Altiok / Melamed Simulation Modeling and Analysis with Arena

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SIMULATION MODELING AND ANALYSIS WITH ARENA T.
Altiok and B. Melamed Chapter 7 Input
Analysis
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Input Analysis Activities

• Input Analysis activities consist of the
  following stages
• Stage 1 data collection
• Stage 2 data analysis
• Stage 3 modeling time series data
• Stage 4 goodness-of-fit testing
• Random variables with negligible variability are
  simplified
• and modeled as deterministic quantities.
• Unknown distributions are postulated to have a
  particular
• functional form that incorporates any available
  partial
• information.

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Data Collection

• To illustrate data collection activities,
  consider modeling a painting station,
  where
• jobs arrive at random, wait in the buffer until
  the sprayer is available
• having been sprayed, they leave the station
• suppose that the spray nozzle can get clogged
  an event that results in a stoppage during which
  the nozzle is cleaned or replaced.
• suppose further that the measure of interest is
  the expected job delay in the buffer.
• The data collection activity in this simple case
  would consist of the following tasks
• collection of job inter-arrival times
• collection of painting times
• collection of times between nozzle clogging
• collection of nozzle cleaning/replacement times
  

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Data Analysis

• Data Analysis deals with statistics of empirical
  data
• statistics related to moments (mean, standard
  deviation, coefficient of variation, etc.)
• statistics related to distributions (histograms)
• statistics related to temporal dependence
  (autocorrelations within an empirical time
  series, or cross-correlations among two or more
  distinct time series)
• For example, consider the sample of 100 repair
  time observations
• 12.9 27.7 13.5 13.7 22.2
• 20.9 26.6 29.1 22.4 10.7
• 30.0 27.4 18.8 25.3 15.0
• 17.0 21.7 13.7 15.5 23.2
• 11.0 27.5 22.5 27.1 25.2
• 10.3 18.0 11.5 14.1 24.0
• 10.9 27.0 24.2 25.6 22.4
• 21.0 21.3 23.1 15.8 13.2
• 22.8 25.9 22.4 13.8 16.6
• 10.8 10.3 15.1 19.0 27.9
• 20.5 19.4 10.9 24.1 10.9

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Data Analysis Example

• Data Analysis of the repair time data produced
  the histogram and summary statistics
  shown below

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Modeling Time Series Data

• Independent observations are modeled as a renewal
  time series, namely, a sequence of iid
  random variables. In this case, the analysts
  task is to merely identify (fit) a good
  distribution and its parameters to the empirical
  data.
• Arena provides built-in facilities for fitting
  distributions to empirical data.
• Dependent observations are modeled as random
  processes with temporal
• dependence. In this case, the analysts task is
  to identify (fit) a good probability law
  to empirical data. This is a far more difficult
  task
• than the previous one, and often requires
  advanced mathematics.
• Arena does not provide facilities for fitting
  dependent random processes
• An advanced method is described, however, in
  Chapter 10
• Examples
• Observed sequences of arrival times to a queue
  are often modeled as iid exponential
  inter-arrival times (i.e., Poisson processes)
• For observed sequence of times to failure and the
  corresponding repair times, the associated
  uptimes may be modeled as a Poisson process, and
  the downtimes as a renewal process or as a
  dependent process (e.g., Markov process)

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Modeling Empirical Distributions

• The simplest approach is to construct a histogram
  from the empirical data
• (sample), and then normalize it to a step pdf or
  a pmf, depending on the underlying state
  space. The obtained pdf or pmf is then declared
  to be the fitted distribution.
  The main advantage of this approach is that no
  assumptions are required on the
  functional form (shape) of the fitted
  distribution.
• The previous approach may reveal (by inspection)
  that the histogram pdf has a particular
  functional form (e.g., decreasing, bell shape,
  etc.). The analyst may then try to
  obtain a better fit, by postulating a particular
  class of distributions having that
  shape, and then proceeding to estimate
  (fit) its parameters from the sample, using such
  common techniques as the method of
  moments and the maximum likelihood estimation
  (MLE) method. This approach can be
  further generalized to multiple functional forms
  by searching for the best fit among a
  number of postulated classes of
  distributions.
• The Arena Input Analyzer provides facilities for
  both fitting approaches.

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Method of Moments

• The method of moments fits the moments of a
  candidate model to sample
• moments, using appropriate empirical statistics
  as constraints on the
• candidate model parameters.
• As an example, consider a random variable X and a
  data sample whose first
• two moments, and are estimated as
  and .
• Write the formulas for the mean and variance of a
  gamma distribution, connecting the first two
  moments of a gamma distribution with its
  parameters, and , namely
• Substitute into the above the previous estimates
• Solve the above equation to obtain

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Maximal-likelihood Estimation (MLE)

• The Maximal-likelihood Estimation (MLE) method
  postulates a particular class of
  distributions (e.g., normal, uniform,
  exponential, etc.), and then estimates
  their parameters from the sample, such that the
  resulting parameters give rise to the
  maximal likelihood (highest probability or
  density) of obtaining the sample. More
  precisely,
• Let be the postulated pdf, as a
  function of its ordinary argument, , as well
  as the unknown parameter (possibly be a
  vector of parameters, but here is assume a
  scalar for simplicity)
• Let be a sample of independent
  observations
• The MLE method estimates
  via the likelihood function
  

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MLE Method Examples

• For the exponential distribution Expo( ) with
  parameter ,
• the corresponding maximal likelihood function is
• the log-likelihood function is
• the value of that maximizes
  is obtained by
  differentiating it with respect to and
  setting the derivative to zero, that is
• solving the above in yields the maximal
  likelihood estimate
• For the uniform distribution Unif(a,b), a similar
  computation yields the MLE estimates

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The Arena Input Analyzer

• The Arena Input Analyzer is a tool that fits a
  distribution to sample data.

Distribution Arena Name Arena Parameters
Exponential EXPO Mean
Normal NORM Mean, StdDev
Triangular TRIA Min, Mode, Max
Uniform UNIF Min, Max
Erlang ERLA ExpoMean, k
Beta BETA Beta, Alpha
Gamma GAMM Beta, Alpha
Johnson JOHN G, D, L, X
Log Normal LOGN LogMean, LogStdDev
Poisson POIS Mean
Weibull WEIB Beta, Alpha
Continuous CONT P1, V1,
Discrete DISC P1, V1,
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Best-fit uniform distribution for the repair
time data
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Best-fit beta distribution for the repair time
data
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Best-fit gamma distribution for a sample of lead
time data
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Fit All Summary for a sample of lead time data
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Goodness-of-Fit Tests for Distributions

• Tests of goodness-of-fit for distributions
  determine the likelihood that an empirical
  sample is drawn from a given distribution
  
• a statistical hypothesis is formulated
• a statistic is computed from the empirical data
• the distribution of the statistic is assumed
  known under the null hypothesis, allowing the
  computation of the probability that it
  exceedsthe observed value
• rejection or acceptance decisions can be taken at
  a given significance level, but these are subject
  to Type I and Type II statistical errors
• Common goodness-of-fit tests for distributions
• Chi-Square test
• Kolmogorov-Smirnov test

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Chi-Square Test

• The Chi-Square test compares the empirical
  histogram density, constructed from sample
  data, to a candidate theoretical density
• assume that the empirical sample
  is a set of iid realizations from an
  underlying (unknown) random variable, .
• this sample is used to construct an empirical
  histogram with cells, where cell
  corresponds to the interval
• The estimator of the probability
  of cell is
• is the number of observations in cell
• it is commonly suggested to take
  for statistical reliability)

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Chi-Square Test (Cont.)

• Let be some theoretical candidate
  distribution of the random variable
  whose goodness-of-fit is to be assessed
• Compute the corresponding theoretical
  probabilities
• for continuous data we have
• where is the density of
• The Chi-square test statistic is then given by

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Chi-Square Test Example

• As an example, consider the repair time sample
  data of size N 100, given earlier, for
  which a histogram with J 10 cells was
  constructed by the Input Analyzer
• The table below displays the elements of the
  Chi-Square test for the repair data

Cell Number Cell Interval Number of Observations Relative Frequency Theoretical Probability
1 10,12) 13 0.13 0.10
2 12,14) 9 0.09 0.10
3 14.16) 8 0.08 0.10
4 16,18) 9 0.09 0.10
5 18,20) 12 0.12 0.10
6 20,22) 8 0.08 0.10
7 22,24) 13 0.13 0.10
8 24,26) 10 0.10 0.10
9 26,28) 10 0.10 0.10
10 28,30) 8 0.08 0.10
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Chi-Square Test Example (Cont.)

• The histogram of the repair data suggests that a
  uniform distribution Unif(a,b) is an acceptably
  good fit to the sample repair data
• The parameters of the uniform distribution are
  estimated as
• The Chi-Square statistic computation yields
• A Chi-Square table shows that for significance
  level and
  degrees of freedom, the critical value is
• Since the test statistic computed above is
  , we accept the null
  hypothesis that the uniform distribution
  Unif(10,30) is an acceptably good fit to the
  sample repair data

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Kolmogorov-Smirnov Test

• The Kolmogorov-Smirnov (K-S) test compares the
  empirical cdf to a theoretical
  counterpart
• while, the Chi-Square test requires a
  considerable amount of data (at least to set up
  a reasonably smooth histogram), the K-S test
  can get away with smaller samples, since it does
  not require a histogram
• The K-S test procedure proceeds as follows
• sort the sample is ascending
  order as
• constructs the empirical cdf
• construct the K-S test statistic
• The smaller is the observed value of KS,
  the better is the fit

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Multi-Modal Distributions

• A mode of a distribution is that value of its
  associated pdf or pmf at which the
  respective function attains a maximal value
• A uni-modal distribution has exactly one mode
• A multi-modal distribution is one whose
  associated pdf or pmf is of the following
  form
• It has more than one mode
• It has only one mode, but it is either not
  monotone increasing to the left of its mode, or
  not monotone decreasing to the right of its mode
• Thus, a multi-modal distribution has a pdf or pmf
  with multiple humps
• One approach to Input Analysis of multi-modal
  samples is
• Separate the sample into mutually exclusive
  uni-modal sub-samples
• Fit a separate distribution to each sub-sample
• The fitted models are then combined into a final
  model according to the relative frequency of each
  sub-sample

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Multi-Modal Distribution Example

• Consider a sample of observations such that
• observations appear to form a uni-modal
  distribution in an interval
• observations appear to form a uni-modal
  distribution in an interval
• Suppose that the theoretical distributions,
  and , are fitted separately to
  the respective sub-samples
• The combined distribution to be fitted the entire
  sample is defined by
• The distribution above is a legitimate
  distribution, formed as a probabilistic mixture
  of the two distributions, and