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Statistics

Notes Since these three components connect in series, the system A consisting of these components has
reliability
r
system_A
= p · p · p
controller server transformer
= .90
Suppose that we want to increase the reliability of system A. What are our options?
Suppose that we have two controllers, two servers, and two transformers.

Theorem 2. For any reliability function r and vectors,
p , p , r[1 – (1 – p )(1 – p )]  1 – [1 – r(p )][1 – r(p )].
1 2 1 2 1 2

Note 1(1 – p )(1 – p ) = (1 – (1 – p )(1 – p ), …, 1 – (1 – p )(1 – p ))
1 2 11 21 1n 2n

17.4 Bounds on Reliability

Let A , …, A be the minimal path sets of a system. Since the system will function if and only if
1 s
all the components of at least one minimal path set are functioning, then

s
r( ) P {all components i A function} 


p
 j 
 j 1 

s

  P{all components i A function}
j
j 1

s
  p .
i
j 1 i A j

This bound works well only if p is small (< 0.2) for each component.
i
Similarly, let C , …, C be the minimal cut sets of a system. Since the system will not function if
1 k
and only if all the components of at least one minimal cut set are not functioning, then
 k 
r( ) 1 P  {all components i C are not functioning}

p


 j 
 j 1 

k
 1   P{all components i C are not functioning}

j

j 1
k  


 1    (1 p ) . 
i
j 1   i j C  

This bound works well only if p is large (> 0.8) for each component.
i
Example: The Bridge Structure
The minimal cut sets are:
{1, 2}, {1, 3, 5}, {4, 5}, {2, 3, 4}.
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