Consider a simple manufacturing operation whose completion
time has a mean equal to 10 minutes. We ask, what is the probability
that the completion time will not exceed 11 minutes? We first
investigate this question assuming the time has an Exponential
(a special case of the gamma), then we divide the operation
into several steps, each with an exponential distribution.
When the operation is performed in one step, we use the exponential
distribution with =1/10.
The add-in computes the probability as 0.667.
Now assume that we can divide the operation into two steps
such that each step has a mean time for completion of 5 minutes
and the steps are done in sequence. In this situation, the
total completion time has a gamma (or Erlang) distribution
with r = 2 and =
Continuing in this fashion for different values
of r with = r /
10, gives the results below. As r increases, the distribution
has less variance and the probability of completing the operation
in 11 minutes increases.