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Statistics

Notes Its structure function is given by

 ( )  max x .
x
i 1,...,n i

17.1.4 The k-out-of-n Structure

A k-out-of-n system 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

17.2 Order and Monotonicity
A partial order 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
Furthermore,
x < y if x  y and x < y for some i.
i i
We assume that if x  y then (x)  (y). In this case we say that the system is monotone.

17.2.1 Minimal Path Sets

 A state vector x is call a path vector if (x) = 1.

 If (y) = 0 for all y < x, then x is a minimal path vector.
 If x is a minimal path vector, then the set A = {i : x = 1} is a minimal path set.
i
Examples:

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

 n 
3. The k-out-of-n System: There are   minimal path sets, namely all of the sets consisting
k
 
of exactly k components.
Let A , …, A be the minimal path sets of a system. A system will function if and only if all the
1 s
components of at least one minimal path set are functioning, so that
 ( ) max  x .

x
i
j
i A j

This expresses the system as a parallel arrangement of series systems.

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