Math 419/592: Adv. Math Modeling

Winter Semester 2009

Basic Information

This version posted on: 2009-01-06

The effective, unofficial course title for Winter 2009 is "Stochastic Math Models". Stochastic means Random. An unofficial catalog entry would read:

Models of randomness in a variety of fields: actuarial studies, economics, biology, engineering, and others as appropriate for student population. Discrete time Markov chains, Poisson processes and generalizations, time series, Brownian motion, and dynamic programming. An important part of the course should include an opportunity for a student to become involved in an actual modeling problem.

Official Course Catalog Entry

(remember that we have adapted this, see above)

A course involving an in-depth study of mathematical models of greater complexity than is possible in MATH319, including both deterministic and probabilistic models. An important part of the course should include an opportunity for a student to become involved in an actual modeling problem.


The official prereq is Math 319, but that will not be enforced for Winter 2009. Real prereqs are Math 120, 122, and 360/370. Please contact Andrew Ross in the math department for an override of the 319 prereq.
Some experience using Excel, VBA, Mathematica, Maple, or Matlab will also be VERY helpful, but it is not strictly a prerequisite.

Follow-up courses: Math 436 Numerical Analysis, various statistics classes

Class Meetings

Tue, Thu 5:30pm-6:45pm in Pray-Harrold 305 or 503 (PC Lab)
"Final Exam" schedule: Tue Apr 21st, regular class time
Math 419: CRN 24211, 3 credit hours.
Math 592: CRN 26712, 3 credit hours.

Class meetings will be mostly interactive lectures, with some time to discuss homework.

Instructor information

Professor Andrew Ross
Pray-Harrold 515b
(734) 487-1064, but I strongly prefer e-mail instead of phone contact.
Math department main office: Pray-Harrold 515, (734) 487-1444

Office Hours and other help

Office Hours:
M,T,W,R: 8:30-9:00am, 11:00-11:30am
T/Th: 4:30-5:30pm

I am also happy to make appointments if you cannot come to the general office hours. Please send me e-mail to arrange an appointment.

The Mathematics Student Services Center (or "Math Lab") is also here to help you, in Pray-Harrold 220. Please give them a call at 734-487-0983 to find out their hours.

Many assignments in this course will be in the form of papers, which I want to be well written. Please consult with The Writing Center for help in tuning up your writing.

Required materials

Our required textbook is "Introduction to Probability Models", by Sheldon Ross (no relation to your instructor), published by Academic Press. Either the 8th edition (ISBN 0125980558) or 9th edition (ISBN 0125980620) Amazon link

The textbook should be available at all the usual bookstores on and around campus. The library has a page about class textbooks that includes bookstore addresses, and also information about the student government's Bookswap.

Course Web Page

We will use the WebCT system. You are expected to keep an eye on your scores using the system, and get extra help if your scores indicate the need.

Supplementary Materials

Here is a list of books that I have found interesting and related to math modeling. Perhaps some of them will strike your fancy, too. I own the ones that are starred (*) and can lend them to you. Others you will have to find at the library or on the usual Internet booksellers. Links are given to Amazon, but I do not specifically endorse them or any particular bookseller. Of course, if you like a book you can see what similar books the online bookseller recommends. Other Stuff:

Course Content

Course Goals

Our primary goal is to teach you to be a good (or great!) stochastic math modeler. To be a good modeler, you need:

We have a few secondary goals, which may be more or less applicable to your personal situation:


We will start by reviewing basic probability ideas. We will also learn how to simulate a variety of random variables using Excel or Matlab (your choice)--doing little simulations will help understand a fair amount of the theory we will learn.

Time Series are used for a variety of things in economics and the various sciences. This will be the most statistics-oriented part of the class.

Dynamic Programming is a method of optimizing one's decisions as they unfold in time. It often includes some model of randomness, because we don't know what the future will hold. It is also used in some pattern recognition problems, such as speech recognition and genomic searches/ DNA alignment.

After that, we will talk about Discrete Time Markov Chains (DTMCs), which are used to model a wide variety of phenomena, from people moving between socio-economic classes to babies learning where one word ends and the next begins. Then, we will talk about Poisson Processes, which are useful for modeling the arrival of demands (like phone calls or customers) or other time-based phenomena (radiation particles, asteroids, etc.)

We will also study Renewal Theory, in which many of the results are completely intuitive, but there is one important result (called the Inspection Paradox) that takes some getting used to.

Queueing Theory is the study of how long people (or items) have to wait to be served. We will only see the briefest peek at it.

Reliability Theory is in the book, but we will not cover it in this course unless there is a demand for it and some extra time.

Brownian Motion is the basis of a lot of stock market models. It is essentially a random walk. We will also look at some generalizations.

Course Topic Outline (the schedule is approximate, and the order may change based on the interests of people in the class)
Week(s) Topic Chapter
1 Queueing none
2 Preliminaries 1,2,3
3 Time Series none
4 Dynamic Programming none
5-6 Markov Chains 4
7-8 Poisson Processes, etc. 5
9-10 Renewal Theory 7
11-12 Queueing Theory 8
13-14 Brownian Motion 10

Grading Policies


Regular attendance is strongly recommended. There will be material presented in class that is not in the textbook, yet will be very useful. Similarly, there are things in the textbook that are might not be covered in class, but are still very useful. If you must miss a class, arrange to get a copy of the notes from someone, and arrange for someone to ask your questions for you.

My lectures and discussions mostly use the chalkboard, along with demonstrations in Excel and other mathematical software. I do not usually have PowerPoint-like presentations, and thus cannot hand out copies of slides.


Homework will be assigned about once a week. It will sometimes be a small problem set designed to help you understand the behavior of math models. Other times, it will involve writing up a little paper on an assigned topic. All homework should be typed.

Homework papers should be submitted on-line, where they may be checked by or a similar service. This is partly to help keep you honest, and partly to help you learn acceptable ways to cite the work of others. A side benefit is that sometimes TurnItIn finds papers relevant to your work that you would not have found otherwise!


There will be no exams, unless the class demonstrates an unwillingness to be motivated any other way.


Instead of a mid-term and a final exam, you will do a mid-term and a final project. Your results will be reported in a paper and a presentation to the class. You may work by yourself or in a team of 2 people, but no groups larger than 2 will be allowed. You may switch project partners at your will. Your project grades will each be split into roughly: 10 percent for the project proposal (due 2 weeks before the project), 80 percent for the written paper and actual work, and 10 percent for the presentation (subject to change). The final presentations will be made during the time slot reserved for the final exam.

Overall Grades

No scores will be dropped, unless a valid medical excuse with evidence is given. In the unfortunate event of a medical need, the appropriate grade or grades will be dropped entirely, rather than giving a make-up. You are highly encouraged to still complete the relevant assignments and consult with me during office hours to ensure you know the material.

Your final score will be computed as follows: Once final scores are computed, a curve will be applied.

General Caveat

The instructor reserves the right to make changes to this syllabus throughout the semester. Notification will be given in class or by e-mail or both. If you miss class, it is your responsibility to find out about syllabus and schedule changes, especially the due dates and times of projects, assignments, or presentations.

Advice from Other Math Modelling Students

In the last two semesters, I've asked my math modeling students to give advice to you, future math modeling students, based on their experiences in my course. Here are some of the highlights: See any common themes?

Standard University Policies

Religious Holy Days

I support students' right to observe religious holidays without penalty. To the best of my ability, I will schedule exams to not conflict with major religions' holidays. Students are to provide advance notice to the instructor in order to make up work, including examinations that they miss as a result of their absence from class due to observance of religious holidays. If satisfactory arrangements cannot be made, the student may appeal to the head of the department.

Academic Honesty

Academic dishonesty, including all forms of cheating and/or plagiarism, will not be tolerated in this class. Penalties for an act of academic dishonesty may range from receiving a failing grade for a particular assignment to receiving a failing grade for the entire course. In addition, you may be referred to the Office of Student Judicial Services for discipline that can result in either a suspension or permanent dismissal. The Student Conduct Code contains detailed definitions of what constitutes academic dishonesty, but if you are not sure about whether something you’re doing would be considered academic dishonesty, consult with the instructor.

Classroom Behavior

Students are expected to abide by the Student Conduct Code and assist in creating an environment that is conducive to learning and protects the rights of all members of the University community. Incivility and disruptive behavior will not be tolerated and may result in a request to leave class and referral to the Office of Student Judicial Services (SJS) for discipline. Examples of inappropriate classroom conduct include repeatedly arriving late to class, using a cellular telephone, or talking while others are speaking. You may access the Code online at

Special Needs Accomodations

If you wish to be accommodated for your disability, EMU Board of Regents policy #8.3 requires that you first register with the Access Services Office (ASO) in room 203 King Hall. You may contact ASO by telephone at (734) 487-2470. Students with disabilities are encouraged to register with ASO promptly as you will only be accommodated from the date you register with them forward. No retroactive accommodations are possible.

Student and Exchange VISitors (SEVIS)

The Student Exchange Visitor Information System (SEVIS) requires F and J students to report the following to the Office of International Students, 229 King Hall within ten (10) days of the event: Prior permission from OIS is needed for the following: Failure to report may result in the termination of your SEVIS record and even arrest and deportation. If you have questions or concerns, contact the OIS at 487-3116, not your instructor. Also, see the EMU SEVIS page.