=min(t1,t2)

Open Close

T=max(t1,t2)

d1(I) d2(I) d3(I) d4(I) d5(I) d6(I)

w11 w1

2 w13 w1

4 w15 w1

6

I

D(I)

w01 w0

2 w03 w0

4 w05 w0

6

unit 1 unit 2 unit 3 unit 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 SystemsWeighted voting systems

Example of Weighted Voting System

Undersea target detection system

r

k-out-of-r-from-n:

11,1

rnikg

ri

imm

nr

Types of Multi-state SystemsSliding window systems

r1=2, w1

r2=6, w2

r3=3, w3

g1 g2 g3 g4

Types of Multi-state SystemsMultiple sliding window systems

r1=3r2=5

G1 … …Gn

Linear

Circular

Types of Multi-state SystemsConsecutively connected systems

Single terminal

Types of Multi-state SystemsMulti-state networks

Multiple terminals

Tree structure

Node states

HardwareHardware SoftwareSoftware

SuccessSuccess

FailureFailure

InputInput

OutputOutput

Types of Multi-state SystemsSoftware systems

N-Version Programming

Fault-Tolerant Programming

Version 1

Version 2

Version N

…

Voter

M Identical Outputs

Correct Result

Failure

Recovery Blocks Scheme

Version 1 ATCorrect Result

+

-

Version 2 ATCorrect Result

+

-…

Version N ATCorrect Result

+

-Failure

Types of Multi-state Systems

t1+t3

1 3

2 4 512

3

4

5

3

21 5

43

12

5

4

Effect of Versions Sequencing

t2+t4+t5

t3+t4 t1+t2+t5

3-out-of-5 system

Fault-Tolerant Programming

Types of Multi-state Systems

Types of Multi-state SystemsMultiprocessor systems

…

SS

GG

Types of Multi-state SystemsGrid computing services

ResourceRMSRMS

Resource

Resource

Resource

Resource

Request for service

ResourceResource

Resource