Choice of Distributions

[rwilcox]

In cases where there is little or no input data to fit (i.e. collection of cycle times or repair times), how might the modeler decide which of various "textbook" distributions (e.g., triangular, normal, lognormal, Weibull, beta, gamma, Erlang) to use?

[ewilliams]

One interesting point that has served me well over the years came originally from Dr. Gene Coffman, operations research specialist at Ford. He argued strongly that the normal distribution is almost NEVER an appropriate distribution to use in a simulation (e.g., for a cycle time, time to next breakdown, time to repair, time to next arrival....). In practice, the distribution used for these situations should typically be positive (right) skewed, not symmetric. Also, the normal distribution is not bounded below, whereas all these situations are surely bounded below by zero.

[ewilliams]

Whereas the normal distribution should be strongly avoided for the distribution of downtime (repair time), good distributions to consider are those bounded below by zero, having a long upper tail, and right-skewness (mean > median). Commonly used distributions of this ilk are the gamma, the Erlang (a subtype of the gamma), the log-normal, and the Weibull.

[rwilcox]

On the one hand: "The triangular is good when you don't have much data -- just ask a person familiar with the operation what the minimum plausible, most common, and maximum plausible times to do X are." On the other hand: "The tails of the triangular are too thick." Which hand do I pick when? When I don't choose the triangular, what do I choose -- a beta distribution?

[ewilliams]

Yes, the beta is often a good refinement for what might otherwise be modeled as a triangular distribution. Its suitability as a refinement of the triangular is based largely on the fact that both distributions are bounded on both ends. A beta distribution can have much thinner tails, depending on how its shape parameters (often denoted alpha-1 and alpha-2) are chosen. The beta, like the triangular, can be positively skewed, negatively skewed, or symmetric. If the two alpha parameters are equal, it will be symmetric. If alpha-1 > alpha-2, the distribution will be left-skewed, and vice versa.