CSC401 Simulation Techniques - Ship



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. Collect data from the real system of interest.

2. Identify a probability distribution to represent the input process

3. Choose parameters that determine a specific instance of the distribution family.

4. 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

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