Transcript Models
. Analysis of Simulation Input Simulation Machine Simulation can be considered as an Engine with input and output as follows: Input Simulation Engine Output 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. 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. Empirical Distributions Consider the following 30-data numerical example Example Continue We might take these data and construct an empirical distribution by developing a histogram Frequency More 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. 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. Summary Statistics Mean Median Standard Deviation(SD) Coefficient of Variation (SD divided by the Mean) Skewness index 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. 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 Hypothesizing a Theoretical Distribution Use the Summary statistics to hypothesize a family of distributions. 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. Goodness of Fit (Chi Square method) Determine how well the given distribution is representing the data. 1. 2. 3. 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 k ( N np ) 2 Compute the Chi-square j j 2 = j =1 np j 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 k 1,1a 2 2 Then accept the distribution with significance (1a)100%.