# B Probability Distribution Modeling

**Learning Objectives**

To be able model discrete distributions based on data

To be able model continuous distributions based on data

To be able to explain the key issues in pseudo-random number testing

To be able to perform basic statistical tests on uniform pseudo-random numbers

When performing a simulation study, there is no substitution for actually observing the system and collecting the data required for the modeling effort. As outlined in Section 1.7, a good simulation methodology recognizes that modeling and data collection often occurs in parallel. That is, observing the system allows conceptual modeling which allows for an understanding of the input models that are needed for the simulation. The collection of the data for the input models allow further observation of the system and further refinement of the conceptual model, including the identification of additional input models. Eventually, this cycle converges to the point where the modeler has a well defined understanding of the input data requirements. The data for the input model must be collected and modeled.

Input modeling begins with data collection, probability, statistics, and analysis. There are many methods available for collecting data, including time study analysis, work sampling, historical records, and automatically collected data. Time study and work sampling methods are covered in a standard industrial engineering curriculum. Observing the time an operator takes to perform a task via a time study results in a set of observations of the task times. Hopefully, there will be sufficient observations for applying the techniques discussed in this section.

Work sampling is useful for developing the percentage of time associated with various activities. This sort of study can be useful in identifying probabilities associated with performing tasks and for validating the output from the simulation models. Historical records and automatically collected data hold promise for allowing more data to be collected, but also pose difficulties related to the quality of the data collected. In any of the above mentioned methods, the input models will only be as good as the data and processes used to collect the data.

One especially important caveat for new simulation practitioners: do not rely on the people in the system you are modeling to correctly collect the data for you. If you do rely on them to collect the data, you must develop documents that clearly define what data is needed and how to collect the data. In addition, you should train them to collect the data using the methods that you have documented. Only through careful instruction and control of the data collection processes will you have confidence in your input modeling.

A typical input modeling process includes the following procedures:

Documenting the process being modeled: Describe the process being modeled and define the random variable to be collected. When collecting task times, you should pay careful attention to clearly defining when the task starts and when the task ends. You should also document what triggers the task.

Developing a plan for collecting the data and then collect the data: Develop a sampling plan, describe how to collect the data, perform a pilot run of your plan, and then collect the data.

Graphical and statistical analysis of the data: Using standard statistical analysis tools you should visually examine your data. This should include such plots as a histogram, a time series plot, and an auto-correlation plot. Again, using statistical analysis tools you should summarize the basic statistical properties of the data, e.g. sample average, sample variance, minimum, maximum, quartiles, etc.

Hypothesizing distributions: Using what you have learned from steps 1 - 3, you should hypothesize possible distributions for the data.

Estimating parameters: Once you have possible distributions in mind you need to estimate the parameters of those distributions so that you can analyze whether the distribution provides a good model for the data. With current software this step, as well as steps 3, 4, and 6, have been largely automated.

Checking goodness of fit for hypothesized distributions: In this step, you should assess whether or not the hypothesized probability distributions provide a good fit for the data. This should be done both graphically (e.g. histograms, P-P plots and Q-Q plots) and via statistical tests (e.g. Chi-Squared test, Kolmogorov-Smirnov Test). As part of this step you should perform some sensitivity analysis on your fitted model.

During the input modeling process and after it is completed, you should
document your process. This is important for two reasons. First, much
can be learned about a system simply by collecting and analyzing data.
Second, in order to have your simulation model accepted as useful by
decision makers, they must *believe* in the input models. Even
non-simulation savvy decision makers understand the old adage “Garbage
In = Garbage Out.” The documentation helps build credibility and allows
you to illustrate how the data was collected.

The following section provides a review of probability and statistical concepts that are useful in distribution modeling.