Multi-state System (MSS)Basic Concepts

MSS is able to perform its task with partial performance

“all or nothing” type of failure criterion cannot be formulated

Oil Transportation system

A

C E

1

2 3

1

2

3

D

G1(t) {0,1.5}

G2(t) {0,2}

G3(t) {0, 1.8, 4}

)}(),()(min{)( 321 tGtGtGtG

))(),(),(()( 21 tGtGtGtG n

Generic MSS model

)(tG j },...,{ 1 jjkj gg

Performance stochastic processes for each system element j :

System structure function that produces the stochastic process corresponding to the output performance of the entire MSS

))(),...,(()( 1 tGtGtG n

)(,112)(

,323 )(

,332

)(,121

)(,321

)(,312

4

1.8

1 1.5, 2, 4

3.5

2 0, 2, 4

23

1.5, 0, 41.5

7 1.5, 0, 1.81.5

111.5, 0, 0

0

8

0

6 0, 2, 1.8

1.8

100, 2, 0

0

120, 0, 0

0

5 0, 0, 40

9 0, 0, 1.80

1.5, 2, 1.8

1.5, 2, 0

)(,221

)(,112

)(,121

)(,212

)(,221

)(,212

)(,323

)(,332

)(,212

)(,221

)(,312 )(

,321

)(,121

)(,112

)(,312

)(,212

)(,312

)(,321 )(

,212 )(

,221

)(,112

)(,121

)(,221

)(,332

)(,212

)(,221

)(,112

)(,121 )(

,323

)(,323 )(

,332

)(,332

)(,112 )(

,121

State-space diagram for the flow transmission MSS

Straightforward Reliability Assessmentfor MSS

Stage 1. State-space diagram building or model construction for MSS

Difficult non-formalized process that may cause numerous mistakes even for relatively small MSS

Stage 2. Solving models with hundreds of states

Can challenge the computer resources available

RBD Method: multi-state interpretation

each block of the reliability block diagram represents one multi-state element of the system

each block's j behavior is defined by the corresponding performance stochastic process

logical order of the blocks in the diagram is defined by the system structure function

)(tG j

Combined Universal Generating Function (UGF) and Random Processes

Method

1-st stage: a model of stochastic process should be built for every multi-state element. Based on this model a state probabilities for every MSS's element can be obtained.

2-nd stage: an output performance distribution for the entire MSS at each time instant t should be defined using UGF technique

Multi-state Element Markov Model

k

k-1

2

1

k,k-1

k-1,k-2

3,2

2,1

......k-1,k

k-2,k-1

2,3

1,2

...

Differential Equations forPerformance Distribution

)()()()()(

)()()(

332223211122

2211121

tptptpdt

tdp

tptpdt

tdp

… = …

)()()(

1,1,1 tptpdt

tdpkkkkkk

k

ENTIRE MULTI-STATE SYSTEM RELIABILITY EVALUATION

based on determined states probabilities for all elements, UGF for each individual element should be defined

by using composition operators over UGF of individual elements and their combinations in the entire MSS structure, one can obtain the resulting UGF for the entire MSS

Individual UGF

Individual UGF for element j

jjk

j

jjg

jkg

jg

jj ztpztpztpztu )(...)()(),( 2121

…

…

)(tpjjk

jjkg

)(2 tp j)(1 tp j

2jg1jg

Element j

UGF for Entire MSS

UGF for MSS with n elements and the arbitrary structure function is defined by using composition operator:

1

1

112

2

1

1

11

1

1 1

),...,(

11

111

).z(...

)z,...,z()(

k

i

k

i

ggn

jji

k

i

k

i

gni

k

i

gi

n

n

nnii

j

n

n

nni

n

i

p

ppzU

Example: MSS consists of two elements

1 2

G(t)=min{G1(t),G2(t)}

G1(t) G2(t)

1312111312111 )( ggg zpzpzpzu

222122212 )( gg zpzpzu

UGF for Entire MSS

3

1

2

1

),(2

1

2

12

3

11

1 2

2211

2

22

21

11

1

)z(

)z,z()(

i i

gg

jji

i

gi

i

gi

ii

j

ii

p

ppzU

Polynomials “Multiplication”

},min{2213

},min{2113

},min{2212

},min{2112

},min{2211

},min{2111

22132113

22122112

22112111)(

gggg

gggg

gggg

zppzpp

zppzpp

zppzppzU

Numerical Example

1

2

3

G(t)=min{G1(t)+G2(t), G3(t)}

Entire MSS

}5.1,0{)(1 tG

}2,0{)(2 tG

}4,8.1,0{)(3 tG

State-space diagrams of the system elements.

Element 1 Element 2 Element 3

)(,323

)(,332

)(,312 )(

,321

g22=2.0

g21=0

g33=4.0

g32=1.8

g31=0

)(,112 )(

,121

1

2

(2)2,1λ

(2)1,2μ

1

2

1

3

2

g12=1.5

g11=0

Differential Equations

For element 1:

)()(/)(

)()(/)(

11)1(2,112

)1(1,212

12)1(1,211

)1(2,111

tptpdttdp

tptpdttdp

For element 2:

)()(/)(

)()(/)(

21)2(2,122

)2(1,222

22)2(1,221

)2(2,121

tptpdttdp

tptpdttdp

For element 3:

)()(/)(

)(

)()()(/)(

)()(/)(

32)3(3,233

)3(2,333

31)3(2,1

32)3(3,2

)3(1,233

)3(2,332

32)3(1,231

)3(2,131

tptpdttdp

tp

tptpdttdp

tptpdttdp

Individual UGF

0.222

0212 )()( zpztpzu

5.112

0111 )()( zpztpzu

0.433

8.132

0313 )()( zpzpztpzu

UGF for Entire MSS

)}()},(),({{ 321 zuzuzups

)}(),(),({)( 321 zuzuzuzU

UGF for parallel connected elements 1 and 2

)}(),({)( 2112 zuzuzu p

5.32212

22211

5.12112

02111 )()()()()()()()( ztptpztptpztptpztptp

UGF for elements connected in series

)}(),({)()( 312123 zuzuzuzU s

5

1

)()(i

gi

iztpzU

Resulting Performance Distribution for the Entire MSS

min8.13 tonsg )()()( 22323 tptptp

min0.24 tonsg

,01 g )()()()()()()()( 221131123121111 tptptptptptptptp

min5.12 tonsg )]()()[()()( 333221122 tptptptptp

min5.35 tonsg

)()()()( 2211334 tptptptp

)()()()( 2212335 tptptptp

Probabilities of different performance levels

0

0.25

0.5

0.75

1

0 0.04 0.08 0.12 0.16 0.2

time (years)

p1(t)p3 (t)p4(t)

p2(t)

p5(t)

CONCLUSIONS The presented method extends classical

reliability block diagram method to repairable multi-state system.

The procedure is well formalized and based on natural decomposition of entire multi-state system.

Instead of building the complex model for the entire multi-state system, one should built n separate relatively simple models for system elements.

Instead of solving one high-order system of differential (for Markov process) or integral (for semi-Markov process) equations one has to solve n low-order systems for each system element.