1. SIMULATION MODELING AND ANALYSIS WITH ARENA
T. Altiok and B. Melamed
Chapter 7
Input Analysis
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2. 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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3. Altiok / Melamed Simulation Modeling and Analysis with Arena
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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4. Altiok / Melamed Simulation Modeling and Analysis with Arena
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
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5. Altiok / Melamed Simulation Modeling and Analysis with Arena
Data Analysis Example
Data Analysis of the repair time data produced the histogram and summary statistics shown below
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6. Modeling Time Series Data
Independent observations are modeled as a renewal time series, namely, a sequence of iid random variables. In this case, the analyst’s 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 analyst’s 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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7. 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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8. Altiok / Melamed Simulation Modeling and Analysis with Arena
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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9. 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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10. Altiok / Melamed Simulation Modeling and Analysis with Arena
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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11. The Arena Input Analyzer
The Arena Input Analyzer is a tool that fits a distribution to sample data.
Arena-supported distributions and their parameters
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
Johnson
JOHN
G, D, L, X
Log Normal
LOGN
LogMean, LogStdDev
Poisson
POIS
Weibull
WEIB
Continuous
CONT
P1, V1, …
Discrete
DISC
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12. Best-fit uniform distribution for the repair time data
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13. Best-fit beta distribution for the repair time data
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14. Best-fit gamma distribution for a sample of lead time data
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15. Fit All Summary for a sample of lead time data
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16. 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 exceeds the 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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17. Altiok / Melamed Simulation Modeling and Analysis with Arena
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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18. 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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19. 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
Interval
Number of
Observations
Relative
Frequency
Theoretical
Probability 1
[10,12) 13
0.13
0.10 2
[12,14) 9
0.09 3
[14.16) 8
0.08 4
[16,18) 5
[18,20) 12
0.12 6
[20,22) 7
[22,24)
[24,26) 10
[26,28)
[28,30)
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20. 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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21. 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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22. 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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23. Multi-Modal Distribution Example
Consider a sample of observations such that
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
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