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.,
if component i is functioning
1
xi  
0 if component i is in a failed state
Similarly the state of a system can be described by a
binary function
if the system is functioning
1
 ( x)  
0 is the system is in a failed state
where x  [ x1 , x2 ,
, xn ]T
Series and Parallel Structures
• Series
n
 (x)   xi
i 1
• Parallel
n
 (x)  1   (1  xi ) 
i 1
 max xi
i 1,2, , n
n
xi
i 1
k-out-of-n Structure

1
if
x

k

i


i 1
 ( x)  
n
0 if
xi  k


i 1
n
2-out-of-3 Structure
 (x)  x1 x2
x1 x3
x2 x3
 1  (1  x1 x2 )(1  x1 x3 )(1  x2 x3 )
 x1 x2  x2 x3  x3 x1  x12 x2 x3  x1 x22 x3  x1 x2 x32  x12 x22 x32
 x1 x2  x2 x3  x3 x1  2 x1 x2 x3
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:  (0i , x)  0   (1i , x)  1 ( i , x)
Irrelevant:  (0i , x)   (1i , x) (i , x)
Non-decreasing structure function:
x  x  x1  x1, x2  x2 ,
  (x)   (x)
, xn  xn
Definitions
(1i , x ) represents a state vector where the state of the ith
component is 1.
(0i , x ) represents a state vector where the state of the ith
component is 0.
(i , x ) represents a state vector where the state of the ith
component is 0 or 1.
Example
• Component 2 is irrelevant
1
2
a
b
1
Some Theorems for Coherent
Structures
 (0)  0  (1)  1
n
x
i 1
i
  ( x) 
n
xi
i 1
 (x
y )   ( x)
 (y )
 ( x  y )   ( x)   ( y )
where
x
y   x1
y1 , x2
x  y   x1 y1 , x2 y2 ,
y2 ,
, xn yn 
, xn
yn 
Redundancy at System Level
Redundancy at Component Level
We obtain a better system by introducing redundancy at component level than
by introducing redundancy at system level.
 (x
y)   (x)
 (y)
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
The minimal path sets
P1  1, 2
P2  1,3
The minimal cut sets
K1  1
K 2  2,3
Example 2
The minimal path sets
P1  1, 4
P3  1,3,5
P2  2,5
P4  2,3, 4
The minimal cut sets
K1  1, 2
K3  1,3,5
K 2  4,5
K 4  2,3, 4
Structures Represented by Paths
 (x )   x ; j  1, 2, , p
j
i
iPj
= the jth path series structure
= the structure function of a series structure
composed of components in Pj
 (x) 
p
j 1
 j (x) 
p
x
j 1 iPj
i


= 1-  1   j  x    1-  1   xi 
j 1
j 1 
 iPj 
p
p
Example 2
1 (x)  x1 x4
  (x)  x2 x5
3 (x)  x1 x3 x5
 4 (x)  x2 x3 x4
4
 (x)  1   1   j ( x) 
j 1
 1  (1  x1 x4 )(1  x2 x5 )(1  x1 x3 x5 )(1  x2 x3 x4 )
Structures Represented by Cuts
 j ( x) 
iK j
j  1, 2,
xi  1   (1  xi );
iK j
,k
k
 ( x)    ( x)
j 1
j
Example 2
1 (x)  1  (1  x1 )(1  x2 )
 2 (x)  1  (1  x4 )(1  x5 )
 3 (x)  1  (1  x1 )(1  x3 )(1  x5 )
 4 (x)  1  (1  x2 )(1  x3 )(1  x4 )
 (x)  1  (1  x1 )(1  x2 )   1  (1  x4 )(1  x5 )  
1  (1  x1 )(1  x3 )(1  x5 )   1  (1  x2 )(1  x3 )(1  x4 ) 
Critical Path
A critical path vector for component i is a state vector
(1i , x)
Such that
 (1i , x)   (0i , x)  1
A critical path set corresponding to the critical path
vector for component i is defined by
C (1i , x)  i
 j; x
j
 1, j  i
Structural Importance
B (i ) 
 (i )
2n 1
where
 (i) 
  (1 , x)   (0 , x)
( i , x )
i
i
= the total number of critical path vectors
for component i
Example
Consider 2-out-of-3 structure
 (1,0,1)   (0,0,1)    (1,1,0)   (0,1,0)  1

B (1) 


231
2
 (1,1,0)   (1,0,0)    (0,1,1)   (0,0,1)  1

B (2) 

31
2
2
 (1,0,1)   (1,0,0)    (0,1,1)   (0,1,0)  1

B (3) 

31
2
2
Example
Given
 ( x1 , x2 , x3 )  x1   x2
x3   x1 1  1  x2 1  x3 
Then
3
B (1) 
4
1
B (2) 
4
1
B (3) 
4
Pivotal Decomposition
 (x )  xi 1i , x   (1  xi )  0i , x 
x

yj
1 y j

 ( x )   y  j x j (1  x j )   y 


where the summation is taken over all n-dimentional
binary vectors.
Example – Bridge Structure
  x   x3 13 , x   1  x3    03 , x 
 13 , x    x1
x2  x4
  03 , x   x1 x4
x2 x5
x5 
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
C  1, 2,
A  i1 , i2 ,
, n  1, 2,
,10
, i   5, 6, 7  C
A  C  A  1, 2,3, 4,8,9,10
c
, xi    x5 , x6 , x7 
x   xi1 , xi2 ,
A
  x A     xi , xi ,
1
2

, xi   x5
x6  x5
x7 
Coherent Modules
Let the coherent structure (C ,  ) be given, and let A  C
Then ( A,  ) is said to be a coherent module of (C ,  )
A
Ac
if  (x) can be written as a function  (  (x ), x )
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
A

(
x
) . When we
“component” with state variable
interpret the system in this way, the structure function
A
Ac
can be written as  (  (x ), x )
Example
A  5,6,7

  x  , x
A
Ac

 4 
  x5 , x6 , x7    xi   x8 x9 x8 x10 x9 x10 
 i 1 
Since A  C , ( A,  ) is referred to as a proper module of (C ,  ).
Modular Decomposition
A modular decomposition of a coherent structure is a
set of disjoint modules together with an organizing
structure such that
C
r
Ak
k 1
where Ai
Aj   for i  j
A1
A2

  x     1  x  ,  2  x  ,
, r  x
Ar

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.