Reliability Block Diagrams

• A reliability block diagram is a success-oriented network describing the function of the system.

• If the system has more than one function, each function is considered individually, and separate reliability block diagram is established for each system function.

• Each component is illustrated by a block. When there is a connection between the end points, we say that component i is functioning.

Example

Consider a pipeline with two independent safety valves that are physically installed in series. These valves are supplied with a spring-loaded, fail-safe, close by hydraulic actuator. The valves are opened and held open by hydraulic control pressure and is closed automatically by spring force whenever the control pressure is removed or lost. In normal operation both valves are held open. The main function of the valves is to act as a safety barrier, i.e., to close and stop the flow in the pipeline in case of an emergency.

Example

It is usually an easy task to convert a fault tree to a reliability block diagram. In this conversion, we start from the top event and replace the gate successfully. An OR-gate is replaced by a series structure of the “components” directly beneath the gate, and an AND-gate is replaced by a parallel structure of the “components” directly beneath the gate.

Structure Function

The state of component i can be described by a binary state variable, i.e.,

Similarly the state of a system can be described by a binary function

1 if component i is functioning

0 if component i is in a failed stateix

1 2

1 if the system is functioning( )

0 is the system is in a failed state

where [ , , , ]Tnx x x

x

x

Series and Parallel Structures

• Series

• Parallel

1

( )n

ii

x

x

1 1

1,2, ,

( ) 1 (1 )

max

nn

i ii i

ii n

x x

x

x

k-out-of-n Structure

1

1

1 if

( )

0 if

n

ii

n

ii

x k

x k

x

2-out-of-3 Structure

1 2 1 3 2 3

1 2 1 3 2 3

2 2 2 2 2 21 2 2 3 3 1 1 2 3 1 2 3 1 2 3 1 2 3

1 2 2 3 3 1 1 2 3

( )

1 (1 )(1 )(1 )

2

x x x x x x

x x x x x x

x x x x x x x x x x x x x x x x x x

x x x x x x x x x

x

Coherent Structures

Definition: A system is said to be coherent if all its components are relevant and the structure function is non-decreasing in each argument.

Relevant:

Irrelevant:

Non-decreasing structure function:

(0 , ) 0 (1 , ) 1 ( , )i i i x x x

(0 , ) (1 , ) ( , )i i i x x x

1 1 2 2, , ,

( ) ( )n nx x x x x x

x x

x x

Definitions

(1 , ) represents a state vector where the state of the ith

component is 1.

(0 , ) represents a state vector where the state of the ith

component is 0.

( , ) represents a state vect

i

i

i

x

x

x or where the state of the ith

component is 0 or 1.

Example

• Component 2 is irrelevant

1 2

1

a b

Some Theorems for Coherent Structures

1 1

1 1 2 2

1 1 2 2

( ) 0 ( ) 1

( )

( ) ( ) ( )

( ) ( ) ( )

where

, , ,

, , ,

nn

i ii i

n n

n n

x x

x y x y x y

x y x y x y

0 1

x

x y x y

x y x y

x y

x y

Redundancy at System Level

Redundancy at Component Level

( ) ( ) ( ) x y x y

We obtain a better system by introducing redundancy at component level than by introducing redundancy at system level.

Path Sets and Cut Sets

• A structure of order n consists of n components numbered from 1 to n. The set of all components is denoted by C.

• A path set P is a set of components in C which by functioning ensures that the system is functioning. A path set is said to be minimal if it cannot be reduced without loosing its status as a path set.

• A cut set K is a set of components in C which by failing causes that the system to fail. A cut set is said to be minimal if it cannot be reduced without loosing its status as a cut set.

Example 1

1 2

1 2

The minimal path sets

1,2 1,3

The minimal cut sets

1 2,3

P P

K K

Example 2

1 2

3 4

1 2

3 4

The minimal path sets

1,4 2,5

1,3,5 2,3,4

The minimal cut sets

1,2 4,5

1,3,5 2,3,4

P P

P P

K K

K K

Structures Represented by Paths

1 1

( ) ; 1,2, ,

= the jth path series structure

= the structure function of a series structure

composed of components in

( ) ( )

j

j

j ii P

j

p p

j ii Pj j

x j p

P

x

x

x x

1 1

= 1- 1 1- 1j

p p

j ij j i P

x

x

Example 2

1 1 4

2 5

3 1 3 5

4 2 3 4

4

1

1 4 2 5 1 3 5 2 3 4

( )

( )

( )

( )

( ) 1 1 ( )

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

jj

x x

x x

x x x

x x x

x x x x x x x x x x

x

x

x

x

x x

Structures Represented by Cuts

1

( ) 1 (1 );

1, 2, ,

( ) ( )

jj

j

j i ii Ki K

k

j

x x

j k

x

x x

Example 2

1 1 2

2 4 5

3 1 3 5

4 2 3 4

1 2 4 5

1 3 5 2 3 4

( ) 1 (1 )(1 )

( ) 1 (1 )(1 )

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

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

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

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

x x

x x

x x x

x x x

x x x x

x x x x x x

x

x

x

x

x

Critical Path

A critical path vector for component i is a state vector

Such that

A critical path set corresponding to the critical path vector for component i is defined by

(1 , )i x

(1 , ) (0 , ) 1i i x x

(1 , ) ; 1,i jC i j x j i x

Structural Importance

1

( , )

( )( )

2where

( ) (1 , ) (0 , )

= the total number of critical path vectors

for component i

i

n

i i

iB i

i

x

x x

Example

Consider 2-out-of-3 structure

3 1

3 1

3 1

(1,0,1) (0,0,1) (1,1,0) (0,1,0) 1(1)

2 2(1,1,0) (1,0,0) (0,1,1) (0,0,1) 1

(2)2 2

(1,0,1) (1,0,0) (0,1,1) (0,1,0) 1(3)

2 2

B

B

B

Example

1 2 3 1 2 3 1 2 3

Given

( , , ) 1 1 1

Then

3 (1)

41

(2)41

(3)4

x x x x x x x x x

B

B

B

Pivotal Decomposition

1

( ) 1 , (1 ) 0 ,

( ) (1 )

where the summation is taken over all n-dimentional

binary vectors.

j j

i i i i

y yj jj

x x

x x

y

x x x x

x y

Example – Bridge Structure

3 3 3 3

3 1 2 4 5

3 1 4 2 5

1 , 1 0 ,

1 ,

0 ,

x x

x x x x

x x x x

x x x

x

x

Structure of Composed Components

Partition into subsystems is done in such a way that each component never appears within more than one of the subsystems.

Some Notations

1 2

1 2

1 2

5 6 7

5 6 5 7

1,2, , 1, 2, ,10

, , , 5,6,7

1,2,3,4,8,9,10

, , , , ,

, , ,

c

Ai i i

Ai i i

C n

A i i i C

A C A

x x x x x x

x x x x x x x

x

x

Coherent Modules

Let the coherent structure be given, and let

Then is said to be a coherent module of

if can be written as a function

where is the structure function of a coherent system.

What we actually do here is to consider all the components with the index belonging to A as one “component” with state variable . When we interpret the system in this way, the structure function can be written as

( , )C A C( , )A ( , )C

( ) x ( ( ), )cA A x x

( )A x

( ( ), )cA A x x

Example

4

5 6 7 8 9 8 10 9 101

5,6,7

,

, ,

Since , ( , ) is referred to as a proper module of ( , ).

cA A

ii

A

x x x x x x x x x x

A C A C

x x

Modular Decomposition

A modular decomposition of a coherent structure is a set of disjoint modules together with an organizing structure such that

1 2

1

1 2

where for

, , , r

r

k i jk

A A Ar

C A A A i j

x x x x

Prime Module

A module that cannot be partitioned into smaller modules without letting each component represent a module, is called a prime module.

III represents a prime module. II is not since it may be described in Fig 3.35 and hence can be partitioned into IIa and IIb.