Analysis of Simulation Input

1. Analysis of Simulation Input .

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

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

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

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

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

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

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

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

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

11. Example Cont. Use Data Analysis of Microsoft Excel • Mean 5.654198 • Median 5.486928 • Standard Deviation 0.910188 • Skewness 0.173392 • Range 3.475434 • Minimum 4.132489 • Maximum 7.607923 The given summary statistics suggest a Normal Distribution

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

13. MLE • Use the Maximum Likelihood Estimator (MLE) to estimate the parameters involved with the hypothesized distribution. • Suppose that q is the parameter involve in the distribution then construct • Let L(q) = fq(X1)fq (X2) . . . fq(Xn) • Find q that maximize L(q) to be the required parameter. • Example: the exponential distribution.

14. Goodness of Fit (Chi Square method) • Determine how well the given distribution is representing the data. • Divide the range of the fitted distribution into k intervals [a0, a1), [a1, a2), … [ak-1, ak] Let Nj = the number of data that belong to [aj-1, aj) • Compute the expected proportion of the data that fall in the jth interval using the fitted distribution call them pj • Compute the Chi-square

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