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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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5.7

8.1

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cumulative probability

Interpolated Empirical cdf

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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

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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

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6.14607

7.29297

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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

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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

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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