# Modeling and Simulation Graduate Program

Document Sample

```					            Ph.D. DIAGNOSTIC EXAMINATION
SPRING 2007

Batten College of Engineering and Technology
Old Dominion University

This examination contains seven problems; you are to complete six of the seven
problems. The problem that you choose to omit must be indicated clearly on the
front cover sheet. The format for this examination is closed-book, closed-notes
and the use of a calculator is not permitted. The time allotted for this examination
is three hours.

Problem              Value              Score

One                 20

Two                 20

Three                20

Four                20

Five                20

Six                 20

Seven                20

Total         120 (Omit one)

NAME:___________________________________________
Problem One (20 points) – Computer Science Concepts
Name:_____________________________

Part A (14 points)
Object oriented programming (OOP) provides for three characteristics over the more
traditional functional programming approach. These three characteristics are
polymorphism, reuse, and data protection. Define and describe these three
characteristics in sufficient detail to clearly illustrate their use in OOP.

Part B (6 points)
The terms “queue” and “heap” arise in the context of programming and data structures.
Define each of these terms. Explain the difference between the two terms.
Problem Two (20 points) – Mathematical Concepts
Name:______________________________

Part A (10 points)
Linear Algebra. An important concept of linear algebra is the “linear transformation”.

1) Complete the following definition of a linear transformation.
Definition. Let A and B be vector spaces. A linear transformation T from A to B,
written T : A  B , is a function that assigns to each vector a  A a vector
T(a)  b  B such that the following two properties hold:
(1) Property 1_____________________________________________
(2) Property 2______________________________________________.

2) Determine if each of the following functions T : R 3  R 2 are linear transformations.
(1) T(x, y, z) = (z, x+y)
(2) T(x, y, z) = (x+y+z, 1)

Part B (10 points)
Differential Equations. Solve the following differential equation for t > 0 with the initial
condition x(0) = 2.

dx(t )
 2x( t )  e  t .
dt
Problem Three (20 points) – Probability and Statistics
Name:_____________________________

Relevant statistical tables are included as attachments.

Part A (5 points)
In a certain communications system, there is an average of 1 transmission error per 10
seconds. Let the distribution of transmission errors be Poisson. What is the probability
of more than 1 error in a communication that has a 30-second duration?

Part B (5 points)
List the underlying assumptions that must be satisfied for Analysis of Variance to be
used.

Part C (5 points)
In regression, what is collinearity and what are its effects? Describe one collinearity
diagnostic and discuss how you would use it.

Part D (5 points)
Several measurements of air pollutant (NOx) levels were made in four different cities.
The measurements were as follows:

City 1: 438   619    732    638
City 2: 857   1014   1153   883   1053
City 3: 925   786    1179   786
City 4: 893   891    917    695   675    595

A one-way ANOVA was run on this data. Fill in the open cells of the ANOVA table
below. What conclusion can you make about the NOx levels in the different cities based
on your completed ANOVA table? Use an  = 0.05.

Degrees of
Source                         Mean Square            F
Freedom
City                            126203

Error                            20322
Total
Problem Four (20 points) – Discrete Event Simulation
Name:_______________________________

Part A (6 points)
Draw and label a flowchart diagram that explains the operation of a discrete event
simulation engine. Briefly explain the operation.

Part B (14 points)
A simple queuing model consists of a FIFO queue having infinite capacity followed by a
server having capacity two. Entities arrival at the queue with random interarrival times;
the entities are named by numbering them sequentially by order of arrival. Service
times also are random times drawn from some distribution as an entity begins service.

A discrete event simulation run is conducted for this system. The system begins empty
and idle; the event “arrive” corresponds to an entity arriving at the queue and the event
“depart” corresponds to an entity departing the server. The following event record is
recorded.

Event Number         Entity Affected         Event Type              Event Time
1                     1                   Arrive                     1
2                     2                   Arrive                     2
3                     3                   Arrive                     4
4                     4                   Arrive                     5
5                     1                   Depart                     6
6                     5                   Arrive                     7
7                     2                   Depart                     8
8                     6                   Arrive                     8
9                     3                   Depart                     10
10                    7                   Arrive                     11
11                    4                   Depart                     12
12                    8                   Arrive                     13
13                    9                   Arrive                     14
14                    5                   Depart                     15
15                    6                   Depart                     15
16                    --                  Terminate                  15

Answer the following questions concerning this simulation run.

1. Determine the maximum entity flowtime.

2. Determine the average entity flowtime.
3. Determine the maximum number of entities in the queue.

4. Determine the average number of entities in the queue.

5. Determine the maximum delay in queue.

6. Determine the average delay in queue.

7. Determine the percent utilization of the server.
Problem Five (20 points) – Modeling Concepts
Name:______________________________

Fill in the blanks by choosing the best answer from the following: Aggregation,
Association, Class, Class Diagram, Collaboration Diagram, Computer Simulation,
Contention, Dependency, FEA, Formalism, FSA, Inheritance, Interaction Diagram,
Markov Model, Model, Multimodel, Multiplicity, Prioritization, Sequence Diagram,
Simulation, System, UML, Uniform, Use Case.

Note that some words may be used more than once and some may not be used at all.

a)                        The mechanism by which more specific elements
incorporate structure and behavior of more general elements related by behavior.
b)                       The semantic relationship between two or more classes that
specify connections among their instances.
c)                       Something that is real or potentially real and is concerned
with space-time effects and causal relationships among parts
d)                       A special form of association that specifies a whole-part
relationship between the aggregate and the component part.
e)                       An abstraction from reality that describes a dynamic system
f)                      An applied methodology in which the behavior of complex
systems is described using mathematical or symbolic symbols
g)                        The discipline of designing a model of a system, executing
the model on a digital computer, and analyzing the execution output
h)                       A distribution used in generating random numbers
i)                       A generic term that applies to several types of diagrams that
emphasize object interactivity.
j)                     Something we use in lieu of the real thing in order to
understand something about that real thing
k)                       A collection of individual models – each characterizing an
abstraction level – connected together in a seamless fashion to promote level
traversal.
l)                       A diagram that shows interactions organized around the
structure of the model using either classifiers and associations or instances and
m)                        A description of a set of objects that share the same
attributes, operations, and semantics.
n)                       An unambiguous description of the semantics of a model
o)                     A graphical modeling         language    used   in   aiding   the
unambiguous description of a model.
p)                   A diagram that shows a collection of declarative and static
model elements such as classes, datatypes, their contents and their
relationships.
q)                        A relationship between two modeling elements in which a
change to one will affect change in the other.
r)                       A specification of the range of allowable cardinalities that a
set may assume.
s)                        A specification of a sequence of actions, including variants,
that a system or entity can perform while interacting with actors of the system.
t)                        A diagram that shows object interactions arranged
chronologically. It shows the objects participating in the interaction and the
messages exchanged.
Problem Six (20 points) – Analysis Concepts
Name:______________________________

Part A (10 points)
Maximize Z = 3x + 2y, subject to 2x + y < 6 and x + 2y < 6 (no negative solutions
allowed).
(a) Use the graphical method to solve the problem.
(b) Identify the pair of constraint boundary solutions for each corner of the
feasible region.
(c) Use the Simplex-algorithm to solve the problem and describe for every step
what this means in your graphical solution.
(d) How does the requirement for integer solutions change your approach?

Part B (10 points)
Given is the following one-step transition matrix of a Markov chain.

State      0         1        2         3        4
0       0.25      0.75      0         0        0
1       0.75      0.25      0         0        0
2       0.33      0.33     0.33       0        0
3        0         0        0        0.75     0.25
4        0         0        0        0.25     0.75

(a) Identify the classes of the Markov chain.
(b) Which states are recurrent, transient, and absorbing?
Problem Seven (20 points) – Visualization Concepts
Name:_______________________________

Part A (6 points)
A graphics pipeline converts points on a 3D object to pixels on the screen and during
the transformation process, a number of various coordinate systems are used by the
graphics pipeline. List and briefly describe the coordinate systems used in the graphics
pipeline.

Part B (6 points)
Homogeneous coordinates are used in modern graphics systems because all
transformations can be presented by 4 x 4 matrices in homogeneous coordinate
systems that facilitate pipeline operations.

(1) Represent a point P (1, 2, 3) and a vector v (3, 8, 2) in homogeneous coordinates.

(2) A point is translated by a displacement vector (-3, 2, 10). Represent this translation
by a 4 x 4 matrix using homogeneous coordinate system.

(3) A point Q (x, y, z) on a 3D object is transformed to a new point Q’ (x’, y’, z’) using the
following equations.

x’ = 10.5x,   y’ = 2.3y,    z’ = 1.3z.

Derive the 4 x 4 matrix representing the transformation from point Q to point Q’.

Part C (8 points)
In OpenGL, the modelview matrix represents viewing and modeling transformations and
it is a state in the OpenGL machine. Write the current modelview matrix after executing
each line in the following code segment. You don’t need to compute the result of matrix
multiplication if it is involved. If you cannot determine the modelview matrix, write down
“undetermined”.

1. glMatrixMode(GL_MODELVIEW);