CSC401 Simulation Techniques
Exam 2 is scheduled to be given on Monday, November 22, 2010
This exam will cover Chapters 7 – 11 and Section 12.1 of the textbook, and the
computer scheduling project.
It will be designed as a 1.25 hour in-class exam. You will be allowed to take up to the
full 100 minutes of the period.
You will be allowed to bring two pages of notes to the in-class portion of the exam.
Everything you should know for the exam
This type of course builds off of the previous material.
I expect you to be familiar with the concepts in Chapters 1 – 6.
For example the terminology and concepts involved in Chapter 5 are
always important.
All the following should be familiar at this point
pdf
cdf
discrete distribution
Bernoulli, binomial, geometric, Poisson
continuous distribution
uniform, exponential, Erlang, normal, lognormal, Weibull, triangular
empirical distributions
expectation, mode, variance
queueing models
performance measures
Chapter 7 Random-Number Generation
7.1 Properties of Random Numbers
What is a random number?
What is a random integer?
7.2 Generation of Pseudo-Random Numbers
Certain problems or errors can occur when generating pseudo-random numbers
Properties of methods or routines for computer-generated random numbers.
7.3 Techniques for Generating Random Numbers
Linear Congruential Method
Mixed Congruential Method
Multiplicative Congruential Method
Combined Linear Congruential Generators
Random-Number Streams
7.4 Tests for Random Numbers
Frequency Test
Kolmogorov-Smirnov or chi-square to compare to uniform distribution
Autocorrelation Test
Tests the correlation between numbers and compares the sample correlation to the
expected correlation.
Chapter 7 Homework problems #3, 4, 6, 7, 8, 10, 12, 14, 16
Chapter 8 Random-Variate Generation
A distribution has been completely specified, and ways are sought to generate samples
from this distribution to be used as input to a simulation model.
It is assumed that a source of uniform random numbers is available
8.1 Inverse-Transformation Technique
The inverse-transform technique is useful when the cdf F(x) is of a form simple
enough so that its inverse F-1 can be computed easily.
step 2: set F(X) = R on the range of X
step 3: solve F(x) = R for X in terms of R
step 4: generate uniform random numbers and compute the desired random variates
from X = F-1 (R)
Exponential distribution is done in detail.
Uniform distribution example
Weibull distribution example
Triangular distribution example
Empirical Continuous Distributions – we had an problem using this
Discrete Distributions – problem
Geometric Distribution
8.2 Acceptance-Rejection Technique
Need to generate random variates, X, uniformly distributed between 1/4 and 1.
Generate the number R and accept if R ≥ 1/4, reject if R < 1/4.
Poisson Distribution acceptance-rejection
Nonstationary Poisson Process using acceptance-rejection called thinning.
Gamma Distribution
8.3 Special Properties
Variate generation based on features of a particular family of probability functions,
rather than being general-purpose techniques.
Direct Transformation for the Normal an Lognormal Distributions
Using Polar coordinates – multivariable calc.
Convolution Method
Erlang Distribution
Chapter 8 homework problems: 1, 2, 5, 9, 13, 15, 16.
Chapter 9 Input Modeling
Input models provide the driving force for a simulation model.
In the simulation of a queueing system, typical input models are the distributions of
time between arrivals and of service times.
In real-world simulation applications, coming up with the appropriate distributions for
the input data is a major task form the standpoint of time and resource requirements.
1.
2.
3.
4.
Collect data from the real system of interest.
Identify a probability distribution to represent the input process
Choose parameters that determine a specific instance of the distribution family.
Evaluate the chosen distribution and associated parameters for goodness of fit,
Kolmogorov-Smirnov or chi-square
9.1 Data collection
This is always the hard part – suggestions please!
9.2 Identifying the Distribution with Data
Histograms and shape
See page 341 for instructions for constructing a histogram.
Selecting the Family of Distributions
One aid to selecting distributions is to use the physical basis of the distributions as
a guide, see pages 346 – 347 for descriptions
Binomial, geometric, Poisson, normal, lognormal, exponential, gamma, beta,
Eralng, Weibull, discrete or continuous uniform, triangular, empirical
Quantile-Quantile Plots
If the Q-Q plot comes out a straight line, it must be normal
9.3 Parameter Estimation
Software packages are available to do a lot of this
Preliminary Statistics: Sample Mean and Sample Variance
Use the following as a guide:
Table 9.3: Suggested Estimator for Distributions Often Used in Simulation
Examples given of the various distributions.
9.4 Goodness-of-Fit Tests
Chi-square test, H0 and H1 (figure this out)
Applied to Poisson Assumption
Test with Equal Probabilities
Test for Exponential Distribution
Kolmogorov-Smirnov Goodness-of-Fit Test
Applied to test for Exponential Distribution, H0 and H1
p-values and Best Fits – should be read over but not important to test at this point
9.5 Fitting a Nonstationary Poisson Process
-should be read over, but not important to the test at this point.
9.6 Selecting Input Models Without Data
Engineering data, Expert opinion, Physical or conventional limitations, the nature of
the process.
9.7 Multivariate and Time-Series Input Models
Variables may be related, and if the variables appear in a simulation as inputs, the
relationship should be investigated and taken into consideration.
Quite often investigators will assume that the random variables are independent to
avoid this.
Covariance and correlation are measures of linear dependence between random
variables.
Multivariate Input Models
Time-Series Input Models
The Normal-to-Anything Transformation (NORTA)
Chapter 9 homework problems: 6, 7, 8, 9, 10, 11, 20, and CPU Simulation Problem
Chapter 10 Verification and Validation of Simulation Models
Definitions of verification and validation
10.1 Model Building, verification, and validation
See figure 10.1 for the loop
10.2 Verification of Simulation Models
Suggestions given, how did you do in the assignment for the CPU simulation?
Use of a trace
10.3 Calibration and Validation of Models
Figure 10.3 Iterative process of calibrating a model
Face Validity – seem reasonable
Validation of Model Assumptions - collect data at different times
validate the input data via Chapter 9
Validating Input-output Transformations
Some version of the system under study must exit
Watch out for changes in the operational model
The Fifth National Bank of Jasper
Input-Output Validation Using Historical Input Data
This is the approach I like best
Input-Output Validation using a Turing Test.
What is a Turing Test??
Chapter 10 homework: problem 8 and the worksheet related to the CPU simulation.
Chapter 11 Estimation of Absolute Performance
What is a terminating simulation?
Appropriate analysis for across replication data output data
What is a non-terminating simulation?
Appropriate output analysis for across replication output data
Section 12.1 Comparison of Two System Designs
Independent Sampling
Common Random Numbers