1. Analysis of Simulation Input.

2. Simulation Machine n Simulation can be considered as an Engine with input and output as follows: Simulation Engine Input Output

3. Realizing Simulation n Input Analysis: is the analysis of the random variables involved in the model such as: u The distribution of IAT u The distribution of Service Times n Simulation Engine is the way of realizing the model, this includes: u Generating Random variables involved in the model u Performing the required formulas. n Output Analysis is the study of the data that are produced by the Simulation engine.

4. Input Analysis n collect data from the field n Analyze these data n Two ways to analyze the data: u Build Empirical distribution and then sample from this distribution. u Fit the data to a theoretical distribution ( such as Normal, Exponential, etc.) See Chapter 3 of Text for more distributions.

5. Empirical Distributions n Consider the following 30-data numerical example

6. Example Continue n We might take these data and construct an empirical distribution by developing a histogram

7. Disadvantages of Empirical distribution n The empirical data may not adequately represent the true underlying population because of sampling error n The Generated RV’s are bounded n To overcome these two problems, we attempt to fit a theoretical distribution.

8. Fitting a Theoretical Distribution n Need a good background of the theoretical distributions. Histogram is useful but may not provide much insight into the nature of the distribution. n Need Summary statistics: Use Data Analysis that Microsoft Excel can do.

9. Summary Statistics n Mean n Median n Standard Deviation(SD) n Coefficient of Variation (SD divided by the Mean) n Skewness index

10. Summary Stats. Cont. n If the Mean and the Median are close to each others, and low Coefficient of Variation, we would expect a Normally distributed data. n If the Median is less than the Mean, and SD is very close to the Mean, we expect an exponential distribution. n If the skewness is very low then the data are symmetric.

11. Example Cont. Use Data Analysis of Microsoft Excel n Mean5.654198 n Median5.486928 n Standard Deviation0.910188 n Skewness0.173392 n Range3.475434 n Minimum4.132489 n Maximum7.607923 The given summary statistics suggest a Normal Distribution

12. Hypothesizing a Theoretical Distribution n Use the Summary statistics to hypothesize a family of distributions.

13. MLE n Use the Maximum Likelihood Estimator (MLE) to estimate the parameters involved with the hypothesized distribution. Suppose that  is the parameter involve in the distribution then construct Let L(  f   (X 1 )  f  (X 2 )  f  (X n ) Find  that maximize L(  ) to be the required parameter. n Example: the exponential distribution.

14. Goodness of Fit (Chi Square method) n Determine how well the given distribution is representing the data. 1. Divide the range of the fitted distribution into k intervals [a 0, a 1 ), [a 1, a 2 ), … [a k-1, a k ] Let N j = the number of data that belong to [a j-1, a j ) 2. Compute the expected proportion of the data that fall in the jth interval using the fitted distribution call them p j 3. Compute the Chi-square

15. Chi-square cont. n Note that np j represents the expected number of data that would fall in the jth interval if the fitted distribution is correct. n If Then accept the distribution with significance (1-  )100%.