Selecting Input Probability Distribution

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 RVs 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 (klt30) 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 npj represents 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 l2.5
• Since k 5, we have pi0.2
• 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
Probability, p Probability, p Probability, p Probability, p Probability, p Degrees of Freedom
0.001 0.01 0.05 0.95 0.99  
10.83 6.64 3.84 0.004 0.000 1
13.82 9.21 5.99 0.103 0.020 2
16.27 11.35 7.82 0.352 0.115 3
18.47 13.28 9.49 0.711 0.297 4
20.52 15.09 11.07 1.145 0.554 5
22.46 16.81 12.59 1.635 0.872 6
24.32 18.48 14.07 2.167 1.239 7
26.13 20.09 15.51 2.733 1.646 8