Page 260 - DMTH404_STATISTICS

Statistics

Notes Structure function: Assume that whether the system as a whole is functioning is completely
determined by the state vector x. Define

 1, if the system isfunctioning when the state vector is x
 x
( )  
 0, if the system has failed when the state vector is x
The function f(x) is called the structure function of the system.

The Series System: There is only one minimal path set, namely the entire system.
The Parallel System: There are n minimal path sets, namely the sets consisting of one component.

 n 
The k-out-of-n System: There are   minimal path sets, namely all of the sets consisting of
k
 
exactly k components.

17.12 Self Assessment

1. A series system functions if and only if all of its n components are functioning:

...
1 2 3 n
Its structure function is given by ...................
(a) k-out-of-n system (b) stated conditions

n
x
(c)  ( )   x i (d) partial order

i 1
2. A ................... functions if and only if at least k of its n components are functioning: Its
structure function is given by

 n
  1, if  x  k
i

 x i 1
( )  
n
 0, if  x  k.
  i 1 i

(a) k-out-of-n system (b) stated conditions
n
x
(c)  ( )   x i (d) partial order
i 1

3. A ................... is defined on the set of state vectors as follows. Let x and y be two state
vectors. We define
x  y if x  y , i =1, …, n.
i i
(a) k-out-of-n system (b) stated conditions
n
x
(c)  ( )   x i (d) partial order

i 1

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