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Selecting Input Probability Distribution

Selecting Input Probability Distribution. Simulation Machine. Simulation can be considered as an Engine with input and output as follows:. Simulation Engine. Output. Input. Realizing Simulation. Input Analysis: is the analysis of the random variables involved in the model such as:

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Selecting Input Probability Distribution

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  1. Selecting Input Probability Distribution

  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 requiring 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 6 of Text for more distributions.

  5. How to select an Input Probability distribution Hypothesize a family of distributions. Estimate the parameters of the fitted distributions Determine how representative the fitted distributions are Repeat 1-3 until you get a fitted distribution foe the collected data. Otherwise go with an empirical distribution.

  6. Hypothesizing a Theoretical Distribution To Fit a Theoretical Distribution • Need a good background of the theoretical distributions (Consult your Text: Section 6.2) • Histogram may not provide much insight into the nature of the distribution. • Need Summary statistics

  7. Summary Statistics • Mean • Median • Variance s2 • Coefficient of Variation (cv = s/m) for continuous distributions • Lexis ration (t = s2/m) for discrete distributions • Skewness index

  8. 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 s is very close to the Mean (cv close to 1), we expect an exponential distribution. • If the skewness (n close to 0) is very low then the data are symmetric.

  9. Example • Consider the following data

  10. Example Cont. • Mean 5.654198 • Median 5.486928 • Standard Deviation 0.910188 • Skewness 0.173392 • Range 3.475434 • Minimum 4.132489 • Maximum 7.607923

  11. Example Continue • We might take these data and construct a histogram The given summary statistics and the histogram suggest a Normal Distribution

  12. Empirical Distribution

  13. 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.

  14. Estimation of Parameters of the fitted distributions Suppose we hypothesized a distribution, then use the Maximum Likelihood Estimator (MLE) to estimate the parameters involved with the hypothesized distribution. • Suppose that q is the only parameter involve in the distribution then construct (for example the mean 1/l in the exponential distribution) • Let L(q) = fq(X1)fq (X2) . . . fq(Xn) • Find q that maximize L(q) to be the required parameter. • Example: the exponential distribution. Do in class

  15. Determine how representative the fitted distributions are • Goodness of Fit (Chi Squared method)

  16. Goodness of Fit (Chi Square method) • Divide the range of the fitted distribution into k (k<30) 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

  17. Chi-square cont. • Note that npjrepresents the expected number of data that would fall in the jth interval if the fitted distribution is correct. • If • Where r is the number of parameters in the distribution (in Exponential dist. r = 1 which is l) • Then do not reject distribution with significance (1-a)100%.

  18. Example: • Consider the following data: 0.01, 0.07, 0.03, 0.23, 0.04, 0.10, 0.31, 0.10, 0.31, 1.17, 1.50, 0.93, 1.54, 0.19, 0.17, 0.36, 0.27, 0.46, 0.51, 0.11, 0.56, 0.72, 0.39, 0.04, 0.78 Suppose we hypothesize an exponential distribution, Use Chi-square test by dividing the range into 5 subintervals.

  19. The estimate of l=2.5 • Since k = 5, we have pi=0.2 • For the exponential distribution • Therefore

  20. Therefore chi-square = 0.4 • From the tables of chi-square • we can accept the hypothesis With significance level 5%

  21. The Chi-square table

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