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Chapter 6:Simulation Input
©Barry L. NelsonNorthwestern University
July 2017
Simulation input
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An input modeling story
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Poisson arrival process
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Service process
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View through the queue
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Univariate input models
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Inference vs. Matching
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Inference approach
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Example: Weibull vs. gamma
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Matching approach
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Where the action is
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Skewness vs. Kurtosis for common univariate distributions
Vargo et al. 2010. Journal of Quality Technology 42 (3): 1-11.
http://www.math.wm.edu/~leemis/2010jqt.Plot4.pdf
Moment matching approach
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Other things to match
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Empirical distributions
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Interpolated ecdf
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Empirical cdf and interpolated empirical cdf for datapoints X = {5.7, 2.1, 3.4, 8.1}
Interpolated Empirical cdf
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Notice that the interpolated ecdf fills in the gaps between observedvalues, but the domain is still limited to the smallest and largestobserved values (no tails).
Chart1
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3.4
5.7
8.1
X
cumulative probability
Interpolated Empirical cdf
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PalisadeFitLinks
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17.81312.10
16.66283.40.33
12.24935.70.67
25.80828.11
12.5654
13.6552
11.2216
18.0409
13.8831
12.2389
7.92051
17.5956
10.8816
6.85679
6.96787
23.7353
8.72906
7.39484
17.322
9.92081
5.90393
6.56084
18.3849
10.7006
5.21952
16.81
9.39875
5.92102
13.9892
10.5747
17.5359
21.9005
9.23805
14.8009
15.9596
13.8663
10.9201
12.0032
11.858
12.6421
5.86839
6.09994
11.7543
7.27576
5.11592
15.2684
8.26433
36.0882
13.768
24.7047
15.9146
6.95071
9.30046
13.6803
16.8774
9.10011
17.43
4.89005
7.39285
6.46003
11.0129
16.7184
12.166
34.589
6.96658
13.3778
19.8246
14.0694
11.6552
5.04598
13.3738
12.0672
12.6112
16.8387
18.9688
29.1235
20.9062
6.45961
23.0282
8.95795
8.22313
6.86053
9.52594
7.31804
9.5768
8.24021
6.50805
14.8416
6.79665
6.3143
5.74378
7.59784
11.6312
5.92281
8.8118
12.1073
7.03363
6.73072
9.28468
7.38051
7.87993
8.68973
15.2016
8.62998
15.5514
12.8942
9.75738
15.0474
5.5585
9.30431
8.37564
17.7841
7.07468
8.40482
16.5856
5.31423
18.1118
4.58817
13.106
9.04574
7.36035
5.66359
6.46512
15.6304
7.13286
8.24625
7.20159
10.5417
9.07331
9.71315
6.14607
7.29297
7.5216
5.14397
12.9351
13.2009
16.0769
8.6135
10.3957
9.25669
5.67924
6.02855
21.6863
19.491
3.3756
16.4475
8.41839
9.01562
11.1791
17.5054
15.5246
6.76007
10.81
10.0024
6.01516
14.2357
19.4222
15.9189
27.8629
10.6286
10.3045
7.38641
9.83675
8.65903
14.5418
6.03736
11.2477
5.81866
6.0948
15.9469
14.7817
28.0322
5.05782
7.05947
11.4363
13.0711
10.1045
20.3943
17.3261
8.18846
16.4599
5.49893
10.3363
7.79521
30.806
14.9481
4.47346
21.2096
14.1517
10.5207
29.0123
7.80544
14.3999
10.2949
19.166
16.8675
10.8073
14.7765
7.94037
18.6209
Sheet1
X
cumulative probability
Interpolated Empirical cdf
Properties of the interpolated ecdf
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Input modeling without data
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Nonstationary arrival processes
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Renewal arrivals
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Nonstationary arrivals
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Inverting Λ(t)
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λ(t) = 2t Λ(t) = t2 Λ-1(s) = √s
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Proof
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Thinning
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λ(t) = 6 + 4sin(t)
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Thinning with exponential base
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Exponential interarrival time
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Fitting λ(t) or Λ(t)
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Fitting Λ(t)
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Linearly interpolated Λ(t)
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Algorithm for inversion of Λ(t)
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Fitting piecewise constant λ(t)
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Piecewise constant λ(t)
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Generating random variates
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Rejection
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An approach for fX(x) a density
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Intuition
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Beta majorized by uniform
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m(x)
g(x)
fX(x)
Beta distribution example
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Key proof steps
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Rejection notes
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Particular properties
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An interactive version of this chart can be found at w
ww
.math.w
m.edu/~leem
is/chart/UDR/U
DR.html
http://www.math.wm.edu/%7Eleemis/chart/UDR/UDR.html
Generating pseudorandom numbers
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Building block: MCG
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Extending the period
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L’Ecuyer’s MRG32k3a
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Proper use of RNGs
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Chapter 6:�Simulation InputSimulation inputAn input modeling storyPoisson arrival processService processView through the queueUnivariate input modelsInference vs. MatchingInference approachExample: Weibull vs. gammaMatching approachWhere the action isSlide Number 13Moment matching approachSlide Number 15Other things to matchEmpirical distributionsInterpolated ecdfEmpirical cdf and interpolated empirical cdf for data points X = {5.7, 2.1, 3.4, 8.1}Properties of the interpolated ecdfInput modeling without dataNonstationary arrival processesRenewal arrivalsNonstationary arrivalsInverting (t)(t) = 2t (t) = t2 -1(s) = sProofThinning(t) = 6 + 4sin(t)Thinning with exponential baseExponential interarrival timeFitting (t) or (t)Fitting (t)Linearly interpolated (t) Algorithm for inversion of (t)Fitting piecewise constant (t)Piecewise constant (t)Generating random variatesRejectionAn approach for fX(x) a densityIntuitionBeta majorized by uniformBeta distribution exampleKey proof stepsRejection notesParticular propertiesSlide Number 47Generating pseudorandom numbersSlide Number 49Building block: MCGExtending the periodL’Ecuyer’s MRG32k3aProper use of RNGs