Rockwell Automation Arena Users Guide - Manual (Page 125)

A • S TATISTICAL D ISTRIBUTIONS
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Stati
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Erlang(
β
, k)ERLANG(ExpMean, k) or ERLA(ExpMean, k)
If X1, X2, . . . , Xk are independent, identically distributed exponential random variables,
then the sum of these k samples has an Erlang-k distribution. The mean (
β ) of each of the
component exponential distributions and the number of exponential random variables (k)
are the parameters of the distribution. The exponential mean is specified as a positive real
number, and k is specified as a positive integer.
[0, +
k β
k β 2
The Erlang distribution is used in situations in which an activity occurs in successive
phases and each phase has an exponential distribution. For large k, the Erlang approaches
the normal distribution. The Erlang distribution is often used to represent the time
required to complete a task. The Erlang distribution is a special case of the gamma
distribution in which the shape parameter,
α, is an integer (k).
Probability
Density
Function
f(x) =
β
k –
x
k 1 –
e
x β ⁄ –
k 1 – ( )!
---------------------------------
f(x) =
for x > 0
0 otherwise
Parameters
Range ∞ )
Mean
Variance
Applications