Page 259 - DMTH404_STATISTICS

Unit 17: System Reliability

 A state vector x is call a path vector if (x) = 1. Notes
 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
 For a continuous distribution F with density f, the failure (or hazard) rate function of F, l(t), is
given by

f(t)
 (t)  .
F(t)
 If the lifetime of a component has distribution function F, then l(t) is the conditional
probability that the component of age t will fail.

 The exponential distribution with parameters l > 0 has distribution function

G(t) 1 e  ( t) , t  0.


Its failure rate function is given by
 e  t
 (t)   . 
e  t
It is considered both IFR and DFR.
The Weibull distribution with parameters  > 0,  > 0 has distribution function




( t)

G(t) 1 e , t  0.

Its failure rate function is given by

 (t)   ( t)  1
It is IFR if a  1 and DFR if 0 < a  1.
The gamma distribution with parameters  > 0,  > 0 has density function
 e  t ( t)  1  e  t ( t)  1


g(t)   , t  0.

 ( )   x  1
 e x dx
0
Its failure rate function is given by
1  x  x   1
  e   1   dx.
 (t)  t 
0
It is IFR if a  1 and DFR if 0 < a  1.

17.11 Keywords

State vector: Consider a system comprised on n components, where each component is either
functioning or has failed. Define
th
  1, if the i component isfunctioning
x   th
i
  0, if the i componenthasfailed
The vector x = {x , …, x } is called the state vector.
1 n

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