Users Guide

ExpertFit Version 8

User’s Guide

by

Averill M. Law, Ph.D. Averill M. Law & Associates, Inc.

4729 East Sunrise Drive, #462 Tucson, AZ 85718

520-795-6265 [email protected] www.averill-law.com

Copyright 1995-2011 by Averill M. Law. All rights reserved.

©

Table of Contents

1. Introduction ................................................................................................................ 1

1.1. Types of ExpertFit Analyses............................................................................... 3

1.2. Type and Amount of Data Required by ExpertFit ............................................... 6

1.3. Installation Instructions ....................................................................................... 8

1.4. ExpertFit Help System........................................................................................ 9

1.5. ExpertFit Software Architecture ........................................................................ 10

2. Data Analysis Module – Standard Mode .................................................................. 11

2.1. Confirmation of the Best Distribution ................................................................. 15

2.2. Examples .......................................................................................................... 16

Example 2.1: Customer Service Times ............................................................. 17

Example 2.2: Ship-Loading Times .................................................................... 32

Example 2.3: Repair Times for a Machine ........................................................ 38

Example 2.4: Weekly Product Sales ................................................................. 43

3. Data Analysis Module – Advanced Mode ................................................................. 50

3.1. Example ............................................................................................................ 56

Example 3.1: Testing the Homogeneity of Two Data Sets ............................... 57

4. Task-Time Models Module ....................................................................................... 62

4.1. Organization and Options ................................................................................. 63

4.2. Example ............................................................................................................ 65

Example 4.1: Modeling a Task Time................................................................. 66

5. Machine-Breakdown Models Module ....................................................................... 71

5.1. Organization and Options ................................................................................. 76

5.2. Examples .......................................................................................................... 78

Example 5.1: Modeling Machine Downtimes in the Absence of Data ............... 79

Example 5.2: Continuation of Previous Example .............................................. 83

6. Distribution Viewer ................................................................................................... 87

7. Batch Mode .............................................................................................................. 88

References .................................................................................................................... 89

Appendix A: Distributions Included in ExpertFit ........................................................... 90

Index ........................................................................................................................... 133

1

1. Introduction

ExpertFit allows you to determine automatically and accurately which probability

distribution best represents a data set. In many cases a complete analysis can be done

in less than 5 minutes. A secondary goal is to provide simulation analysts with

assistance in modeling a source of randomness (e.g., a service time) in the absence of

data.

ExpertFit is extensively used by analysts performing discrete-event simulation

studies of real-world systems in application areas such as defense, manufacturing,

transportation, healthcare, call centers, and communications networks. For these

users, ExpertFit will take the selected distribution and put it into the proper format for

direct input to a large number of different simulation-software products. ExpertFit is

also used for data analysis in such diverse disciplines as actuarial science, agriculture,

chemistry, economics, environmental analysis, finance, forestry, hydrology, medicine,

meteorology, mining, physics, psychology, reliability engineering, and risk analysis.

ExpertFit is the result of 33 years of statistical research.

When determining what distribution best fits a data set, there are two modes of

operation (see the Mode pull-down menu): Standard and Advanced. Standard Mode

(the default) is sufficient for 95 percent of all analyses and is easier to use. It focuses

the user on those features that are the most important at a particular point in an

analysis. Advanced Mode contains a large number of additional features for the

sophisticated user. A user can switch from one mode to another at any time. There

are also two levels of precision when fitting distributions (see the Precision pull-down

menu): Normal and High. Normal Precision provides good estimates for many data

sets of the parameters of a distribution and has a small execution time. High

Precision (the default) provides better parameter estimates for most data sets, but it

may have a large execution time for data sets containing many observations.

ExpertFit has extensive context-dependent online help for every options and

results screen, and there is also a Feature Index. There is a glossary of key terms and

also tutorials on a number of general topics such as the available probability

distributions. A set of data and all corresponding analyses performed by ExpertFit can

2

be stored in a Project for later reuse. All ExpertFit results (i.e., charts and tables) can

be printed or copied to the Windows Clipboard for use in other applications (e.g.,

Microsoft Word or Excel).

3

1.1. Types of ExpertFit Analyses

ExpertFit can perform the three main types of analyses given in Table 1.1.

Table 1.1. Main types of ExpertFit analyses.

Module Description

Data Analysis Used to determine what probability distribution best

represents a data set. You can either have ExpertFit

determine the best distribution automatically or specify

the distributions for consideration manually. See

Chapters 2 and 3 for further discussion and examples.

Task-Time Models Used to specify a probability distribution for a task time

when no data are available. Based on subjective

estimates of the minimum task time, the most-likely

task time (the mode), and, say, the 90th percentile of

the task time, ExpertFit specifies a Weibull, lognormal,

or triangular distribution as a model for the task time.

(For a triangular distribution, it is possible to use the

maximum task time instead of the 90th percentile.)

See Chapter 4 for further discussion and examples.

Machine-Breakdown Models

Used to model the random downtimes of a machine

when no downtime data are available. Based on

subjective estimates of such parameters as machine

efficiency (e.g., 0.90) and mean downtime, ExpertFit

specifies a busy-time distribution and a downtime

distribution. See Chapter 5 for further discussion and

examples.

4

The probability distributions available in ExpertFit are given in Table 1.2.

Table 1.2. Probability distributions available in ExpertFit.

Continuous

Discrete

beta Cauchy chi-square Erlang error exponential exponential power extreme value (2 types) F gamma inverse Gaussian inverted Weibull Johnson SB

Johnson SU Laplace logistic

log-Laplace log-logistic lognormal normal Pareto Pearson type 5 Pearson type 6 power function random walk Rayleigh Student’s t triangular uniform Wald Weibull

Bernoulli binomial geometric hypergeometric logarithmic series negative binomial Poisson uniform

If you are using the Simulation Professional or University versions of ExpertFit,

then the probability distributions resulting from an ExpertFit analysis are placed in the

proper format for the discrete-event simulation-software products given in Table 1.3

and, also, for Crystal Ball and @Risk, which are Monte Carlo simulation add-ins for

spreadsheets.

Table 1.3. Discrete-event simulation-software products supported by ExpertFit.

AnyLogic Arena AutoMod AutoSched AP ExtendSim Flexsim

MedModel Micro Saint OPNET Modeler ProcessModel ProModel ServiceModel

Simio SIMPROCESS SIMUL8 SLX WITNESS

5

If you are using a single-vendor version of ExpertFit, then the resulting probability

distributions are put into the format for that vendor’s software. The Analyst versions of

ExpertFit support Crystal Ball and @RISK.

6

1.2. Type and Amount of Data Required by ExpertFit

ExpertFit requires that your data set be in ASCII format and contain between 10

and 100,000 observations (larger data sets are truncated with an indication given).

There may be one or more data values per line. In the latter case, observations should

be separated by blanks and, if desired, by commas. In general, data files should follow

the simple ASCII file format created by editors like Notepad.

You can copy data from an active spreadsheet (e.g., Excel) to the Clipboard and

then paste it into ExpertFit (see the tutorial in general Help).

If all of the data values in the file are integers in the range –214,748,345 to

214,748,345, then the sample will be considered to be integer; otherwise, it will be

taken to be real. Real values must have magnitudes between -1.0E+99 and 1.0E+99.

If all of the data values are integers, then you will be asked whether the data should be

considered to be real valued.

We recommend that the following ideas be used in collecting or analyzing a data

set:

1. If, at all possible, collect at least 100 observations on the random phenomenon of

interest, with 200 observations providing more ability to discriminate between two

distributions. In general, the benefit from increasing the sample size from 200 to

300 will be less than that provided by increasing the sample size from 100 to 200,

etc.

2. If you are collecting observations on a continuous random variable (e.g., a service

time), then the data values should have enough resolution so that the sample will

have a “large” number of distinct values. Otherwise, it will be difficult, in general, to

find a continuous distribution that provides a good representation.

3. If the available data values are integer, then you may want to convert them to real

numbers. ExpertFit contains many more continuous distributions than discrete

distributions.

4. You should understand the process that produced the data, rather than treating the

observations as just abstract numbers. For example, suppose your data set

contains a few extremely large observations – these are called outliers. If you don't

7

understand the problem context, then it will be difficult to know whether these large

observations are really legitimate or, perhaps, the result of measuring or recording

errors.

5. If you have collected times of arrival of “customers,” then these can be converted to

interarrival times using the ExpertFit transformation “DIFF.”

8

1.3. Installation Instructions

ExpertFit is shipped on a CD and the security key (dongle) must be attached to

your computer to run the software. To install ExpertFit, insert the enclosed CD into the

corresponding drive. The Setup program should start within a few seconds. If not use

START, RUN to invoke SETUP.EXE on the CD. When the Setup program finishes, a

Windows program group will be created to hold an icon for ExpertFit and for reading a

README.WRI file. (This file contains any available notifications and also clarifications

for this User’s Guide.) Click on the icon to start ExpertFit.

ExpertFit saves information on the user organization in the security key. If this

information has not been set before the software was shipped to you, then on the first

execution you will be prompted for a permanent user name.

9

1.4. ExpertFit Help System

There are two main types of help available in ExpertFit: context-dependent and

general. Context-dependent help is available for every options and results screen, and

is accessed by clicking on the displayed Help button.

General help is accessed by clicking on the Help pull-down menu in the Menu

Bar at the top of the screen. This menu contains the following entries:

• Contents

• Context-Dependent

• Glossary

• Search

• Tutorials

• User’s Guide

• About ExpertFit

The Contents entry includes the following categories:

• Introduction to ExpertFit

• Types of ExpertFit Analyses

• Standard Mode Versus Advanced Mode

• Normal Precision Versus High Precision

• ExpertFit Software Architecture

• Tutorials

• Glossary

• Additional Documentation

• Customer Support

10

1.5. ExpertFit Software Architecture

ExpertFit uses the concept of a Project, which is a file containing one or more

analysis items. In the case of a simulation study, a Project could contain several items

corresponding to different data sets and their corresponding ExpertFit analyses, and

other items corresponding to Task-Time Models or Machine-Breakdown Models. A

Project allows you to save the results of an ExpertFit analysis for future reuse.

When an ExpertFit Project is created or read from a file, a Project Window is

created to represent it. This window acts as a directory for the analysis items contained

in the Project. Buttons at the bottom of the window allow you to create new elements,

to change an element’s name or description, to begin a new (or return to an existing)

analysis, and to delete an element.

11

2. Data Analysis Module – Standard Mode

There are two modes of operation (see the Mode pull-down menu) for the Data

Analysis module: Standard and Advanced. Standard Mode (the default), which is

described in this chapter, is sufficient for 95 percent of all analyses and is easier to use.

It focuses the user on those features that are the most important at a particular point in

an analysis. Advanced Mode, which is discussed in Chapter 3, contains a large

number of additional features for the sophisticated user. A user can switch from one

mode to another at any time.

The use of the Data Analysis module to determine what probability distribution

best represents a data set is based on the sequential application of the four tabs shown

in Table 2.1.

Table 2.1. Tabs for the Data Analysis module.

Tab Overall Purpose

Data Used to read in a data set from a file, to enter a data set at the keyboard, or to paste in a sample from the Clipboard

Models Used to “fit” probability distributions to a data set

Comparisons Used to compare the fitted distributions to the data set

Applications Used to determine or display characteristics of a distribution (e.g., its moments or density function) or to represent the distribution in a simulation-software product

The options available in these four tabs are shown in Tables 2.2 through 2.5,

respectively.

12

Table 2.2. Options for the Data tab.

Option Specific Purpose

Enter Data Read Data from File

Enter/Edit Data Values (enter values at the keyboard, delete values, paste in values from Clipboard, copy all values to Clipboard for export)

Delete Data Set

View/Modify Data View Data (either sorted or unsorted)

Create a Subset

Perform a Transformation

Data Summary Summary statistics for the data set

Histogram Histogram Plot

Frequency Table

Assess Independence Scatter Plot

Lag-Correlation Plot

Lag-Correlation Table

Table 2.3. Options for the Models tab.


Automated Fitting

Automated fitting, ranking, and evaluation of models based on a default characterization of the range of the random variable

Fit Individual Models Manual fitting of specific distributions – parameter values can be estimated from data or user specified

View/Delete Models Show Model Parameters

Delete Models

13

Table 2.4. Options for the Comparisons tab.


Graphical Comparisons Density-Histogram Plot (real data only)

Frequency-Comparison Plot

Distribution-Function-Differences Plot

P-P Plot

Goodness-of-Fit Tests Anderson-Darling Test (real data only)

Kolmogorov-Smirnov Test (real data only)

Chi-Square Test

Evaluate a Model Evaluation Report


Table 2.5. Options for the Applications tab.


Use a Specified Distribution (Model)

Characteristics (density function plot, moments, etc.) of a distribution

Representation of a distribution in a simulation-software product

Use an Empirical Distribution

Characteristics [distribution function plot, percentiles (only for real data), etc.] of an empirical distribution

Representation of an empirical distribution in a simulation-software product

14

Although there are different ways that these four tabs could be used to

determine the best distribution for a data set, the following are the explicit steps that we

recommend for real data (see Example 2.4 for integer data):

1. Obtain a data set using the Data tab – see Section 1.2 for a discussion of the type

and amount of data required.

2. View the resulting Data-Summary Table (at the Data tab) – provides information on

the center, shape, and range of the true density (mass) function.

3. Make a histogram of your data (used in Step 5) using the Data tab – see the

Constructing a Histogram from Your Data tutorial in the online help.

4. Select the distribution that is the best representation for your data using the

Automated Fitting option at the Models tab.

5. Confirm using the Comparisons tab that the best distribution as determined by

ExpertFit is, in fact, satisfactory in an absolute sense – See Section 2.1 for

recommendations.

6. If you are doing simulation modeling, then either represent the best-fitting

distribution (if good in an absolute sense) or an empirical distribution based on your

data (if the best distribution is not satisfactory) in your simulation software using the

Applications tab.

Four examples of the use of the Data Analysis module are given in Section 2.2.

15

2.1. Confirmation of the Best Distribution

Before actually using the best model, we recommend that some amount of

confirmation of this model be done using the options in the Comparisons tab. We

suggest that a Density-Histogram Plot and/or a Frequency-Comparison Plot be

made with an appropriately constructed histogram. When the maximum of the density

function of the best model is “far” from x = 0, then the former plot is probably preferable.

If, on the other hand, the maximum occurs “close” to x = 0, then the latter plot will often

be more useful. This is because it may be difficult in this case to determine from a

histogram whether the true underlying density function strictly decreases as x increases

(similar to an exponential density) or whether the density function has its mode (x-value

where the maximum occurs) close to x = 0 (similar to a lognormal density with shape

parameter = 3/2). Care must be taken when using these plots since the choice of the

histogram intervals is somewhat subjective.

We also recommend that a Distribution-Function-Differences Plot and a

P-P Plot be used to confirm the quality of the best model.

Finally, one might perform the Anderson-Darling Test, the Kolmogorov-

Smirnov Test, and/or the Chi-Square Test in order to get a formal evaluation of the

best-fitting model.

16

2.2. Examples

We now present four examples of the use of the Data Analysis module,

following the six-step approach outlined at the beginning of this chapter. For the first

example, we give the ExpertFit commands necessary to accomplish a particular part of

the analysis and then the actual results of the analysis. For the other examples, we

only discuss the results.

Examples 2.1 through 2.3 use High Precision (the default) – see the “Overview”

in the Precision pull-down menu. It doesn’t matter which option you use for the integer

data of Example 2.4, since the two options are identical in this case.

17

Example 2.1: Customer Service Times

A: The Data-Summary Table for this set of n = 450 service times (read in the Data

tab) is given in Table 2.6. The positive value of the sample skewness indicates that the

underlying distribution of the data is skewed to the right (i.e., it has a longer right tail

than left tail). This is supported by the sample mean being larger than the sample

median.

Steps for Action A: At window: Do:

Project 1 Click on New.

Project-Element Editing Select Fit distributions to data.

In the Project-Element Name edit box, enter Example 2.1.

Click on OK.

Project 1 Click on Analyze.

Data tab Click on Enter Data.

Enter-Data Options Click on Apply.

Open Select Example 21 in the ExpertFit folder. Examine the Data-Summary Table. Data-Summary Table Click on Done.

18

Table 2.6. Data summary for the service-time data.

Data Characteristic Value

Source file Exam21 Observation type Real valued Number of observations 450 Minimum observation 0.06438 Maximum observation 3.11115 Mean 1.18250 Median 1.15192 Variance 0.32394 Coefficient of variation 0.48131 Skewness 0.48796

19

B: In Figure 2.1 we present the default ExpertFit histogram for the service-time data.

Note that the histogram is quite “ragged,” since the interval width is too small. Using a

trial-and-error approach discussed in the Constructing a Histogram from Your Data

tutorial, we determined that a better histogram is obtained by using an interval width of

0.25. The improved “smooth” histogram is shown in Figure 2.2. In general we

recommend that you construct your own histogram rather than rely on the ExpertFit

default. There is no definitive prescription for choosing histogram intervals!

Note that the histogram interval width can also be changed by using the

two buttons with arrowheads at the top of the histogram screen. The left (right)

Steps for Action B: At window: Do:

Data tab Click on Histogram.

Histogram Options Click on Apply. Examine the Histogram Plot. Click on Done.

The following shows how to change the lower endpoint of the first interval from 0.064 to 0: Histogram Options Click on the equal-sign ("=") button next to 0.06400.

Change the value to 0.0 in the edit box.

Click on Apply.

Perform similar actions to change the interval width to 0.25 and the number of intervals to 13. Histogram Options Click on Apply. Histogram Plot Examine the Histogram Plot. Click on Done. Histogram Options Click on Done.

20

23 intervals of w idth 0.133

0.00

0.02

0.04

0.06

0.08

0.10

0.13 0.53 0.93 1.33 1.73 2.13 2.52 2.92

Pro

port

ion

HistogramHistogram

Interval Midpoint

Figure 2.1. Default ExpertFit histogram of the service-time data.

21


0.00

0.03

0.07

0.10

0.13

0.17

0.13 0.63 1.13 1.63 2.13 2.63 3.13

Pro

port

ion

HistogramHistogram

Interval Midpoint

Figure 2.2. Histogram of the service-time data with an interval width of 0.25.

22

button decreases (increases) the interval width by 5 percent, and can be applied

repeatedly.

23

C: We begin the actual process of finding a distribution that is a good representation

for our data by selecting the Automated Fitting option at the Models tab. Based on

certain heuristics, ExpertFit determined that the “best” representation for the data is

provided by a Weibull distribution (see Table 2.7) with location, scale, and shape

parameters of 0, 1.334, and 2.183, respectively. This best model received a Relative

Score of 100.00 and its Absolute Evaluation message is “Good,” indicating no reason

for concern. (See the context-dependent online help for a discussion of the terms in

boldface.) Furthermore, the model mean and the sample mean are almost identical.

Note that the third-best fitting model is a Rayleigh distribution with an estimated

location parameter (denoted by “E”) of 0.062. (If we were to click on View/Delete

Models at the Models tab, we would see that the normal distribution was not

automatically fit to our non-negative service-time data. This is because the normal

distribution can take on negative values. However, the normal distribution could, if

desired, be fit to our data using the Fit Individual Models option at the Models tab.)

Steps for Action C: At window: Do:

Data tab Click on the Models tab.

Models tab Click on Automated Fitting. Examine Automated-Fitting Results. Automated-Fitting Results Click on Done.

24

Table 2.7. Evaluation of the candidate models.

Relative Evaluation of Candidate Models

Relative

Model Score Parameters

1 - Weibull 100.00 Location Scale Shape

0.00000 1.33367 2.18304

2 - Beta 95.00 Lower endpoint Upper endpoint Shape #1 Shape #2

6.46210 e -4

3.72935 2.53832 5.47937

3 - Rayleigh(E) 88.75 Location Scale

0.06196 1.25652

21 models are defined with scores between 3.75 and 100.00

Absolute Evaluation of Model 1 - Weibull

Evaluation: Good

Suggestion: Additional evaluations using Comparisons Tab might be informative. See Help for more information.

Additional Information about Model 1 - Weibull

"Error" in the model mean

relative to the sample mean 0.00139 = 0.12%

25

D: We now do some additional confirmation of the best-fitting Weibull distribution

using the Comparison tab, as suggested by the latter part of the Absolute Evaluation

message. The Density-Histogram Plot based on the final histogram is shown in

Figure 2.3. The closeness of the density function to the histogram visually confirms the

quality of the Weibull representation. Note that we could have simultaneously

plotted the density functions of several distributions in Figure 2.3.

Steps for Action D: At window: Do:

Comparisons tab Click on Graphical Comparisons. Graphical-Comparisons Options Select Density-Histogram Plot.

Click on Apply. Examine Density-Histogram Plot. Density-Histogram Plot Click on Done.

26

13 intervals of w idth 0.25 1 - Weibull

0.00

0.03

0.07

0.10

0.14

0.17

0.13 0.63 1.13 1.63 2.13 2.63 3.13

Density

/Pro

port

ion

Density-Histogram PlotDensity-Histogram Plot

Interval Midpoint

Figure 2.3. Density-Histogram Plot for the fitted Weibull distribution and the service-time data.

27

E: We present a Distribution-Function-Differences Plot for the Weibull distribution

in Figure 2.4. The plot shows the differences between the Weibull distribution function

and the sample distribution function over the range of the data. [The sample

distribution function, which is an estimate of the true underlying distribution function of

the data, is defined at a particular value of x as (approximately) the proportion of

observations in the sample that is less than or equal to x.] Since the vertical differences

in the plot are close to 0, this is further indication that the Weibull distribution is a good

model for the data.

Steps for Action E:

At window: Do:

Graphical-Comparisons Options Select Distribution-Function-Differences Plot.

Click on Apply. Examine Distribution-Function-Differences Plot. Distribution-Function- Differences Plot Click on Done. Graphical-Comparisons Options Click on Done.

28

Use caution if plot crosses line 1 - Weibull (mean diff. = 0.00501)

0.00

0.07

0.13

0.20

-0.07

-0.13

-0.200.06 0.50 0.93 1.37 1.81 2.24 2.68 3.11

Diff

ere

nce (

Pro

port

ion)

Distribution-Function-Differences PlotDistribution-Function-Differences Plot

x

Figure 2.4. Distribution-Function-Differences Plot for the fitted Weibull distribution and the service-time data.

29

F: We conclude the confirmation process by performing an Anderson-Darling Test

(the most-powerful test available in ExpertFit) to see formally whether our data could

have been generated from the specified Weibull distribution. (You may want to read

the discussion of goodness-of-fit tests in the Goodness-of-Fit Tests and Their

Interpretation tutorial in the software before proceeding.) We will perform the test at a

level (alpha) of 0.05. Since the Anderson-Darling statistic, 0.205, is less than critical

value, 0.750, we do not reject the Weibull distribution. You should keep in mind that

failure to reject by this test does not necessarily mean that the Weibull distribution is

exactly the distribution that produced the data; this test tends to have low power for

small to moderate sample sizes. (We also performed the Kolmogorov-Smirnov Test

and Chi-Square Test and they also did not reject the Weibull distribution.)

In summary, there is no reason to believe based on the above heuristics and

tests that the Weibull distribution does not provide a good model for the service-time

data.

Steps for Action F: At window: Do:

Comparisons tab Click on Goodness-of-Fit Tests.

Options for Goodness-of-Fit Tests Select Anderson-Darling Test.

Click on Apply. Examine Anderson-Darling Test. Anderson-Darling Test Click on Done. Options for Goodness-of-Fit Tests Click on Done.

30

G: If you are using ExpertFit in the context of simulation modeling, we now see how to

put the selected Weibull distribution into the proper format for several different

simulation-software products using the Applications tab. In particular, the above

actions show how to represent the Weibull distribution in the software product of your

choice; the actual representations for selected products are shown in Table 2.8.

Steps for Action G: At window: Do:

Comparisons tab Click on Applications tab.

Applications tab Click on Simulation Representation in the Use a Specified Distribution (Model) section.

Simulation-Representation Options Select the simulation software of your choice.

Click on Apply. Examine Simulation-Software Representation. Simulation-Software Representation Click on Done.

Simulation-Representation Click on Done. Options

Applications tab In the File menu, select Close Data Analysis.

31

Table 2.8. Simulation-software representations for the Weibull distribution.

Software Product Representation

AnyLogic weibull(2.183039, 1.333669, 0.000000)

Arena WEIB(1.333669, 2.183039, <stream>)

AutoMod weibull 2.183039, 1.333669

ExtendSim Distribution Weibull

Scale 1.333669

Shape 2.183039

Location 0.000000

Flexsim Distribution Weibull

Location 0.000000

Scale 1.333669

Shape 2.183039

ProModel W(2.183039, 1.333669, <stream>)

Simio Random.Weibull(2.183039, 1.333669, <stream>)

WITNESS WEIBULL(2.183039, 1.333669, <stream>)

32

Example 2.2: Ship-Loading Times

This sample of n = 856 observations corresponds to loading times (in hours) for

an oil tanker. The Data-Summary Table is given in Table 2.9; the positive skewness

and the fact that the mean is larger than the median both suggest the underlying

distribution of the data has a longer right tail than left tail.

A histogram of the data with a lower endpoint of 0, an interval width of 0.1, and

23 intervals is given in Figure 2.5 – the latter two choices were obtained by trial and

error. Note that the histogram is reasonably smooth, skewed to the right, and is

definitely shifted away from the origin.

The process of finding a distribution that is a good representation for the data

once again begins by selecting Automated Fitting. The log-logistic distribution was

found by ExpertFit to provide the best representation (see Table 2.10), with a Relative

Score of 100.00. The Absolute Evaluation for the log-logistic distribution is “Good,”

which indicates that there is no reason for concern. Also, the model mean and sample

mean differ by only 0.34 percent.

Table 2.9. Data summary for the ship-loading times.



33


0.00

0.05

0.10

0.15

0.20

0.25

0.05 0.35 0.65 0.95 1.25 1.55 1.85 2.15

Pro

port

ion

HistogramHistogram

Interval Midpoint

Figure 2.5. Histogram of the ship-loading times.

34



Relative


1 – Log-Logistic 100.00 Location Scale Shape

0.00000 0.82199 8.84087

2 – Pearson Type VI 91.35 Location Scale Shape #1 Shape #2

0.00000 0.25314 99.97455 31.06366

3 – Pearson Type V 88.46 Location Scale Shape

0.00000 19.18409 23.78474


Absolute Evaluation of Model 1 – Log-Logistic

Evaluation: Good

Suggestion: Additional evaluations using Comparisons Tab might be informative.

See Help for more information.

Additional Information about Model 1 – Log-Logistic

"Error" in the model mean

relative to the sample mean 0.00290 = 0.34%

35

We continue our evaluation of the log-logistic distribution by displaying the

Density-Histogram Plot in Figure 2.6. This plot seems to indicate that the log-logistic

distribution provides a good fit for the ship-loading times.

23 intervals of w idth 0.1 1 - Log-Logistic

0.00

0.05

0.11

0.16

0.22

0.27

0.05 0.35 0.65 0.95 1.25 1.55 1.85 2.15

Density

/Pro

port

ion


Interval Midpoint

Figure 2.6. Density-Histogram Plot for the fitted log-logistic distribution and the ship-loading times.

36

The Distribution-Function-Differences Plot for the log-logistic distribution is

displayed in Figure 2.7. The small vertical differences (errors) suggest that this model

provides a good fit. The P-P Plot in Figure 2.8 also indicates a good fit, since the plot

is close to the straight line with a slope of 1 and a y-intercept of 0.

Use caution if plot crosses line 1 - Log-Logistic (mean diff. = 0.00517)

0.00

0.07

0.13

0.20

-0.07

-0.13

-0.200.37 0.63 0.89 1.14 1.40 1.66 1.92 2.18

Diff

ere

nce (

Pro

port

ion)


x

Figure 2.7. Distribution-Function-Differences Plot for the fitted log-logistic distribution and the ship-loading times.

37

Range of sample 1 - Log-Logistic (discrepancy = 0.01424)

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Model V

alu

eP-P PlotP-P Plot

Sample Value

Figure 2.8. P-P Plot for the fitted log-logistic distribution and the ship-loading times.

We performed an equal-probability chi-test with 40 intervals and a Kolmogorov-

Smirnov test each at level α = 0.05, and these tests failed to reject the log-logistic

distribution.

In summary, the log-logistic distribution appears to be a good model for the ship-

loading times. It is interesting to note that the “esoteric” log-logistic distribution provides

a better model than the well-known gamma, lognormal, and Weibull distributions.

Suppose that we want to compute a characteristic of the log-logistic distribution

such as the probability that it takes on a value less than or equal to 1. From the

Probability for an x calculation in the [Use a Specified Distribution (Model)]

Characteristics option at the Applications tab, we get 0.850.

38

Example 2.3: Repair Times for a Machine

This example discusses a data set where no model provides a good fit and

where the use of an empirical distribution is recommended.

This sample of n = 288 observations corresponds to repair times for a machine

used for manufacturing household products. Since the repair times were generally

rounded to the nearest 5 minutes, we converted the observations to real numbers to

allow a greater number of distribution choices. The Data-Summary Table is given in

Table 2.11, from which it appears that the underlying distribution of the data is skewed

to the right.

A histogram of the data with a lower endpoint for the first interval of 0, an interval

width of 11, and 17 intervals is given in Figure 2.9. It is clear that the histogram is

positively skewed and has a long right tail.

Using Automated Fitting, we find that the “best” fitting model is a Pearson type

V distribution, which has a Relative Score of 96.00. However, the Absolute

Evaluation is “Bad.” Finally, there is a significant error in the model mean relative to

the sample mean of 6.74 percent.

Table 2.11. Data summary for the repair-time data.



39

17 intervals of w idth 11

0.00

0.08

0.16

0.24

0.32

0.40

5.50 38.50 71.50 104.50 137.50 170.50

Pro

port

ion

HistogramHistogram

Interval Midpoint

Figure 2.9. Histogram of the repair-time data.

40

The poor quality of the Pearson type V representation is confirmed by the

Density-Histogram Plot in Figure 2.10. (You might also try the Frequency-

Comparison Plot.)

17 intervals of w idth 11 1 - Pearson Type V(E)

0.00

0.08

0.16

0.24

0.32

0.40

5.50 38.50 71.50 104.50 137.50 170.50

Density

/Pro

port

ion


Interval Midpoint

Figure 2.10. Density-Histogram Plot for the fitted Pearson type V distribution and the repair-time data.

41

The Distribution-Function-Differences Plot for the Pearson type V distribution

in Figure 2.11 crosses the blue-dashed rectangle, which strongly indicates that this

distribution is not a good representation for the data. Goodness-of-fit tests are not

applicable to the inverse Gaussian distribution since, for example, the location

parameter was not estimated by the method of maximum likelihood.

Use caution if plot crosses line 1 - Pearson Type V(E) (mean diff. = 0.01272)

0.00

0.07

0.13

0.20

-0.07

-0.13

-0.205.00 30.71 56.43 82.14 107.86 133.57 159.28 185.00

Diff

ere

nce (

Pro

port

ion)


x

Figure 2.11. Distribution-Function-Differences Plot for the fitted Pearson Type V distribution and the repair-time data.

42

In summary, none of the fitted continuous distributions appears to provide a

good representation for the repair-time data. (This is not surprising since there are only

32 distinct values in a sample with a range of [5, 185] – no discrete distribution works

either.) Therefore, if we are doing simulation modeling, we must resort to the use of an

empirical distribution. The empirical distribution function based on the distinct sample

values is shown in Figure 2.12. You can employ the (Use an Empirical Distribution)

Simulation Representation option at the Applications tab to put this empirical

distribution into the proper format for your simulation software. Furthermore, you can

use the Copy button to place the simulation-software representation into the Windows

Clipboard.

Empirical using 288 sample values

0.00

0.20

0.40

0.60

0.80

1.00

5.00 30.71 56.43 82.14 107.86 133.57 159.28 185.00

F(x

)

Distribution Function PlotDistribution Function Plot

x

Figure 2.12. Empirical distribution function for the repair-time data based on the unique data values.

43

Example 2.4: Weekly Product Sales

In this example we illustrate how ExpertFit can be used to analyze an integer data set.

The Data-Summary Table for n = 156 weekly sales of a product over a 3-year period

[see Law (2007, p. 325)] is given in Table 2.12. (Note that values range from 0 to 11.)

A histogram of the sales data, starting at 0 and using 12 intervals that each contain one

value, is given in Figure 2.13. Its shape is similar to the probability mass function of a

geometric distribution.

Using Automated Fitting, ExpertFit found that the geometric distribution with

ˆ( ) .=0 0 346p provides the best representation, receiving a Relative Score of 83.33 (see

Table 2.13). This score is low because the second-best model is the negative binomial

distribution with ˆ = 1s and ˆ( ) .=0 0 346p , which is the same as the above geometric

distribution. If the negative binomial distribution is deleted (see View/Delete Models at

the Models tab), then the Relative Score of the geometric distribution is 100.00. The

Absolute Evaluation for the geometric distribution is “Indeterminate,” which indicates

that additional evaluations using the Comparisons Tab are strongly recommended.


Source file Exam24 Observation type Integer valued Number of observations 156 Minimum observation 0 Maximum observation 11 Mean 1.89103 Median 1.00000 Variance 5.28482 Lexis ratio (var./mean) 2.79469 Skewness 1.68749

Table 2.12. Data summary for the sales data.

44

12 intervals of w idth 1

0.00

0.08

0.15

0.23

0.30

0.38

0 2 4 6 8 10

Pro

port

ion

HistogramHistogram

Value

Figure 2.13. Histogram of the sales data.

45



Relative


1 - Geometric 83.33 Probability 0.34590

2 - Negative Binomial 83.33 Probability Success

0.34590 1

3 - Poisson 33.33 Lambda 1.89103


Absolute Evaluation of Model 1 - Geometric

Evaluation: Indeterminate

Suggestion: Additional evaluations using Comparisons Tab are strongly recommended.

See Help for more information.

Additional Information about Model 1 - Geometric

""Error" in the model mean

relative to the sample mean 0

46

To determine whether the geometric distribution is a good representation for the

sales data, we present a Frequency-Comparison Plot for the geometric distribution in

Figure 2.14. The agreement between the histogram and the expected proportions for

the geometric distribution is good except possibly for the interval corresponding to x = 1.

12 intervals of w idth 1 1 - Geometric

0.00

0.08

0.15

0.23

0.30

0.38

0 2 4 6 8 10

Pro

port

ion

Frequency-Comparison PlotFrequency-Comparison Plot

Value

Figure 2.14. Frequency-Comparison Plot for the fitted geometric distribution and the sales data.

47

A Distribution-Function-Differences Plot and P-P Plot for the geometric

distribution are given in Figures 2.15 and 2.16, respectively. Neither of these plots

gives us any particular reason to think that the geometric distribution is not a good

representation for the sales data.

Use caution if plot crosses line 1 - Geometric (mean diff. = 0.00884)

0.00

0.07

0.13

0.20

-0.07

-0.13

-0.200 2 4 6 8 10

Diff

ere

nce (

Pro

port

ion)


x

Figure 2.15. Distribution-Function-Differences Plot for the fitted geometric distribution and the sales data.

48

Range of sample 1 - Geometric (discrepancy = 0.04460)

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

Model V

alu

e

P-P PlotP-P Plot

Sample Value

Figure 2.16. P-P Plot for the fitted geometric distribution and the sales data.

49

We conclude this example by performing an Equal-Width Chi-Square Test for

the geometric distribution, since equal-probable intervals are not available for discrete

distributions. It is recommended for discrete distributions that the intervals be chosen

so that the probabilities (expected numbers or counts) under the hypothesized model

are approximately equal for all intervals. One way to do this is to note that the mode

(most-likely value) of the geometric distribution is 0; furthermore, ˆ( ) .=0 0 346p . The

large value for the mode limits our choice of intervals and we end up with the three

intervals given in Table 2.14, where most of the calculations for the chi-square test are

also presented. (These intervals were obtained using the View/Group Cells button.)

Note that the expected count for each interval is at least 5, as is recommended. Since

the chi-square statistic value of 1.930 is less than the critical value of 5.991

corresponding to a level of 0.05 and 2 degrees of freedom, we do not reject the

geometric distribution at level 0.05.

In summary we have no reason to believe that the geometric distribution is not a

good model for our data.

Table 2.14. Intervals and results for the chi-square test.

Cell Structure Counts

Contents Upper Endpoint Sample Model (Expected) Contribution to Statistic

1: 1..1 0 59 53.96009 0.47073

2: 2..3 2 50 58.38219 1.20347

3: 4..12 infinity 47 43.65772 0.25587

50

3. Data Analysis Module – Advanced Mode

In this chapter we discuss Advanced Mode for the Data Analysis module,

which is accessed from the Mode pull-down menu at the top of the screen. Advanced

Mode contains a large number of features that are not in Standard Mode – these

features will be of interest to the sophisticated user. However, a user can switch from

one mode to another at any time.

The use of Advanced Mode for the Data Analysis module to determine what

probability distribution best represents a data set is based on the sequential application

of the four tabs shown in Table 3.1. (These tabs are similar to those in Table 2.1.)

Table 3.1. Tabs for the Data Analysis module.

Tab Overall Purpose

Data Used to read in a data set from a file, enter a data set at the keyboard, or paste in a sample from the Clipboard

Models Used to “fit” probability distributions to a data set

Comparisons Used to compare the fitted distributions to the data set

Applications Used to determine or display characteristics of a distribution (e.g., its moments or density function) or to represent the distribution in a simulation-software product

The options available in these four tabs are shown in Tables 3.2 through 3.5, respectively.

51

Table 3.2. Options for the Data tab.


Enter Data Read Data from File

Enter/Edit Data Values (enter values at the keyboard, delete values, paste in values from Clipboard, copy all values to Clipboard for export)

Delete Data Set

View/Modify Data View Data (either sorted or unsorted)

Create a Subset

Perform a Transformation

Data Summary Summary statistics for the data set

Histogram Histogram Plot

Frequency Table

Additional Data Summaries Scatter Plot

Lag-Correlations Plot

Lag-Correlations Table

Moving-Average Plot

Box Plot

Box-Plot Table

Homogeneity Tests Perform Kruskal-Wallis Test

Histogram Comparisons

Distribution Function Comparisons

Box-Plot Comparisons

Merge Selected Data Sets

52



Automated Fitting

Automatic fitting, ranking, and evaluation of models based on a default characterization of the random variable range – user may change the default range

Fit Individual Models Manual fitting of specific distributions – parameter values can be estimated from data or user specified

Fit a Class of Models Automatic fitting of all models in a particular class (i.e., non-negative continuous, bounded continuous, and unbounded continuous) – parameters are typically estimated from the data

View/Delete Models Show Model Parameters

Delete Models

53

Table 3.4. Options for the Comparisons tab.


Histogram Comparisons Density-Histogram Plot (real data only)

Frequency-Comparison Plot

Frequency-Comparison Table

Raw-Error Plot

Absolute-Error Plot

Table of Errors

Distribution Comparisons Distribution-Function-Differences Plot

Table of Errors

Distribution Function Plot

Survivor Function Plot

Probability Plots P-P Plot

Q-Q Plot (real data only)

Relative-Discrepancies Table

Goodness-of-Fit Tests Anderson-Darling Test (real data only)

Kolmogorov-Smirnov Test (real data only)

Equal-Probable Chi-Square Test

Equal-Width Chi-Square Test

Test-Statistics Comparison (real data only)

Additional Comparisons Moment-Comparison Table

Box-Plot Comparisons Plot (real data only)

Box-Plot Comparisons Percentile Table (real data only)

Likelihood-Function Table

Evaluate a Model Evaluation Report


54



Use a Specified Distribution (Model)

Characteristics (density function plot, moments, etc.) of a distribution

Representation of a distribution in a simulation-software product

Generate random values from a distribution

Use an Empirical Distribution

Characteristics [distribution function plot, percentiles (only for real data), etc.] of an empirical distribution

Representation of an empirical distribution in a simulation-software product

55

Although there are different ways that these four tabs could be used to

determine the best distribution for a data set, the following are the explicit steps that we

recommend (same as for Standard Mode):

1. Obtain a data set using the Data tab – see Section 1.2 for a discussion of the type

and amount of data required.

2. View the resulting Data-Summary Table (at the Data tab) – provides information on

the center, shape, and range of the true density (mass) function.

3. Make a histogram of your data (used in Step 5) using the Data tab – see the

Constructing a Histogram from Your Data tutorial in the online help.

4. Select the distribution that is the best representation for your data using the

Automated Fitting option at the Models tab.

5. Confirm using the Comparisons tab that the best distribution as determined by

ExpertFit is, in fact, satisfactory in an absolute sense – see Section 2.1 for

recommendations.

6. If you are doing simulation modeling, then either represent the best-fitting

distribution (if good in an absolute sense) or an empirical distribution based on your

data (if the best distribution is not satisfactory) in your simulation software using the

Applications tab.

An example of the use of Advanced Mode for the Data Analysis module is

given in Section 3.1.

56

3.1. Example

We now present an example of the use of Advanced Mode for the Data

Analysis module, following the six-step approach outlined at the beginning of this

chapter. We use Normal Precision in fitting distributions to the data.

57

Example 3.1: Testing the Homogeneity of Two or More Data Sets

It is sometimes of interest to determine whether two or more “similar” data sets

are homogeneous. If the data sets are homogeneous, they can be merged. We can

then attempt to fit a single distribution to the merged data set.

Consider processing times corresponding to two different machines from the

same vendor. The data from machine 1, Example 3.1-1, contains 910 observations

and the data from machine 2, Example 3.1-2, contains 838 observations. We would

like to determine whether these data sets are homogeneous and, thus, can be merged.

Select the data set Example 3.1-1 in the Project EXAMPLES.EFP that comes with

ExpertFit. Now select Homogeneity Tests at the Data tab (for Advanced Mode) and

data set Example 3.1-2 from the scroll list of available data sets. We are now ready to

determine if the two selected data sets are homogeneous.

We first perform the Kruskal-Wallis test [see Law (2007, p. 380)] at level 0.05.

Since the test statistic, 0.004, is less than the critical value, 3.841 for 1 degree of

freedom, we cannot reject the hypothesis that the two data sets are homogeneous.

We next display a Frequency-Comparison Plot (see Histogram Comparisons

for Homogeneity Tests) for the two data sets, which plots histograms of both data sets

on the same graph. The common histograms, which start at 10 and have 14 intervals of

width 4.75, are shown in Figure 3.1. The similarity of the two histograms supports the

homogeneity of the two data sets.

Finally, we display a Distribution Function Plot and a Box-Plot Comparisons

for the two data sets in Figures 3.2 and 3.3, respectively. These plots also support

homogeneity.

In conclusion, there is no reason to believe that the two data sets are not

homogeneous, and we merge them by clicking on the Merge Selected Data Sets

option. The merged data set is added to the current Project and is named Merged

data set. However, the name of this data set can be changed by closing the current

data analysis (i.e., for the data set Example 3.1-1) and clicking on the Edit button.

58

Example 3.1-1: Processing Times 1 Example 3.1-2: Processing Times 2

0.00

0.04

0.09

0.13

0.18

0.22

12.38 21.88 31.38 40.88 50.38 59.88 69.38

Pro

port

ion

Frequency-Comparison PlotFrequency-Comparison Plot

Interval Midpoint

Figure 3.1. Frequency-Comparison Plot for the two data sets.

59


0.00

0.20

0.40

0.60

0.80

1.00

17.00 24.83 32.66 40.49 48.31 56.14 63.97 71.80

F(x

)Distribution Function PlotDistribution Function Plot

x

Figure 3.2. Distribution Function Plot for the two data sets.

60


17.00 24.83 32.65 40.48 48.31 56.14 63.96 71.79

Box-Plot ComparisonsBox-Plot Comparisons

Figure 3.3. Box-Plot Comparisons for the two data sets.

61

We now use Automated Fitting (with Normal Precision) to determine what

distribution best represents the Merged data set, which consists of 1748 observations.

It turns out that the Pearson type V distribution provides the best fit with a Relative

Score of 97.92. The Absolute Evaluation is “Indeterminate,” which means that more

evaluations need to be performed at the Comparisons tab before the quality of the

representation provided by the Pearson type V distribution can be determined. The

Anderson-Darling test says to reject the Pearson type V distribution at level 0.05;

however, for this large sample size, the test is very powerful and may reject a

hypothesized distribution whose error may be practically insignificant. In fact, the

Density-Histogram Plot, the Distribution Function Plot, the Distribution-Function-

Differences Plot, and the P-P Plot all indicate that the Pearson type V distribution

provides a reasonably good representation of the data.

62

4. Task-Time Models Module

In some simulation studies it is not possible to obtain good data on the random

variables of interest, so the usual statistical techniques are not applicable to the

problem of selecting corresponding probability distributions. For example, if the system

being studied does not currently exist in some form, collecting data from the system is

obviously not possible. This difficulty can also occur for existing systems, if the number

of required probability distributions is large and the time available for the simulation

study prohibits the necessary data collection and analysis. In addition, sometimes there

are data available from an existing system, but the data are not in a format suitable for

use in a simulation model (e.g., the data were collected by an automated system). For

such situations ExpertFit provides guidance on modeling a task time in the Task-Time

Models module. Note that the use of no-data models is not a substitute for a careful

analysis of data collected from your system, if this is possible.

Consider the continuous random variable corresponding to the time to complete

some task (e.g., a machine repair time or a customer service time in a bank). ExpertFit

allows you to model such a random variable by a Weibull, lognormal, or triangular

distribution. For a particular distribution, you must give subjective estimates of the

minimum task time, the most-likely task time (the mode), and the 100pth percentile of

the task time. Allowable percentiles for the Weibull and lognormal distributions are the

90th (the default), 95th, and 99th; the 100th percentile (the maximum value) is also

available for the triangular distribution. More information on the use of the Task-Time

Models module can be found in the Modeling Task Times in the Absence of Data

tutorial, which is accessed from the Help pull-down menu in ExpertFit.

After completely specifying a task-time model, you can display characteristics of

the model such as its density function or percentiles. You can also represent the task-

time model in the simulation-software product of your choice.

63

4.1. Organization and Options

A Task-Time Model is based on the sequential use of the two tabs given in

Table 4.1.

The options for the Models and Applications tabs are given in Tables 4.2 and

4.3, respectively.

Table 4.1. Tabs for the Task-Time Models module.

Tab Overall Purpose

Models Used to construct models for a task time

Applications Used to determine or display characteristics (e.g., density function) of specified models or to represent the models in a simulation-software product



Specify a Model Used to create new (or to modify existing) models for a task time

View/Delete Models Used to display the parameters of currently specified models or to delete models

64



Characteristics Density Function Plot


Moment Table

Percentile Table

Probability for an x

Percentile for a p

Simulation Representation

Representation of a model in a simulation-software product

65

4.2. Example

In this section we present an example of the use of the Task-Time Models module.

66

Example 4.1: Modeling a Task Time



Project-Element Editing Select Construct distributions in the absence of data.

Select Task-Time Models.

In Project-Element Name edit box, enter Example 4.1.

Click on OK.


Models tab Click on Specify a Model.

Specify/Edit Task Time Model Click on Create a New Model.

Assumptions for Task Time Select Triangular Distribution.

Change the Minimum possible value to 1.0.

Change the Most-likely value to 4.0.

Change the Percentile to 100th (max.).

Change the 100th percentile to 11.0.

Click on Apply.

Specify/Edit Task-Time Model Click on Done.

Models tab Click on Applications tab.

Applications tab Click on Characteristics.

Characteristics Options Select Density Function Plot.

Click on Apply. Examine Density Function Plot. Click on Done.

67

A: Suppose it is thought that the minimum and maximum times to perform some task

are 1 and 11 minutes. Furthermore, suppose that the most-likely time to perform the

task is believed to be 4 minutes. Then the density function of the ExpertFit-specified

triangular distribution is given in Figure 4.1. Suppose that we want to know the 95th

percentile of our model. Using the Percentile Table button in the Characteristics

option, we get the percentile table shown in Table 4.4. From this table we get that 95

percent of the time the task-time random variable will take on values less than or equal

to 9.129 minutes.

The use of a triangular distribution is a simple approach for modeling a task time

in the absence of data, and it is usually possible to get estimates of the three

parameters. However, the triangular distribution does not have a very flexible shape

[see Law (2007 p. 370)]. Therefore, one could use a Weibull distribution or a

lognormal distribution instead, which are also supported by ExpertFit.

68

1 - Triangular

0.00

0.04

0.08

0.12

0.16

0.20

1.00 2.43 3.86 5.29 6.71 8.14 9.57 11.00

f(x)

Density Function PlotDensity Function Plot

x

Figure 4.1. Density function of the specified triangular distribution.

69

Steps for Action A (continued): At window: Do:

Characteristics Options Select Percentile Table.

Click on Apply. Examine Percentile Table. Percentile Table Click on Done.

Characteristics Options Click on Done.

Application tab In the File menu, select Close Task-Time Models.

70

Table 4.4. Percentiles of the specified triangular distribution.

Percent 1 - Triangular

0.0 1.00000

0.1 1.17321

0.5 1.38730

1.0 1.54772

2.5 1.86603

5.0 2.22474

10.0 2.73205

25.0 3.73861

50.0 5.08392

75.0 6.81670

90.0 8.35425

95.0 9.12917

97.5 9.67712

99.0 10.16334

99.5 10.40389

99.9 10.73542

100.0 11.00000

71

5. Machine-Breakdown Models Module

Representing the breakdown and repair of a machine in a simulation model is

considerably more complicated than just modeling a task time, since we have both

machine uptimes and downtimes to be concerned with. Also, the machine can be

starved (waiting for parts) or blocked (inability to remove a part from the machine) by

other machines downstream from it.

A machine goes through a sequence of cycles, with the jth cycle consisting of an

up segment (machine is operational) of length Uj, followed by a down segment of length

Dj. During an up segment, a machine will process parts if any are available and if the

machine is not blocked. The first two up-down cycles for a machine are shown in

Figure 5.1. Let Bj and Ij be the amounts of time during Uj that the machine is busy

processing parts and that the machine is idle (either starved for parts or blocked by the

current finished part), respectively. Thus, Uj = Bj + Ij. Note that Bj and Ij may each

correspond to a number of separated time segments and, thus, are not represented in

Figure 5.1.

We will assume for simplicity that cycles are independent of each other and are

probabilistically identical. We will also assume that Uj and Dj are independent for all j.

ExpertFit will help you specify probability distributions for a busy time B and for a

downtime D.

End of cycle End of cycle 0

Time

U1 U2 D1 D2

Figure 5.1. Up-down cycles for a machine.

72

The busy time before failure of the machine, B, is assumed to have a gamma

distribution with a shape parameter α equal to 0.7 and a scale parameter βB to be

specified, as shown in Figure 5.2. We chose the gamma distribution because of its

flexibility (i.e., its density can assume a wide variety of shapes) and because it has the

general shape of many busy-time histograms when α is less than or equal to 1. The

particular shape parameter of 0.7 for the gamma distribution was determined by fitting a

gamma distribution to a number of different sets of busy-time data, with 0.7 being the

average shape parameter obtained.

Note that busy time for a machine is only accumulated when the machine is

doing productive work, not when it is blocked or starved. For example, suppose that

the first busy time generated from the gamma distribution is 60.7 minutes and that each

part takes exactly 1 minute to be processed. Then the machine fails while processing

its 61st part. However, the simulation clock might be somewhat larger than 60.7 when

the machine fails, due to starving or blocking for the machine.

At the instant the machine fails, we assume that the downtime of the machine, D,

begins. The downtime of the machine is assumed to have a gamma distribution with a

shape parameter α equal to 1.3 and a scale parameter βD to be specified, as shown in

Figure 5.3. This particular shape parameter was determined by fitting a gamma

distribution to a number of different sets of downtime data, with 1.3 being the average

shape parameter obtained.

In order to determine the values of the scale parameters βB and βD, ExpertFit

asks you to give subjective estimates for two of the following three basic machine

characteristics:

• Machine efficiency

• Mean downtime for the machine

• Mean number of downs in some time period (e.g., in an 8-hour shift)

The efficiency of a machine is defined to be the long-run proportion of potential

processing time (i.e., parts present and machine not blocked) during which the machine

is actually processing parts. If the machine is never starved or blocked, then the

efficiency is the long-run proportion of time that the machine is processing parts.

73

0.00

2.58

5.17

7.75

10.33

12.91

15.50

0.00 0.87 1.73 2.60 3.46 4.33 5.20 6.06

f(x)


x

Figure 5.2. Busy-time gamma distribution with αααα = 0.7 and ββββB = 1.0.

74

0.00

0.10

0.19

0.29

0.38

0.48

0.58

0.00 1.09 2.19 3.28 4.38 5.47 6.57 7.66

f(x)


x

Figure 5.3. Downtime gamma distribution with αααα = 1.3 and ββββD = 1.0.

75

The mean downtime of a machine is the mean amount of time that elapses from

the instant the machine breaks down until the instant that it is repaired; it includes both

the time spent waiting for a repairman (if any) and the repair time itself.

The mean number of downs in some time period (called the Time Frame in

ExpertFit) such as a shift (possibly, non integral) is, more specifically, the mean number

of busy-time/downtime cycles in a time period. For example, if the mean number of

downs is exactly 2, then the machine fails and is subsequently repaired an average of 2

times in a time period.

If the machine is subject to significant starving or blocking, then you must also

give the mean number of parts produced per time period and the mean part-processing

time in order for ExpertFit to compute βB and βD.

The default gamma distributions used for busy time and downtime have location

parameters of zero; thus, they can take on arbitrarily small positive values. However, in

practice one or both of these distributions might have a minimum possible value (i.e., its

location parameter) that is a positive number. For example, it might be known that the

minimum possible downtime is 10 minutes. Thus, ExpertFit allows you to specify a

positive value for the minimum possible downtime or for the minimum possible busy

time.

76

5.1. Organization and Options

A Machine-Breakdown Model is based on the sequential use of the two tabs

given in Table 5.1.

The options for the Models and Applications tabs are given in Tables 5.2 and

5.3, respectively.

Table 5.1 Tabs for the Machine-Breakdown Models module.

Tab Overall Purpose

Models Used to construct models for the busy-time and downtime distributions

Applications Used to determine or display characteristics (e.g., density function) of the above distributions and to represent these distributions in a simulation-software product



Specify a Model Used to create models for the busy-time and downtime distributions

View/Delete Models Used to display the parameters of currently specified models or to delete models

77



Time-Frame Report Displays the expected total time during a time frame that the machine will be busy, down, and either blocked or starved

Characteristics Density Function Plot


Moment Table

Percentile Table

Probability for an x

Percentile for a probability p

Simulation Representation

Used to put the busy-time and downtime distributions into the proper format for direct input to a large number of different simulation-software products

78

5.2. Examples

In this section we present two examples of the use of the Machine-Breakdown Models module.

79

Example 5.1: Modeling Machine Downtimes in the Absence of Data



Project-Element Editing Select Construct distributions in the absence of data.

Select Machine-Breakdown Models.

In Project-Element Name edit box, enter Example 5.1.

Click on OK.



Specify/Modify Machine-Breakdown Model Click on Create a New Model.

Assumptions about the Machine Specify the Machine efficiency to be 0.9.

Specify the Mean downtime to be 60.0.

Click on Additional Machine Characteristics tab.

Assumptions about the Machine Specify the Minimum downtime to be 10.0.

Click on OK.

Specify/Modify Machine-Breakdown Model Click on Done.

80

A: Consider a machine that is never starved or blocked. Suppose that the machine

has an efficiency of 0.9; that is, it is actually producing parts 90 percent of the time.

When the machine goes down, the mean downtime is 60 minutes. However, the

minimum possible downtime is 10 minutes. These characteristics are entered using

the commands on the previous page. Note that the default values for the Blocking

and/or Starving are Significant checkbox, Characteristics to be Entered, and

Time Unit are correct. Time Frame is not used in this example.

The machine busy-time and downtime distributions have now been completely

specified, and all of the specified and calculated (shown in blue) machine

characteristics are shown on the Specify/Modify Machine-Breakdown Model screen.

In particular, note that the mean number of downs (actually the mean number of busy-

time/downtime cycles) per 8-hour shift has been calculated to be 0.8. This makes

sense since the mean length of a busy-time/downtime cycle is 10 hours.

81

B: We now can display various characteristics of the busy-time and downtime

distributions. For example, a machine Time-Frame Report is given in Table 5.4. Note

from this report that the machine is expected to be busy 90 percent of the time, which is

another way of saying that its efficiency is 0.9 (also see Table 5.5).

Steps for Action B: At window: Do:


Applications tab Click on Time-Frame Report.

Specify Units and Model for Time-Frame Report Click on Apply. Examine Time-Frame Report. Click on Done.

Specify Units and Model for Time-Frame Report Click on Done.

Applications tab Click on Models tab. (This step anticipates proceeding to Example 5.2.)

82

Table 5.4. Machine Time-Frame Report for the specified busy-time and downtime models.

Machine Time-Frame Report for Model 1 - Know e and D Time unit Minutes Time frame 1 8-Hour Shift Blocking and/or starving Not significant Machine efficiency 0.90000 Minimum downtime 10.00000 Mean downtime 60.00000 Mean number of downs <calculated> 0.80000 Minimum busy time 0.00000 Mean busy time <calculated> 540.00000 Mean number of parts produced <not applicable> Mean part-processing time <not applicable>

Expected Total Time Expected Percentage Machine Status During Time Frame of Time Frame

Busy 432.00000 90.00000 Down 48.00000 10.00000 Blocked or Starved 0.00000 0.00000

Total 480.00000 100.00000

83

Example 5.2: Continuation of Previous Example



Specify/Modify Machine-Breakdown Model Click on Create a New Model.

Assumptions about Click on the Blocking and/or Starving are the Machine Significant checkbox.

Specify the Machine efficiency to be 0.9.

Specify the Mean downtime to be 60.0.

Click on Part Production Characteristics tab.

Assumptions about the Machine Specify the Mean number of parts produced to be 100.

Specify the Mean part-processing time to be 4.0.

Click on Additional Machine Characteristics tab.

Assumptions about the Machine Specify the Minimum downtime to be 10.0.

Click on OK.

Specify/Modify Machine-Breakdown Model Click on Done.

84

A: Suppose for the machine of Example 5.1 that blocking/starving is now significant.

For example raw materials might arrive to the machine on an intermittent basis.

Suppose also that the mean number of parts produced per 8-hour shift (the default

value of the Time Frame) is 100 and the mean part-processing time is 4 minutes.

We set the Blocking and/or Starving are Significant checkbox to “on” at the Basic

Machine Characteristics tab, in addition to specifying the Machine Efficiency and

Mean Downtime as in Example 5.1. Because blocking/starving is now significant, we

must specify values for Mean number of parts produced and Mean part-processing

time at the Part Production Characteristics tab. Finally, the Minimum downtime is

specified at the Additional Machine Characteristics tab.

85

B: In Table 5.5 we display a machine Time-Frame Report, from which we can see

that the machine is expected to be busy 83.33 percent of time. Recall in Example 5.1

(see Table 5.4) that the machine was expected to be busy 90 percent of the time. Also

the Mean number of downs per 8-hour shift is different for the two examples. Thus, the

simulation results for the two examples will be different provided that the machine does,

indeed, experience blocking/starving in the second example.

Steps for Action B:

At window: Do:


Applications tab Click on Time-Frame Report.

Specify Units and Model for Time-Frame Report Select model 2 in the Model for Time-Frame Report scroll list.

Click on Apply. Examine Time-Frame Report. Click on Done.

Specify Units and Model for Time-Frame Report Click on Done.

Application tab In the File menu, select Close Machine-Breakdown Models.

86

Table 5.5. Machine Time-Frame Report for the specified busy-time and downtime models.

Machine Time-Frame Report for Model 2 - Know e and D, B and/or S Time unit Minutes Time frame 1 8-Hour Shift Blocking and/or starving Significant Machine efficiency 0.90000 Minimum downtime 10.00000 Mean downtime 60.00000 Mean number of downs <calculated> 0.74074 Minimum busy time 0.00000 Mean busy time <calculated> 540.00000 Mean number of parts produced 100.00000 Mean part-processing time 4.00000

Expected Total Time Expected Percentage Machine Status During Time Frame of Time Frame

Busy 400.00000 83.33333 Down 44.44444 9.25926 Blocked or Starved 35.55556 7.40741

Total 480.00000 100.00000

87

6. Distribution Viewer

The Distribution Viewer is used to display/calculate characteristics (e.g., the

density function or moments) of a distribution without having to enter a data set. It is

accessed from the Menu Bar at the top of the screen.

The distribution of interest is selected from the scroll list in the upper left-hand

corner of the screen and its density (or mass) function is displayed automatically for

default values of the distribution’s parameters. The parameters can be changed in the

following two ways:

• The value for a particular parameter can be entered in the corresponding data box.

[Click on the equal sign (“=”) and then enter a value.] In order to obtain a

meaningful density function plot, there are limits on the value of a parameter.

• A particular parameter can be changed dynamically by clicking on the “up” or “down”

button to the right of the data box. Clicking the up (down) button causes a real-

valued parameter to increase (decrease) by 0.1. (For an integer-valued parameter,

the change is 1 or –1.) Alternatively, a button can be held down to change the

parameter at a faster rate.

You can, in certain cases, choose to plot a density (or mass) function from either

its 0th or ath (e.g., a = 0.1) percentile to either its bth (e.g., b = 99.9) or 100th

percentile. Use of other than the 0th or the 100th percentile may be necessary to

obtain a plot that is not completely concentrated near the origin.

Additional information (e.g., moments, percentiles, and probabilities) about the

selected distribution can be obtained by clicking on the Other Options button in the

lower left-hand corner of the screen.

88

7. Batch Mode

Batch Mode, which is available in the Professional Version of ExpertFit or the

Analyst with Batch Mode Version, is used to fit distributions to several data sets with

only a few keystrokes. It is accessed from the Menu Bar at the top of the screen. Table

7.1 gives the four selections that can be made in Batch Mode.

Table 7.1. Options for Batch Mode.


Data Entry Used to choose the data sets for analysis. Data entry can be performed by reading an ASCII file containing a single data set, by reading an ASCII file that contains several data sets in columns (e.g., from Excel), and by copying a data set from the Clipboard.

Analysis Options Used to specify certain options for the fitting process such as whether all data sets should be treated as real valued and whether to display a simulation-software representation for the best-fitting distribution for each data set

Perform Analyses Used to fit distributions to the selected data sets and to display the results

Review Results Used to review the results from the fitting process

89

References

Evans, M., N. Hastings, and B. Peacock, Statistical Distributions, Third Edition, John Wiley, New York (2000).

Johnson, N. L., S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Volume 1, Second Edition, Houghton Mifflin, Boston (1994).

Johnson, N. L., S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Volume 2, Second Edition, Houghton Mifflin, Boston (1995).

Johnson, N. L., S. Kotz, and A.W. Kemp, Univariate Discrete Distributions, Second Edition, Houghton Mifflin, Boston (1992).

Law, A. M., Simulation Modeling and Analysis, Fourth Edition, McGraw-Hill, New York (2007).

90

Appendix A. Distributions Included in ExpertFit

In this appendix we present important information on the thirty-two continuous and eight

discrete standard distributions available in ExpertFit. (ExpertFit also supports three types of

empirical distributions.) These distributions are organized according to the following categories:

Category Distribution Notation Page

Non-negative

Continuous

Chi-Square

Erlang

Exponential

F

Gamma

Inverse Gaussian

Inverted Weibull

Log-Laplace

Log-Logistic

Lognormal

Pareto

Pearson Type V

Pearson Type VI

Random Walk

Rayleigh

Wald

Weibull

chisq(ν)

m-Erlang(β) or

Erlang(γ, β, m)

expo(γ, β)

F(ν1, ν2)

gamma(γ, β, α)

IG(γ, β, α)

IW(γ, β, α)

LP(γ, β, α)

LL(γ, β, α)

LN(γ, β, α)

Pareto(γ, β)

PT5(γ, β, α)

PT6(γ, β, α1,α2)

RW(γ, β, α)

Rayleigh(γ, β)

Wald(γ, α)

Weibull(γ, β, α)

97

99

101

105

106

109

110

116

117

118

121

122

123

126

127

131

132

Unbounded

Continuous

Cauchy

Error

Exponential Power

Extreme Value Type A

Extreme Value Type B

Johnson SU

Laplace

Logistic

Normal

Student's t

Cauchy(γ, β)

error(h)

expower(γ, β, α)

EVTA(γ, β)

EVTB(γ, β)

JSU(γ, β, α1,α2)

Laplace(γ, β)

logistic(γ, β)

N(γ, β)

t(ν)

96

100

102

103

104

112

113

115

120

128

91

A continuous random variable can take on any value in some interval of the real line [e.g.,

(0, ∞)]. A non-negative continuous distribution restricts the random variable to be strictly larger

than a specified lower-bound value. A bounded continuous distribution restricts the random

variable to be strictly larger than a specified lower-bound value and strictly smaller than a

specified upper-bound value. An unbounded continuous distribution places no restrictions on the

values of the random variable. A discrete random variable can take on some subset of the non-

negative integers; the subset depends upon the specific distribution.

Category Distribution Notation Page

Bounded

Continuous

Beta

Johnson SB

Power Function

Triangular

Uniform

beta(a, b, α1,α2)

JSB(a, b, α1,α2)

power(a, b, α)

triang(a, b, m)

U(a, b)

94

111

125

129

130

Discrete Bernoulli

Binomial

Discrete Uniform

Geometric

Hypergeometric

Logarithmic Series

Negative Binomial

Poisson

Bernoulli(p)

bin(t, p)

DU(i, j)

geom(p)

hyper(N, n, s)

logseries(α)

negbin(s, p)

Poisson(λ)

93

95

98

107

108

114

119

124

92

The following table describes the special symbols used in this Appendix:

Symbol Denotes Definition

!n Factorial function

1

1 i f 0

! i f 1, 2 ,

=

=

= = …

∏

n

k

n

nk n

t

x

Binomial coefficient !

!( )!

=

−

t t

x x t x

( )Γ z Gamma function 1

0

( ) fo r 0

∞

− −Γ = >∫z tz t e d t z

Note that ( 1) !Γ + =k k for any non-negative integer k.

( , )B u v

Beta function 11 1

0( , ) (1 ) fo r 0 a n d 0− −= − > >∫

u vB u v t t d t u v

( ) ( )N o te th a t ( , ) ( , )

( )

Γ Γ= =

Γ +

u vB u v B v u

u v.

( )Φ z Standard normal

distribution function

x Floor function x is the integral part of the real number x

IID Independent,

identically distributed

∼ Is distributed as

∈ Is contained in

93

Bernoulli - Bernoulli(p) Discrete

Mass

1 if 0

( ) = if 1

0 otherwise

p x

p x p x

− =

=

Parameter ( 0 ,1)p ∈

Range { }0 ,1

Mean p

Variance (1 )p p−

Mode

0 if 0.5

0 and 1 if 0.5

1 if 0.5

p

p

p

<

= >

Comment:

1. The Bernoulli(p) and bin(1, p) distributions are the same.

Use the Distribution Viewer to display

the probability density or mass

function for a distribution.

94

Beta – beta(a, b,αααα1, αααα2) Bounded Continuous

Density

21

11

1

( ) = if ( ) ( )

0 otherwise

2

−− − − − − < <−

ααx a b x

b a b af x a x b

b a B α ,α

Parameters Lower-endpoint parameter ( , )a ∈ − ∞ ∞ , upper-endpoint parameter ( )b b a> ,

shape parameters 1 0α > and 2 0α >

Range ( , )a b

Mean 1

1 2

( )α

a b aα α

+ −+

Variance 2 1 2

2

1 2 1

( )( ) ( 1)

α αb a

α α α α−

+ + +2

Mode ( ) ( )

( )

11 2

1 2

1 2

1 2

1 2

1( ) if 1 1

2

and if 1 1

if or 1 1 1 1

if o1 1

a b a ,

a b ,

a , ,

b ,

αα α

α α

α α

α α α α

α α

−+ − > >

+ −

< <

< ≥ = >

≥ <

1 2

( )

1 2

r 1 1

does not uniquely exist if 1

,α α

α α

> =

= =

1 2

Comments:

1. The beta( , ,1,1)a b and U( , )a b distributions are the same.

2. If 1X and 2X are independent random variables with gamma(0, , )i i

X β α∼ , then

1 1 2 1 2/( ) beta(0,1, , )Y X X X α α= + ∼ .

3. 1 2beta(0,1, , )X α α∼ if and only if 1 2/ (1 ) PT6(0,1, , )Y X X α α= − ∼

4. The density is symmetric about ( ) / 2a b+ if and only if 1 2α α= .

Also, the mean and mode are equal if and only if 1 2 1α α= > .

5. The beta( , ,1, 2)a b density is a left triangle, and the beta( , , 2,1)a b density is a right triangle.

6. The 1beta( , , ,1)a b α and 1power( , , )a b α distributions are the same.

95

Binomial - bin(t, p) Discrete

Mass (1 ) if {0 1 }

( ) =

0 otherwise

x t xt

p p x , ,...,tp x x

−

− ∈

Parameters t a positive integer, p ∈ (0, 1)

Range {0,1, , )t…

Mean tp

Variance (1 )tp p−

Mode ( ) ( ) ( )

( )

1 and if is an integer1 1 1

otherwise1

p p pt t t

p t

− + + + +

Comments:

1. If 1 2, , ,t

Y Y Y… are independent Bernoulli(p) random variables, then

1 2 bin( , )t

Z Y Y Y t p= + + + ∼� .

2. The bin(1, p) and Bernoulli(p) distributions are the same.

3. If 1 2, , ,m

X X X… are independent random variables with bin( , )i i

X t p∼ , then

1 2 1 2bin( , )m m

Y X X X t t t p= + + + ∼ + + +� � .

4. The bin( , )t p mass function is symmetric if and only if 1/ 2p = .

96

Cauchy - Cauchy(γγγγ, ββββ) Unbounded Continuous

Density

12

( ) [1 for all real numbers

− −

= +

xf x x

γπ β

β

Parameters Location parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0β >

Range ( , )− ∞ ∞

Mean Does not exist

Variance Does not exist

Mode γ

Comment:

1. The Cauchy(0, 1) and t(1) distributions are the same.

97

Chi-Square - chisq(γγγγ, νννν ) Non-negative Continuous

Density / 2

( 2) / 2( ) / 2( )

if ( ) = 2 ( / 2)

0 otherwise

ν

νγγ

γν

−− − −

>Γ

xxe x

f x

Parameter Location (shift) parameter ( , )γ ∈ − ∞ ∞ and degrees of freedom 0ν >

Range ( , )γ ∞

Mean γ ν+

Variance 2ν

Mode 2 if 2

otherwise

γ ν ν

γ

+ − >

Comment:

1. The chisq(0, )ν and gamma(0, 2, / 2)ν distributions are the same.

98

Discrete Uniform - DU(i, j) Discrete

Mass

1 if { 1 }

1( ) =

0 otherwise

∈ +

− +

x i,i ,..., jj ip x

Parameters i and j integers with ≤i j ; i is a location parameter

and −j i is a scale parameter

Range { , 1, , }i i j+ …

Mean 2

i j+

Variance 2( + 1) 1

12

j i− −

Mode Does not uniquely exist

Comment:

1. The DU(0, 1) and the Bernoulli(1/2) distributions are the same.

99

Erlang - m-Erlang(ββββ)

or Erlang(γγγγ, ββββ, m) Non-negative Continuous

Density

1( ) ( )exp if

( ) = ( 1)!

0 otherwise

m

m

x xx

f x m

γ γγ

β β

− − − −> −

Parameters Location (shift) parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0β > ,

shape parameter m ∈ {1, 2, …}

Range ( , )γ ∞

Mean mγ β+

Variance 2mβ

Mode ( )1mγ β+ −

Comments:

1. When γ = 0, the notation m-Erlang(β) is typically used.

2. The expo(γ, β) and Erlang(γ, β, 1) distributions are the same.

3. The Erlang(γ, β, m) and gamma(γ, β, m) distributions are the same.

100

Error - error(h) Unbounded continuous

Density 2( )( ) for all real numbers −= xhh

f x e xπ

Parameter Precision 0h >

Range ( ),− ∞ ∞

Mean 0

Variance 2

1

2h

Mode 0

Comment:

1. The error(h) and 1

N(0, 2 )h−

distributions are the same.

101

Exponential - expo(γγγγ, ββββ) Non-negative Continuous

Density

( )1exp if

( ) =

0 otherwise

− − >

xx

f x

γγ

ββ

Parameters Location (shift) parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0β >

Range ( , )γ ∞

Mean γ β+

Variance 2β

Mode γ

Comments:

1. The expo(γ, β) distribution is a special case of both the gamma and Weibull

distributions (for shape parameter α = 1, scale parameter β, and location

parameter γ in both cases).

2. If 1 2, , ,

mX X X… are independent expo(0, β) random variables, then

1 2

gamma(0, , )m

Y X X X mβ= + + + ∼� , also called the m-Erlang(β)

distribution.

102

Exponential Power - expower(γγγγ, ββββ, αααα) Unbounded Continuous

Density

2/

/ 2

exp2

( ) for all real numbers

2(2 ) 12

−− =

Γ +

x

f x x

α

α

γ

β

αβ

Parameters Location parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0β > , shape parameter 0α >

Range ( , )− ∞ ∞

Mean γ

Variance

2 32

2

2

Γ

Γ

α αβ

α

Mode γ

Comments:

1. The expower(γ, β, 1) and N( , )γ β distributions are the same.

2. The expower(γ, β, 2) and Laplace(γ, β /2) distributions are the same.

103

Extreme Value Type A - ( , )γ βγ βγ βγ βEVTA Unbounded Continuous

Density ( ) (1/ ) exp[( ) / ]exp{ exp[( ) / ]} for all real numbers f x x x xβ γ β γ β= − − −


Range ( , )− ∞ ∞

Mean 0.57722γ β−

Variance 21.64493β

Mode γ

Comments:

1. Weibull(0, , )X β α∼ if and only if ln EVTA(ln ,1/ )Y X β α= ∼

2. EVTA( , )X γ β∼ if and only if exp[( ) / ] expo(0, )Y Xβ γ β β= − ∼

3. EVTA( , )X γ β∼ if and only if Y X= − has an extreme value type B distribution with

parameters γ− and β , denoted EVTB( , )γ β−

104

Extreme Value Type B - ( , )γ βγ βγ βγ βEVTB Unbounded Continuous

Density ( ) (1/ ) exp[ ( ) / ]exp{ exp[ ( ) / ]} for all real numbers = − − − − −f x x x xβ γ β γ β


Range ( , )− ∞ ∞

Mean 0.57722γ β+

Variance 21.64493β

Mode γ

Comments:

1. Weibull(0, , )X β α∼ if and only if ln EVTB( ln ,1/ )Y X β α= − ∼ −

2. EVTB( , )X γ β∼ if and only if exp[ ( ) / ] expo(0, )Y Xβ γ β β= − − ∼

3. EVTB( , )X γ β∼ if and only if EVTA( , )Y X γ β= − ∼ −

105

F - ( , )ν νν νν νν ν1 2F Non-negative Continuous

Density

1 2 1

1 2

/ 2 / 2 ( 2) / 2

1 2

( ) / 2

1 2 2 1

if 0( ) ( / 2, / 2)( )

0 otherwise

xx

f x B x

ν ν ν

ν ν

ν ν

ν ν ν ν

−

+

>

= +

Parameters Degrees of freedom 1 {1, 2, }ν ∈ … and 2 {1,2, }ν ∈ …

Range (0, )∞

Mean 22

2

for 22

νν

ν>

−

Variance 2

2 1 222

1 2 2

2 ( 2) for 4

( 2) ( 4)

ν ν νν

ν ν ν

+ −>

− −

Mode

2 11

1 2

( 2) for 2

( 2)

does not exist on (0, ) otherwise

−>

+ ∞

ν νν

ν ν

Comment:

1. The 1 2( , )F ν ν and 2 1 1 2PT6(0, / , / 2, / 2)ν ν ν ν distributions are the same.

106

Gamma - gamma(γγγγ, ββββ, αααα) Non-negative Continuous

Density

1 ( )( )exp if

( ) = Γ( )

0 otherwise

xxx

f x

α

α

γγγ

ββ α

− − −− >


shape parameter 0α >

Range ( , )∞γ

Mean γ αβ+

Variance 2αβ

Mode ( ) if 11

otherwise

γ β αα

γ

+ ≥ −

Comments:

1. The expo(γ, β) and gamma(γ, β, 1) distributions are the same.

2. For a positive integer m, the gamma(0, β, m) distribution is called the m-Erlang(β)

distribution, and the gamma(γ, β, m) and the Erlang(γ, β, m) distributions are the same.

3. If 1X and 2X are independent random variables with gamma(0, , )i i

X β α∼ , then

1 1 2 1 2/( ) beta(0,1, , )Y X X X α α= + ∼ .

4. gamma( , , )X γ β α∼ if and only if 1/( )Y X γ= − has a Pearson type V distribution with

location parameter 0, scale parameter 1/β , and shape parameter α, denoted PT5(0,1/ , )β α

5. The chisq(0, )ν and gamma(0, 2, / 2)ν distributions are the same for {1, 2, }ν ∈ … .

6. If 1 2, , ,m

X X X… are independent random variables with gamma(0, , )i i

X β α∼ , then

1 2 1 2gamma(0, , )m m

Y X X X β α α α= + + + ∼ + + +� � .

7. If 1X and 2X are independent random variables with 1 1gamma(0, , )X β α∼ and

2 2gamma(0,1, )X α∼ , then 1 2/=Y X X has a Pearson type VI distribution with location

parameter 0, scale parameter β , and shape parameters 1 2 1 2 and , denoted PT6(0, , , ).α α β α α

107

Geometric - geom(p) Discrete

Mass (1 ) if {0,1,...}

( ) = 0 otherwise

xp p x

p x − ∈

Parameter (0,1)p ∈

Range {0,1, }…

Mean 1 p

p

−

Variance 2

1 p

p

−

Mode 0

Comments:

1. If 1 2, ,Y Y … is a sequence of independent Bernoulli(p) random variables and

min{ : 1} 1i

X i Y= = − , then geom( )X p∼ .

2. If 1 2, , ,s

X X X… are independent geom(p) random variables, then 1 2 sY X X X= + + +� has

a negative binomial distribution with parameters s and p, denoted negbin(s, p).

3. The geom(p) and negbin(1, p) distributions are the same.

108

Hypergeometric - hyper( , , )N n s Discrete

Mass ( )

s N s

x n xp x

N

n

−

− =

Parameters {1, 2, }N ∈ … , {1, , }n N∈ … , {1, , }s N∈ …

Range max(0, ) min( , )n N s x s n− + ≤ ≤

Mean ns

N

Variance ( / )(1 / )( )

1

ns N s N N n

N

− −

−

Comment:

1. Suppose that there is a population of N items of which s are successes (i.e., have a certain

property). If we select n of these items without replacement, then the number of successes

obtained has a hypergeometric distribution with parameters N, n, and s.

109

Inverse Gaussian - IG(γγγγ, ββββ, αααα) Non-negative Continuous

Density

2

3

( )exp if

( ) = 2 ( ) 2 ( )

0 otherwise

xx

f x x x

α α γ βγ

π γ β γ

− − − > − −

1

2

2



Range ( , )∞γ

Mean γ β+

Variance 3β

α

Mode ( )2 , where 3 2 1 /γ β θ β αθ θ+ =+ −

Comments:

1. The parameter β has elements of a shape parameter since it affects the skewness

and kurtosis.

2. The IG( ,1, )γ α and Wald( , )γ α distributions are the same.

110

Inverted Weibull - IW(γγγγ, ββββ, αααα) Non-negative Continuous

Density

1( ) exp if ( ) =

0 otherwise

xx x

f x

αα α γ

αβ γ γβ

−

− − −

− >−



Range ( , )γ ∞

Mean 1

Γ for 11γ β αα

+ >−

Variance

2

2 2 1 for 2Γ Γ1 1β α

α α

>−− −

Mode

1

1

ααγ β

α

+

+

Comments:

1. IW(0, , )X β α∼ if and only if 1 Weibull(0, , )Y X β α−= ∼ 1/

2. IW(0, , )X β α∼ if and only if -expo(0, )Y X α αβ−= ∼

3. IW(0, , )X β α∼ if and only if lnY X= has the extreme value type B distribution with

location parameter ln β and scale parameter 1/α , denoted EVTB(ln ,1/ )β α

4. IW(0, , )X β α∼ if and only if lnY X= − has the extreme value type A distribution with

location parameter ln β− and scale parameter 1/α , denoted EVTA( ln ,1/ )β α−

111

Johnson SB - JSB(a, b, αααα1, αααα2) Bounded Continuous

Density

2

1 2

( ) 1exp if ln

( ) = ( )( ) 2 2

0 otherwise

b a x aa x b

f x x a b x b x

αα α

π

− − < <− + − − −

2

Parameters Lower-endpoint parameter ( , )a ∈ − ∞ ∞ , upper-endpoint parameter b (b > a),

shape parameters 1 ( , )α ∈ − ∞ ∞ and 2 0α >

Range (a, b)

Mean All moments exist, but are extremely complicated.

Mode Bimodal when ( )2

2 -1 22 21 2

2

1 21 and 2 tanh ; 1 2

2

−< < − −

αα αα α

α

otherwise unimodal.

Comments:

1. 1 2JSB( , , , )X a b α α∼ if and only if 1 2 ln N(0 1)

X aZ ,

b Xα α

− = + ∼

−

2. The density function is (skewed left, symmetric, skewed right) as the shape parameter

1 is ( 0, 0, 0)> = <α .

3. lim ( ) lim ( ) 0x a x b

f x f x→ →

= = for all values of 1 2 and α α

112

Johnson SU - JSU(γγγγ, ββββ, αααα1, αααα2) Unbounded Continuous

Density

22

1 22 2

1( ) = exp ln 1

22 ( )

for all real numbers

xxf xx

x

α γγα απ γ β ββ

−− − + + + − +

2

Parameters Lower-endpoint parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0β > ,

shape parameters 1 ( , )α ∈ − ∞ ∞ and 2 0α >

Range ( , )−∞ ∞

Mean 1

2

2 2

1exp sinh

2

αγ β

α α

−

Mode ( )2 2 2 21 2 2, where satisfies 1 1 ln 0+ 1y y y y y y yγ β α α α+ + + + + =+

Comments:

1. 1 2

JSU( , , , )X γ β α α∼ if and only if 1 2 ln N(0 1)1XX

Z ,γγα α

ββ

−− = + ∼ + +

2

2. The density function is (skewed left, symmetric, skewed right) as the shape parameter

1 is ( 0, 0, 0)> = <α .

113

Laplace - Laplace (γγγγ, ββββ) Unbounded Continuous

Density 1

( ) exp for all real numbers 2

− = −

xf x x

γ

β β

Parameters Location parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0>β

Range ( , )− ∞ ∞

Mean γ

Variance 22β

Mode γ

Comments:

1. Laplace( , )∼X γ β if and only if = XY e has the log-Laplace distribution with location

parameter 0, scale parameter eγ , and shape parameter 1/ β , denoted LP(0, ,1/ )eγ β

2. The Laplace distribution is also called the double exponential distribution.

3. The Laplace( , )γ β and expower( , 2 , 2)γ β distributions are the same.

114

Logarithmic series - logseries(αααα) Discrete

Mass if {1, 2, }, where 1/ log(1 )

( ) =

0 otherwise

∈ … = − −

x

xp x x

θαθ α

Parameters shape parameter (0,1)∈α

Range {1, 2, }…

Mean 1−

θα

α

Variance 2

(1 )

(1 )

−

−

θα θα

α

Mode 1

115

Logistic - logistic(γγγγ, ββββ) Unbounded Continuous

Density 2

exp[ ( ) / ]( ) = for all real numbers

{1 exp[ ( ) / ]}

xf x x

x

γ β

β γ β

− −

+ − −


Range ( , )− ∞ ∞

Mean γ

Variance 2 2

3

β π

Mode γ

Comments:

1. logistic( )∼X ,γ β if and only if = XY e is distributed as a log-logistic distribution with

location parameter 0, scale parameter eγ , and shape parameter 1/ β , denoted LL(0, ,1/ )eγ β

2. The logistic distribution has greater “tail weight” than the normal distribution.

116

Log-Laplace - LP(γγγγ, ββββ, αααα) Non-negative Continuous

Density

1

1

if 2

( ) = if

2

0 otherwise

−

− −

− > >

−

≥

Xx

f x Xx

α

α

γαβ γ

ββ

γαβ

ββ



Range ( , )γ ∞

Mean 2

2 for 1

1

βαγ α

α+ >

−

Variance 2 2 2

2 2 2

(2 1 for 2

( 1) ( 4)

)β α αα

α α

+>

− −

Mode for 1γ β α+ >

Comment:

1. LP( , , )X γ β α∼ if and only if ln( )Y X γ= − has the Laplace distribution with location

parameter ln β and scale parameter 1/α , denoted Laplace(ln ,1/ )β α

117

Log-logistic - LL(γγγγ, ββββ, αααα) Non-negative Continuous

Density

1

2

if ( ) =

1

0 otherwise

− −

> −

+

X

xf x X

α

α

γα

βγ

γβ

β



Range ( , )γ ∞

Mean cosecant( ) for 1 where + > =,π

γ βθ θ α θα

Variance 2 2{2cosecant(2 ) [cosecant( )] } for 2, where π

β θ θ θ θ α θα

− > =

Mode

1

1 if 1

1

0 otherwise

ααγ β α

α

− + > +

Comment:

1. LL(0, , )X β α∼ if and only if lnY X= has the logistic distribution with location

parameter ln β and scale parameter 1/α , denoted logistic(ln ,1/ )β α

118

Lognormal - LN(γγγγ, ββββ, αααα) Non-negative Continuous

Density

2

2

1 [ln( ) ]exp if

( ) = 2( ) 2

0 otherwise

xx

f x x

γ β γαγ πα

− − −> −

2

Parameters Location (shift) parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0eβ > ,


Range ( ),γ ∞

Mean 2

exp2

+ +

αγ β

Variance 2 2exp(2 )[ exp( ) 1]β α α+ −

Mode ( )2expγ β α+ −

Comments:

1. LN( ) if and only if ln( ) N(ln , )∼ = − ∼X , , Y Xγ β α γ β α

2. lim ( ) 0, regardless of the parameter values→

=x

f xγ

119

Negative Binomial - negbin(s, p) Discrete

Mass

1(1 ) if {0,1, }

( ) =

0 otherwise

s xs x

p p xp x x

+ − − ∈ …

Parameters s a positive integer, (0,1)∈p

Range {0,1, }…

Mean (1 )s p

p

−

Variance 2

(1 )s p

p

−

Mode Let ( ) 11

.s p

yp

−−=

Then the mode is y and y + 1 if y is an integer; it is 1 otherwise.y +

Comments:

1. If 1 2, , ,s

Y Y Y… are independent geom(p) random variables, then

1 2 negbin( , )s

Z Y Y Y s p= + + + ∼� .

2. If 1 2, ,Y Y … is a sequence of independent Bernoulli(p) random variables and

1

min{ : } then negbin( ).i

j

j

X i Y s s, X s, p=

= = − ∼∑

3. If 1 2, , ,m

X X X… are independent random variables with negbin( , )i i

X s p∼ , then

1 2 1 2negbin( , )m m

Y X X X s s s p= + + + ∼ + + +� � .

4. The negbin(1, p) and geom(p) distributions are the same.

120

Normal - N(γγγγ, ββββ) Unbounded Continuous

Density 2

22

1 ( )( ) = exp for all real numbers

22

xf x x

γ

βπβ

− −


Range ( , )− ∞ ∞

Mean γ

Variance 2β

Mode γ

Comments:

1. The N(0,1) distribution is called the standard normal distribution.

2. If 1 2, , ,k

X X X… are independent standard normal random variables, then

2 2 2

1 2 kY X X X= + + +� has a chi-square distribution with k degrees of freedom, denoted

chisq(k).

3. N( if and only if LN(0 )XX , ) Y e ,e ,γγ β β∼ = ∼

4. The N(0, )β and 1error([ 2] )β − distributions are the same.

5. The N( , )γ β and 2expower( , ,1)γ β distributions are the same.

121

Pareto - Pareto(γγγγ, ββββ) Non-negative Continuous

Density

1

if ( ) =

0 otherwise

xf x x

ββ γ

γγ

+ >

Parameters Location parameter 0γ > , scale parameter 0β >

Range ( ),γ ∞

Mean for 11

>−

βγβ

β

Variance ( ) ( )

2

for 21 2

2>

− −

βγβ

β β

Mode γ

Comments:

1. The location parameter γ must be strictly positive and, thus, all data values must be as

well.

2. The parameter γ has attributes of a scale parameter since it affects the variance.

3. The parameter β has attributes of a shape parameter since it affects higher moments

such as the skewness and the kurtosis.

122

Pearson Type V - PT5(γγγγ, ββββ, αααα) Non-negative Continuous

Density

( 1)( )exp if

( ) = Γ( )

0 otherwise

xx

f x x

α

α

γ βγ

β α γ

− +

−

− −>

−

Parameters Location (shift) parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0β > , shape

parameter 0α >

Range ( , )γ ∞

Mean for 11

+ >−

βγ α

α

Variance ( ) ( )

2

2 for 2

1 2>

− −

βα

α α

Mode 1

βγ

α+

+

Comments:

1. PT5( , , ) if and only if 1 ( ) gamma(0 1 ).X Y / X , / ,γ β α γ β α∼ = − ∼ Thus, the Pearson type V

distribution is sometimes called the inverted gamma distribution.

2. Note that the mean and variance only exist for certain values of the shape parameter α.

123

Pearson Type VI - PT6(γγγγ, ββββ, αααα1, αααα2) Non-negative Continuous

Density

[ ] 1

1 2

1

1 2

( )/ if

( ) = ( , ){1 [( ) / ]}

0 otherwise

xx

f x B x

α

α α

γ βγ

β α α γ β

−

+

−>

+ −


parameters 1 0α > and 2 0α >

Range ( , )γ ∞

Mean 12 for 1

1

β αγ α

α+ >

−2

Variance ( )

( ) ( )

2

1 1 2

22

2 2

1 for 2

1 2

+ −>

− −

β α α αα

α α

Mode

( )

( )

1

1

2

1 if 1

1

does not exist on otherwise ,

β αγ α

α

γ

−+ >

+ ∞

Comments:

1. ( )1 2 1 2PT6( , , , ) if and only if ( ) beta(0 1 )X Y / X , , ,Xγ β α α γ β α αγ∼ = − + ∼−

2. If 1X and 2X are independent random variables with 1 1gamma(0, , )X β α∼ and

2 2gamma(0,1, )X α∼ , then 1 2 1 2/ PT6(0, , , )Y X X β α α= ∼ .

3. The 1 2F( , )ν ν and 2 1 1 2PT6(0, / , / 2, / 2)ν ν ν ν distributions are the same for 1 {1,2, }ν ∈ … and

2 {1,2, }ν ∈ … .

4. Note that the mean and variance only exist for certain values of the shape parameter 2α .

124

Poisson - Poisson(λλλλ) Discrete

Mass if {0,1, }

( ) = !

0 otherwise

xex

p x x

λλ−∈

…

Parameters λ > 0

Range {0,1, }…

Mean λ

Variance λ

Mode

1 and if is an integer

otherwise

−

λ λ λ

λ

Comment:

1. Let 1 2, ,Y Y … be a sequence of non-negative IID random variables, and let

1

: 1maxi

j

j

i YX .=

≤=

∑ Then the distribution of the Yi’s is expo(0, 1/λ) if and only if

Poisson( )X λ∼ .

2. If 1 2, , ,m

X X X… are independent random variables with Poisson( )i i

X λ∼ , then

1 2 1 2Poisson( )m m

Y X X X λ λ λ= + + + ∼ + + +� � .

125

Power Function - power(a, b, αααα) Bounded Continuous

Density

1

if ( ) =

0 otherwise

x aa x b

f x b a b a

αα

− − < >

− −

Parameters Lower-endpoint parameter ( , )a ∈ − ∞ ∞ , upper-endpoint parameter ( )>b b a ,


Range ( )a,b

Mean 1

+

+

a bα

α

Variance 2

2

( )

( 1) ( 2)

−

+ +

b a α

α α

Mode

if 1

does not uniquely exist if 1

if 1

<

= >

a

b

α

α

α

Comment:

1. The power( )a,b,α and beta( , , ,1)a b α distributions are the same.

126

Random Walk - RW(γγγγ, ββββ, αααα) Non-negative Continuous

Density ( )( )

( )

12

2 1exp if

( ) = 2 2

0 otherwise

xx

f x x x

α α β γγ

π γ β γ

− − − > − −

2


parameter 0α >

Range ( , )γ ∞

Mean +

+α β

γα β

Variance 2

2α β

α β

+

Mode ( )2 , where 21 / /γ β θ β αθ θ+ =+ −

Comments:

1. The parameter β has attributes of a shape parameter since it affects the skewness and kurtosis.

2. RW(0, , )∼X β γ if and only if 1 IG(0, , )Y X β α−= ∼

127

Rayleigh - Rayleigh(γγγγ, ββββ) Non-negative Continuous

Density

2

2

2( )exp if

( ) =

0 otherwise

x xx

f x

γ γγ

β β

− − − >

Parameters Location (shift) parameter ( , )γ ∈ − ∞ ∞ , scale parameter 0β >

Range ( , )γ ∞

Mean 2

+β π

γ

Variance 2(4 )

4

−β π

Mode 2

+β

γ

Comment:

1. The Rayleigh( , )γ β and Weibull( , , 2)γ β distributions are the same.

128

Student’s t - t(νννν) Unbounded Continuous

Density ( 1) / 2

2

+1

2( ) = for all real numbers

12 2

+

Γ

Γ +

f x xx

ν

ν

νπν

Parameters Degrees of freedom {1 2 }∈ …ν , ,

Range ( )− ∞ ∞,

Mean 0 for 1>ν

Variance ν

for 22

νν

>−

Mode 0

Comment:

1. The t(1) and Cauchy(0, 1) distributions are the same.

129

Triangular - triang(a, b, m) Bounded Continuous

Density

( )

( )

( )

( ) ( )

2 if

( )

( ) = 2 if

0 otherwise

x aa x m

b a m a

f x b xm x b

b a b m

−< ≤ − −

− < < − −

Parameters a, b, and m are real numbers with a < m < b; a is a location parameter, b - a is a

scale parameter, m is a shape parameter.

Range (a, b)

Mean 3

a b m+ +

Variance 2 2 2

18

a b m ab am bm+ + − − −

Mode m

Comment:

1. The triangular distribution is used as a rough model for a continuous random variable in the

absence of data.

130

Uniform - U(a, b) Bounded Continuous

Density

1 if

( )( ) =

0 otherwise

a x bb af x

< <

−

Parameters a and b are real numbers with a < b; a is a location parameter, b - a is a

scale parameter

Range ( , )a b

Mean 2

a b+

Variance ( )

12

b a−2

Mode Does not uniquely exist

Comments:

1. The U(a, b) and beta(a, b, 1, 1) distributions are the same.

131

Wald - Wald(γγγγ, αααα) Non-negative Continuous

Density ( )

1/ 22

3

( 1)exp if

( ) = 2( )2

0 otherwise

− − − > − −

xx

f x xx

α α γγ

γπ γ

Parameters Location (shift) parameter ( )∈ − ∞ ∞γ , , shape parameter 0>α

Range ( )∞γ,

Mean 1+γ

Variance 1

α

Mode 2 31 , where

2+ + − =γ θ θ θ

α

Comment:

1. The Wald( )γ,α and IG( ,1, )γ α distributions are the same.

132

Weibull - Weibull(γγγγ, ββββ, αααα) Non-negative Continuous

Density ( )

1exp if

( ) =

0 otherwise

xxx

f x

α

αα γα β γγ

β

−− − − >−


parameter 0α >

Range ( , )γ ∞

Mean 1

Γβ

γα α

+

Variance

212 1

2Γ Γβ

α αα α

−

2

Mode

1

1 if 1

otherwise

ααγ β α

α

γ

− + ≥

Comments:

1. The Weibull( , ,1)γ β and exp o( , )γ β distributions are the same.

2. The Weibull( , , 2)γ β and Rayleigh( , )γ β distributions are the same.

3. Weibull(0, , )X β α∼ if and only if expo(0, )= ∼Y X α αβ

4. Weibull(0, , )X β α∼ if and only if lnY X= has the extreme value type A distribution with

location parameter ln β and scale parameter 1/α , denoted EVTA( ln ,1/ )β α

5. Weibull(0, , )X β α∼ if and only if lnY X= − has the extreme value type B distribution with

location parameter ln β− and scale parameter 1/α , denoted EVTB( ln ,1/ )β α−

6. The Weibull distribution is skewed to the left if 3.6α > .

133

Index

A

absolute-error plot, 53

absolute evaluation, 23-25, 32, 34, 38, 43, 61

Advanced Mode, 1, 9, 11, 50, 55, 56

AnyLogic, 4, 31

Anderson-Darling test, 13, 15, 29, 41, 53, 61

Applications tab, 11, 13, 14, 50, 54, 63, 64, 76, 77

Arena, 4, 31

ASCII format, 6, 88

Automated Fitting, 12, 14, 23, 32, 38, 43, 52, 55, 61

AutoMod, 4, 31

AutoSched AP, 4

B

Batch Mode, 88

Bernoulli distribution, 4, 93

beta distribution, 4, 94

binomial distribution, 4, 95

blocked machine, 71, 72, 75, 77, 80, 82-86

box plot, 51

box-plot comparisons for homogeneity, 51, 57, 60

box-plot comparisons plot, 53

box-plot comparisons percentile table, 53

busy-time distribution, 3, 71-77, 80-82, 86

C

Cauchy distribution, 4, 96

chi-square distribution, 4, 97

chi-square test, 13, 15, 29, 37, 49, 53

134

Clipboard, 2, 6, 11, 12, 42, 50, 51, 88

coefficient of variation, 18, 32, 38

Comparisons tab, 11, 13, 14, 25, 50, 53, 55

confirmation options, 15

correlation plots, 12, 51

continuous distributions, 4, 6, 90, 91

Crystal Ball, 4, 5

customer support, 9

D

Data Analysis Module, 3, 11-61

data requirements, 6, 7

data summary, 12, 14, 18, 32, 38, 43

Data tab, 11, 12, 14, 50, 51, 55

delete a model, 12, 43, 52, 63, 76

delete data, 12, 51

density function, 15, 25, 68, 76

density function plot, 13, 54, 64, 68, 73, 74, 77

density-histogram plot, 13, 15, 25, 26, 35, 40, 53, 61

discrete distributions, 4, 6, 42, 43, 49, 91

discrete uniform distribution, 4, 98

distribution function comparisons, 51, 59

distribution function, 27, 51, 53, 61

distribution-function-differences plot, 13, 15, 27, 28, 36, 41, 47, 53, 57, 61

distribution function plot, 13, 54, 57, 59, 64, 77

Distribution Viewer, 87

dongle (see security key)

downtime distribution, 3, 71-77, 80-82, 86

E

empirical distribution, 13, 14, 38, 42, 54, 55

Erlang distribution, 4, 99

135

error distribution, 4, 100

evaluation report, 13, 53

Excel, 2, 6, 88

ExpertFit software architecture, 10

exponential distribution, 4, 15, 101

exponential power distribution, 4, 102

ExtendSim, 4, 31

extreme value distribution, 4, 103, 104

F

F distribution, 4, 105

Flexsim, 4, 31

Frequency-comparison plot, 13, 15, 46, 53, 57, 58

frequency-comparison table, 53

frequency table, 12, 51

G

gamma distribution, 4, 24, 37, 72-75, 106

generate random values, 54

geometric distribution, 4, 43-49, 107

glossary, 1

goodness-of-fit tests, 13, 29, 37, 41, 49, 53, 61

Anderson-Darling (see Anderson-Darling test)

chi-square (see chi-square test)

Kolmogorov-Smirnov (see Kolmogorov-Smirnov test)

H

help, online, 1, 9, 14, 29, 55

High Precision, 1, 9, 16

histogram, 12, 19-22, 32, 33, 38, 39, 43, 44, 51

histogram comparisons, 53, 57, 58

homogeneity tests, 51, 57-60

136

hypergeometric distribution, 4, 108

I

identically distributed data (see moving-average plot)

importing data from Excel, 6, 88

independent data (see correlation plots)

index, feature, 1

installation instructions, 8

integer data, 6, 14, 43-49

inverse Gaussian distribution, 4, 109

inverted Weibull distribution, 4, 110

J

Johnson SB distribution, 4, 111

Johnson SU distribution, 4, 112

K

Kolmogorov-Smirnov test, 13, 15, 29, 37

Kruskal-Wallis test, 51, 57

L

Laplace distribution, 4, 113

lexis ratio, 43

likelihood-function table, 53

location parameter, 23, 24, 34, 75

logarithmic series distribution, 4, 114

logistic distribution, 4, 115

log-Laplace distribution, 4, 116

log-logistic distribution, 4, 32-37, 117

lognormal distribution, 4, 3, 15, 37, 62, 67, 118

137

M

Machine-Breakdown Models, 3, 71-86

machine efficiency, 72, 79-84, 86

mass function, 43

mean, 17, 18, 32, 38, 43

mean downtime of a machine, 72, 75, 79, 80, 82, 83, 86

mean number of downs, 72, 75, 80, 82, 85, 86

mean number of parts produced, 75, 83, 84, 86

mean part-processing time, 75, 83, 84, 86

median, 17, 18, 32, 38, 43

MedModel, 4

merge data sets, 51, 57

Micro Saint, 4

mode, 15, 49, 62

Models tab, 11, 12, 14, 23, 50, 52, 63, 76

moment table, 13, 54, 64, 77

moment-comparison table, 53

moving-average plot, 51

N

negative binomial distribution, 4, 43, 45, 119

normal distribution, 4, 23, 120

Normal Precision, 1, 9, 56

O

OPNET Modeler, 4

outliers, 6

P

P-P plot, 13, 15, 36, 37, 47, 48, 53, 61

Pareto distribution, 4, 121

138

Pearson type 5 distribution, 4, 38-41, 61, 122

Pearson type 6 distribution, 4, 123

percentile for a probability p, 13, 54, 62, 64, 66, 67, 77

percentile table, 13, 54, 64, 67, 69, 70

Poisson distribution, 4, 45, 124

positive skewness, 17, 18, 32, 38, 43

power function distribution, 4, 125

probability for an x, 37, 64, 77

probability plots, 13, 15, 36, 37, 47, 48, 53, 61

ProcessModel, 4

Project, 1, 2, 10, 57

ProModel, 4, 31

Q

Q-Q plot, 53

R

random variates (see generate random values)

random walk distribution, 4, 126

Rayleigh distribution, 4, 127

raw-error plot, 53

real data, 6, 14, 17-42, 57-61

relative-discrepancies table, 53

relative score, 23, 24, 32, 34, 38, 43, 45, 61

repair-time data, 38-42

@Risk, 4, 5

S

sales data, 43-49

sample distribution function, 27

scale parameter, 23, 24, 31, 34, 72-75

scatter plot, 12, 51

139

security key, 8

ServiceModel, 4

service-time data, 17-31

shape parameter, 2-13, 72-74

ship-loading data, 32-37

Simio, 4, 31

SIMPROCESS, 4

simulation-software representation 11, 13, 14, 30, 31, 42, 63, 76, 77

skewed to the right, 17, 18, 32, 38, 43

skewness, 17, 18, 32, 38, 43

SLX, 4

Standard Mode, 1, 9, 11, 50

starved machine, 71, 72, 75, 77, 80, 82-86

Student’s t distribution, 4, 128

subset selection, 12, 51

survivor function plot, 53

T

table of errors, 53

Task-Time Models, 3, 62-70

test-statistics comparison, 53

trend analysis (see moving-average plot)

transformations of data, 12, 51

triangular distribution, 3, 4, 62, 66-70, 129

tutorials, 1

U

uniform distribution, 4, 130

uniform distribution, discrete, 4, 98

V

variance, 18, 32, 38, 43

140

W

Wald distribution, 4, 131

Weibull distribution, 3, 4, 23-31, 37, 62, 67, 132

Windows, 8

WITNESS, 4, 31

Date post:	26-Nov-2014
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