Ph.D. DIAGNOSTIC EXAMINATIONExample Two

Modeling and Simulation Graduate ProgramBatten 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 , is a function that assigns to each vector a vector

such that the following two properties hold:(1) Property 1_____________________________________________(2) Property 2______________________________________________.

2) Determine if each of the following functions are linear transformations. Justify your answers.

(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.

.

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 638City 2: 857 1014 1153 883 1053City 3: 925 786 1179 786City 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.

Source Degrees of Freedom Mean Square F

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 Time1 1 Arrive 12 2 Arrive 23 3 Arrive 44 4 Arrive 55 1 Depart 66 5 Arrive 77 2 Depart 88 6 Arrive 89 3 Depart 1010 7 Arrive 1111 4 Depart 1212 8 Arrive 1313 9 Arrive 1414 5 Depart 1515 6 Depart 1516 -- 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 links.

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 40 0.25 0.75 0 0 01 0.75 0.25 0 0 02 0.33 0.33 0.33 0 03 0 0 0 0.75 0.254 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);2. glLoadIdenty();3. glTranslatef(1.0, -2.0, 3.0);4. glScalef(2.0, 3.0, 1.0);