5.3.5 Gamma Distribution
The gamma distribution is used in reliability analysis for cases
where partial failures can exist, i.e., when a given number of partial
failures must occur before an item fails (e.g., redundant systems) or the time
to second failure when the time to failure is exponentially distributed. The
failure density function is


for t > 0, 
(5.39) 


a >
0, l > 0


where:


(5.40) 

m = mean of data a
= standard deviation 

and l is the failure rate (complete failure) and a is the number of partial
failures for complete failure or events to generate a failure. G(a) is the gamma
function:


(5.41) 
which can be evaluated by means of standard tables (See Table
5.33).
When (a1) is a positive integer, G(a) = (a1)!, which is usually the
case for most reliability analysis, e.g., partial failure situation. For this
case the failure density function is


(5.42) 
which, for the case of
a = 1 becomes the
exponential density function, previously described.
If a is an integer, it can be shown by integration by parts
that


(5.44) 

(5.45) 

(5.46) 
The gamma distribution can also be used to describe an
increasing or decreasing hazard (failure) rate. When a > 1, h(t) increases; when
a < 1, h(t)
decreases. This is shown in Figure
5.31.