Discrete Distributions

1. Discrete Distributions • The values generated for a random variable must be from a finite distinct set of individual values. • For example, based on past observations, the annual demand for the FOT-320 is a discrete random variable that is limited to positive integer values in a certain range.

2. Modeling Discrete Distributions • In Excel, use the Vlookup function: • =vlookup(value to look up in column 1, table to look in, column to report result from) • =vlookup(Random number, table with first column containing cumulative probability distribution and second column containing corresponding values for the random variable, 2)

3. Modeling Discrete Distributions • In Crystal Ball: • =CB.Custom(spreadsheet matrix where column 1 lists the distinct set of values for the random variable and column 2 lists the associated probability of each value)

4. Continuous Distributions • The values generated for a random variable are specified from a set of uninterrupted values over a range; an infinite number of values is possible • For example, the fraction of the market that the FOT-320 will gain could be anywhere between 15% to 65%.

5. Common Continuous Distributions • Normal Distribution: A symmetrical bell shaped curve that is centered around a specified mean μ with a spread described by the standard deviation σ • Uniform Distribution: A rectangular curve where it is assumed that all values between a specified minimum and a specified maximum are equally likely to occur

6. Common Continuous Distributions • Triangular Distribution: The situation where the most likely value to occur falls between an identified minimum value and an identified maximum value, forming a triangular shaped distribution as it is assumed that values near the minimum and maximum are less likely to occur than those near the most likely value.

7. Modeling Continuous Distributions • In Excel, for the Normal distribution: • =norminv(random #, μ, σ) • Values will be simulated from a symmetrical bell-shaped curve where the most likely value is μ and 64% of the values have a chance of lying within 1 σ (in either direction) of μ

8. Model Validation • Models based on assumptions which do not accurately reflect real world behavior cannot be expected to generate meaningful results. • Errors in programming can result in nonsensical results. • Validation is generally done by having an expert review the model and the computer code for errors. • If possible, the simulation should be run using actual past data. Predictions from the simulation model should be compared with historical results.

9. Experimental Design • Policies under consideration for implementation in the real system must be identified. • For each policy under consideration by the decision maker, the simulation requires performing many runs. • Whenever possible, different policies should be compared by using the same sequence of random numbers.

10. Time Increments • In a fixed time simulation model, time periods are incremented by a fixed amount. For each time period new random numbers are used to calculate the effects on the model. (Piedmont airline problem) • In a next event simulation model, time periods are not fixed but are determined by the data values from the previous event. (Waiting Line Analysis)

11. Experimental Design Issues for Next Event Simulation • Issues such as the length of time of the simulation, the number of runs and the treatment of initial data outputs from the model must be addressed prior to collecting and analyzing output data. • Normally one is interested in results for the steady state (long run) operation of the system being modeled in a next event model. • The initial data inputs to the simulation generally represent a start-up period for the process and it may be important that the data outputs for this start-up period be neglected for predicting this long run behavior.