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Unit 17: System Reliability

17.3 The Bridge Structure Notes

The system whose structure is shown below is called the bridge system. Its minimal path sets
are: {1, 4}, {1, 3, 5}, {2, 5}, {2, 3, 4}.

1 4
3

2 5
For example, the system will work if only 1 and 4 are working, but will not work if only 1 is
working.

Its structure function is given by


 ( ) max{x x ,x x x ,x x ,x x x }
x
5
4
1
3
1
3
4
2
5
2




 1 (1 x x )(1 x x x )(1 x x )(1 x x x ).

2
1
3
5
2
3
5
4
1
4
17.3.1 Minimal Cut Sets
 A state vector x is call a cut vector if (x) = 0.
 If (y) = 1 for all y > x, then x is a minimal cut vector.
 If x is a minimal cut vector, then the set C = {i: x = 0} is a minimal cut set.
i
Examples:
1. The Series System: There are n minimal cut sets, namely, the sets consisting of all but one
component.
2. The Parallel System: There is one minimal cut set, namely, the empty set.
 n 
3. The k-out-of-n System: There are   minimal cut sets, namely all of the sets


 n k 1 
consisting of exactly n – k + 1 components.
Let C , …, C be the minimal cut sets of a system. A system will not function if and only if all the
1 k
components of at least one minimal cut set are not functioning, so that
k
 ( )   maxx .
x
i

j 1 i j C
This expresses the system as a series arrangement of parallel systems.
The Bridge Structure
The system whose structure is shown below is called the bridge system. Its minimal cut sets are:
{1, 2}, {1, 3, 5}, {4, 5}, {2, 3, 4}.
1 4
3
2 5
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