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# Anatoly Lisnianski - PowerPoint PPT Presentation

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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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

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

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

• 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

• 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

• 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

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 Processes MethodPerformance Distribution

… = …

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

Processes Method

Individual UGF

Element j

Individual UGF for element j

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 Processes Method

G1(t)

G2(t)

1

2

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

UGF for Entire MSS Processes Method

Numerical Example Processes Method

1

3

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

2

Entire MSS

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 Processes Method

• For element 1:

• For element 2:

Individual UGF Processes Method

UGF for Entire MSS Processes Method

Probabilities of different performance levels Processes Method

p5(t)

p2(t)

p3 (t)

p4(t)

p1(t)

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.