Analysis of Simulation Input

.

Simulation Machine Simulation can be considered as an Engine

with input and output as follows:

Simulation EngineInput 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

5.076808 5.050842 6.3984924.895876 5.300643 4.6154946.77878 6.236305 6.091197

6.909572 6.829625 6.1210486.474918 4.524959 4.9275477.607923 4.913438 6.6516876.699065 4.965261 5.9685936.019929 5.505035 4.1475875.249301 5.170052 5.468824.653011 4.132489 6.241657

Example Continue

We might take these data and construct an empirical distribution by developing a histogram

0

2

4

6

8

4 5 6 7 8 More

Frequency

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 Mean5.654198 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 is the parameter involve in the distribution then construct

Let L(f(X1)f(X2)f(Xn) Find that maximize L() 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. 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)

2. Compute the expected proportion of the data that fall in the jth interval using the fitted distribution call them pj

3. Compute the Chi-square

k

j j

jj

np

npN

1

22 )(

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-)100%.

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