IE 513 Fall 2015 Syllabus
Instructor
Dr. Sarah M. Ryan
3017 Black Engineering, 294-4347, smryan@iastate.edu
Office Hours
Wed 2:30 - 4 or by appointment
Time & Place
MWF 11:00 – 11:50, Howe 1242
Text
Introduction to Probability Models, 11th ed., by Sheldon M. Ross, Academic Press,
2014 ISBN: 978-0-12-407948-9. The following sections will be included:
Chapter Title
Sections
1
Introduction to Probability Theory 1-6 Review with emphasis on
conditional probability
2
Random Variables
1-6, 8 (1-4 should be review)
3
Conditional Probability and
1-5 plus supplement on
Conditional Expectation
Conditional Value at Risk
4
Markov Chains
1-4, 5.1, 6
5
The Exponential Distribution and
1-4
the Poisson Process
6
Continuous-Time Markov Chains
1-5
7
Renewal Theory and Its
1-5, 7
Applications
8
Queueing Theory
1-5, 6.2, 9.1, 9.2
10
Brownian Motion and Stationary
1-4
Processes
Description
From the catalog: Prereq: Stat 231. Introduction to modeling and analysis of
manufacturing and service systems subject to uncertainty. Topics include the
Poisson process, renewal processes, Markov chains, and Brownian motion.
Applications to inventory systems, production system design, production scheduling,
reliability, and capacity planning.
Expanded:
• When do initial conditions not matter in the long run?
• What can go wrong if processing rates equal demand rates?
• Why would a process inspected at random times look worse than it really is?
• Why do supermarkets have separate waiting lines for each checkout but
banks have a single waiting line for all the teller windows?
• How can an investor benefit from volatility?
• What are some useful tools for managing a system that changes over time in
an unpredictable way?
IE 513 is an introduction to using probability models to help analyze stochastic
systems. “Stochastic” is a synonym for “random”, which can also mean “variable”,
and a system is any collection of components that interact. Because the analysis
methods and tools rely on mathematics, familiarity with calculus including
differential equations is assumed. However, the focus is on understanding and
learning how to think probabilistically rather than on computation. The successful
student will have good facility with mathematical notation and reasoning. The
course begins with a quick review of probability, including random variables,
distributions, expectation and variance. Students who have not worked with these
topics recently may need to spend additional time on independent review.
Structure
There will be approximately 9 homework assignments to be collected and (partially)
graded. Students are encouraged to work together but the product each student
submits is expected to be his/her own work. Little to no credit will be given for
solutions apparently copied from any source. Individual completion of homework
may be assessed via quizzes in class or online. Homework may not submitted late
and missed quizzes cannot be made up but the lowest homework/quiz score will be
dropped. There will be three tests; the third will be given during the final
examination period, but will not be comprehensive.
Project
A random variable is a real-valued function defined on the sample space of an
experiment (Ross, p. 21). A stochastic process is a collection of random variables,
usually indexed by time (Ross, p. 84). Informally, a random variable is a number
that is unknown until it is observed, and a stochastic process is a random variable
whose value changes over time. The goal of this project is to formulate a stochastic
process model for some quantity of interest based on data collected. This
formulation will include the following elements:
• Describe the nature of the process. Is the time index discrete or continuous?
Is the state space (i.e., the range of the random variable) discrete or
continuous? Is it a counting process? A renewal process? Is it Markovian,
and/or does it have embedded Markovian processes? Does it have some
relationship to a queueing situation?
• Specify the characteristics of the process. Depending on the type of
stochastic process model, these may include the marginal distribution(s) of
the process at specific points in time, the distribution of the time intervals
between events, and transition probabilities or functions. These probabilities
and distributions should be estimated from data.
• Based on these input characteristics, analyze the process to identify limiting
probabilities and/or stationary averages if they exist, and descriptions of
transient behavior such as hitting times or first passage times.
• Draw any useful conclusions you can. These will depend greatly on the
context and model type. They could include: expected waiting times and
queue lengths in a congested system; the probability that an important event
occurs within a specified time interval; the expected value, variance, and
autocovariance of the process; the probability that a future value falls in a
certain range or the expected first time a particular value will be observed,
etc.
Students will work in groups of 3-4 formed according to common interest. Each
group will make a 15-20 minute presentation. These are tentatively scheduled for
Friday, Dec. 4 – Friday, Dec. 11. The presentations should focus on explaining the
context and model, why the stochastic process is interesting and what questions can
be answered by analyzing the model. A short written report will be due by Friday,
December 11.
Grading
75%
15%
10%
Tests (25% each)
Homework/Quizzes
Project
Student
Learning
Outcomes
Upon completion of this course, students are expected to:
1. Apply the technique of conditioning to obtain probabilities of events and
expected values of dependent random variables.
2. Formulate discrete-time Markov chain models by identifying states and
specifying transition probabilities.
3. Find and use multiple step transition probabilities to make probabilistic
statements about future Markov chain states.
4. Distinguish between ergodic and non-ergodic Markov chains and analyze
the limiting behavior of each type.
5. Exploit the memoryless property of the exponential distribution.
6. Use the properties of a Poisson process, including non-homogeneous and
compound variations, to evaluate probabilities and expectations concerning
event times, cumulative counts, and occurrences within intervals.
7. Formulate continuous-time Markov chain models by identifying states and
transition rates.
8. Compute the limiting distributions of a continuous-time Markov chain. In
the particular case of birth-death processes, understand the conditions for a
limiting distribution to exist and use streamlined methods to find it.
9. Exploit the relationship between event counts and inter-event times in
renewal processes.
10. Apply renewal limit theorems, alternating renewal and renewal reward
theory to compute long-run expectations.
11. Analyze the long-run behavior of birth-death queueing systems.
12. Apply Little’s formula, demonstrate understanding of capacity, and analyze
the throughput of a single-server system or network.
13. Apply mean-value analysis to evaluate expected populations and waiting
times of systems with Poisson arrivals.
14. Apply stationary and independent increments to evaluate probabilities and
expectations concerning linear and geometric Brownian motion processes
with or without drift.
Academic
Honesty
The IMSE Department expects that all students will be honest in their actions and
communications. Individuals suspected of committing academic dishonesty will be
directed to the Dean of Students Office as per University policy. For more
information regarding Academic Misconduct see
http://www.dso.iastate.edu/ja/academic/misconduct.html.
Exams will be strictly individual efforts. Cheating on an exam will result in a
failing grade for the course.
Professionalism
The IMSE Department expects that all students will behave in a professional manner
during all interactions with fellow students, faculty, and staff. Treating others with
respect and having constructive communications are examples of being
professional.
Student
Disabilities or
Special
Accommodations
Iowa State University is committed to assuring that all educational activities
are free from discrimination and harassment based on disability status. All
students requesting accommodations are required to meet with staff in
Student Disability Resources (SDR) to establish eligibility. A Student
Academic Accommodation Request (SAAR) form will be provided to
eligible students. The provision of reasonable accommodations in this course
will be arranged after timely delivery of the SAAR form to the instructor.
Students are encouraged to deliver completed SAAR forms as early in the
semester as possible. SDR, a unit in the Dean of Students Office, is located
in room 1076, Student Services Building or online at
www.dso.iastate.edu/dr/. Contact SDR by e-mail at
disabilityresources@iastate.edu or by phone at 515-294-7220 for additional
information.