Analysis of Simulation Input

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Analysis of Simulation Input

• .

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Simulation Machine

• Simulation can be considered as an Engine with
  input and output as follows

Simulation Engine
Output
Input
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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.

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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.

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Empirical Distributions

• Consider the following 30-data numerical example

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Example Continue

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

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Disadvantages of Empirical distribution

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

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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.

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Summary Statistics

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

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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.

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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

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Hypothesizing a Theoretical Distribution

• Use the Summary statistics to hypothesize a
  family of distributions.

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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.

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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

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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.