Simulation Modeling and Analysis

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Simulation Modeling and Analysis

• Input Modeling

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Outline

• Introduction
• Data Collection
• Matching Distributions with Data
• Parameter Estimation
• Goodness of Fit Testing
• Input Models without Data
• Multivariate and Time Series Input Models

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Introduction

• Steps in Developing Input Data Model
• Data collection from the real system
• Identification of a probability distribution
  representing the data
• Select distribution parameters
• Goodness of fit testing

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Data Collection

• Useful Suggestions
• Plan, practice, preobserve
• Analyze data as it is collected
• Combine homogeneous data sets
• Watch out for censoring
• Build scatter diagrams
• Check for autocorrelation

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Identifying the Distribution

• Construction of Histograms
• Divide range of data into equal subintervals
• Label horizontal and vertical axes appropriately
• Determine frequency occurrences within each
  subinterval
• Plot frequencies

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Physical Basis of Common Distributions

• Binomial Number of successes in n independent
  trials each of probability p .
• Negative Binomial (Geometric) Number of trials
  required to achieve k successes.
• Poisson Number of independent events occurring
  in a fixed amount of time and space (Time between
  events is Exponential).

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Physical Basis of Common Distributions - contd

• Normal Processes which are the sum of component
  processes.
• Lognormal Processes which are the product of
  component processes.
• Exponential Times between independent events
  (Number of events is Poisson).
• Gamma Many applications. Non-negative random
  variables only.

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Physical Basis of Common Distributions - contd

• Beta Many applications. Bounded random variables
  only.
• Erlang Processes which are the sum of several
  exponential component processes.
• Weibull Time to failure.
• Uniform Complete uncertainty.
• Triangular When only minimum, most likely and
  maximum values are known.

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Quantile-Quantile Plots

• If X is a RV with cdf F, the q-quantile of X is
  the value ? such that F(?) P(X lt ?) q
• Raw data xi
• Data rearranged by magnitude yj
• Then yj is an estimate of the (j-1/2)/n
  quantile of X, i.e.
• yj F-1(j-1/2)/n

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Quantile-Quantile Plots -contd

• If F is a member of an appropriate family then
  a plot of yj vs. F-1(j-1/2)/n is a straight
  line
• If F also has the appropriate parameter values
  the line has a slope 1.

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Parameter Estimation

• Once a distribution family has been determined,
  its parameters must be estimated.
• Sample Mean and Sample Standard Deviation.

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Parameter Estimation -contd

• Suggested Estimators
• Poisson ? mean
• Exponential ? 1/mean
• Uniform (on 0,b) b (n1) max(X)/n
• Normal ? mean ?2 S2

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Goodness of Fit Tests

• Test the hypothesis that a random sample of size
  n of the random variable X follows a specific
  distribution.
• Chi-Square Test (large n continuous and discrete
  distributions)
• Kolmogorov-Smirnov Test (small n continuous
  distributions only)

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Chi-Square Test

• Statistic
• ?20 ?k (Oi - Ei)2/Ei
• Follows the chi-square distribution with k-s-1
  degrees of freedom (s d.o.f. of given
  distribution)
• Here Ei n pi is the expected frequency while Oi
  is the observed frequency.

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Chi-Square Test -contd

• Steps
• Arrange the n observations into k cells
• Compute the statistic ?20 ?k (Oi - Ei)2/Ei
• Find the critical value of ?2 (Handout)
• Accept or reject the null hypothesis based on the
  comparison
• Example StatFit

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Chi-Square Test - contd

• If the test involves a discrete distribution each
  value of the RV must be in a class interval
  unless combined intervals are required.
• If the test involves a continuous distribution
  class intervals must be selected which are equal
  in probability rather than width.

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Chi-Square Test - contd

• Example Exponential distribution.
• Example Weibull distribution.
• Example Normal distribution.

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Kolmogorov-Smirnov Test

• Identify the maximum absolute difference D
  between the values of of the cdf of a random
  sample and a specified theoretical distribution.
• Compare against the critical value of D
  (Handout).
• Accept or reject H0 accordingly
• Example.

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Input Models without Data

• When hard data are not available, use
• Engineering data (specs)
• Expert opinion
• Physical and/or conventional limitations
• Information on the nature of the process
• Uniform, triangular or beta distributions
• Check sensitivity!

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Multivariate and Time-Series Input Models

• If input variables are not independent their
  relationship must be taken into consideration
  (multivariable input model).
• If input variables constitute a sequence (in
  time) of related random variables, their
  relationship must be taken into account
  (time-series input model).

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Covariance and Correlation

• Measure the linear dependence between two random
  variables X1 (mean ?1, std dev ?1) and X2 (mean
  ?2, std dev ?2)
• X1 - ?1 ?(X2 - ?2) ?
• Covariance
• cov(X1,X2) E(X1 X2) - ?1 ?2
• Correlation
• ? cov(X1,X2)/?1??2

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Multivariate Input Models

• If X1 and X2 are normally distributed and
  interrelated, they can be modeled by a bivariate
  normal distribution
• Steps
• Generate Z1 and Z2 indepedendent standard RVs
• Set X1 ?1 ?1 Z1
• Set X2 ?2 ?2(??Z1 (1-?2)1/2 Z2)

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Time-Series Input Models

• Let X1,X2,X3, be a sequence of identically
  distributed and covariance-stationary RVs. The
  lag-h correlation is
• ?h corr(Xt,Xth) ?h
• If all Xt are normal AR(1) model.
• If all Xt are exponential EAR(1) model.

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AR(1) model

• For a time series model
• Xt ? ? (Xt-1 - ?) ?t
• where
• ?t are normal with mean 0 and var ?2?
• ??????????

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AR(1) model -contd

• 1.- Generate X1 from a normal with mean ? and
  variance ?2? /(1 - ?2). Set t 2.
• 2.- Generate ?t from a normal with mean 0 and
  variance ?2? .
• 3.- Set Xt ? ? (Xt-1 - ?) ?t
• 4.- Set t t1 and go to 2.

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EAR(1) model

• For a time series model
• Xt ? Xt-1 with prob????
• Xt ? Xt-1 ?t with prob??????
• where
• ?t are exponential with mean 1/? and
• ??????????

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EAR(1) model - contd

• 1.- Generate X1 from an exponential with mean ???
  . Set t 2.
• 2.- Generate U from a uniform on 0,1. If U lt ?
  set Xt ? Xt-1 . Otherwise generate from an
  exponential with mean 1/? and set Xt ? Xt-1
  ?t
• 4.- Set t t1 and go to 2.

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