Anatoly Lisnianski
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Anatoly Lisnianski. EXTENDED RELIABILITY BLOCK DIAGRAM METHOD. 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. 1. D. C. E. 3. 2. G 1 ( t ). {0,1.5}.

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Anatoly Lisnianski

EXTENDED RELIABILITY

BLOCK DIAGRAM

METHOD


Multi state system mss basic concepts
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

1

D

C

E

3

2

G1(t)

{0,1.5}

G3(t) {0, 1.8, 4}

1

3

A

2

G2(t)

{0,2}

Oil Transportation system


Generic mss model
Generic MSS model

Performance stochastic processes

for each system elementj:

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


State-space diagram for the flow transmission MSS

1

1.5, 2, 4

3.5

4

2

0, 2, 4

1.5, 2, 1.8

2

1.8

3

1.5, 0, 4

1.5

5

6

8

0, 2, 1.8

1.5, 2, 0

0, 0, 4

1.8

0

0

7

1.5, 0, 1.8

1.5

10

9

0, 0, 1.8

0, 2, 0

0

0

11

1.5, 0, 0

0

12

0, 0, 0

0


Straightforward reliability assessment for 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
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


Combined universal generating function ugf and random processes method
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

l Processes Methodk,k-1

mk-1,k

lk-1,k-2

mk-2,k-1

...

...

...

k

k-1

2

1

l3,2

m2,3

l2,1

m1,2

Multi-state Element Markov Model


Differential equations for performance distribution
Differential Equations for Processes MethodPerformance Distribution

… = …


Entire multi state system reliability evaluation
ENTIRE MULTI-STATE SYSTEM RELIABILITY EVALUATION Processes Method

  • 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

Processes Method

Individual UGF

Element j

Individual UGF for element j


Ugf for entire mss
UGF for Entire MSS Processes Method

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


Example mss consists of two elements
Example: MSS consists of two elements Processes Method

G1(t)

G2(t)

1

2

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


Ugf for entire mss1
UGF for Entire MSS Processes Method



Numerical example
Numerical Example Processes Method

1

3

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

2

Entire MSS


State space diagrams of the system elements

Element 3 Processes Method

Element 1

Element 2

g12=1.5

g33=4.0

g22=2.0

g32=1.8

g21=0

2

3

1

1

1

2

2

g11=0

g31=0

State-space diagrams of the system elements.


Differential equations
Differential Equations Processes Method

  • For element 1:

  • For element 2:



Individual ugf1
Individual UGF Processes Method


Ugf for entire mss2
UGF for Entire MSS Processes Method





Probabilities of different performance levels
Probabilities of different performance levels Processes Method

p5(t)

p2(t)

p3 (t)

p4(t)

p1(t)


Conclusions
CONCLUSIONS Processes Method

  • 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.


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