15-858A: Advanced Stochastic Analysis and Applications (SPRING 07)
T 10:30 - 1:30, Room: Tepper 227
(cross-listed as Tepper 47-774 and 47-775)
STARTING DATE: Tuesday, January 23, 2007
Queueing theory is an old area of mathematics which has recently
become very hot. The goal of queueing theory has always been to
improve the design/performance of systems, e.g. networks, servers,
memory, distributed systems, etc., by finding smarter schemes for
allocating resources to jobs.
This class emphasizes the beautiful mathematical techniques used in
queueing theory and throughout stochastic analysis in general. Topics
covered include: stochastic orderings of random variables,
heavy-tailed workloads, Laplace transforms, z-transforms, scheduling
theory analysis using transforms, advanced generalization of Little's
Law and other conservation laws, fluid anlaysis, heavy-traffic
approximations, delay tail analysis, matrix-geometric methods, and
more. Each topic will be presented first be presented in terms of the
technique being introduced and then in terms of the application of
that technique to solving problems in computer systems and
The techniques studied in this class are useful to students in
Computer Science, Tepper, Mathematics, ACO, Statistics, and Engineering.
This course is packed with open problems -- problems which if solved
are not just interesting theoretically, but which have huge
applicability to the design of computer systems today.
It is strongly recommended that you have taken 15-857A or 15-849 before you take this class. I you haven't you should get special permission from one of the instructors.
TEXTBOOK: We will pass out my our own course notes and some supplementary
handouts and papers at the end of each class.
Some good reference texts are listed
here: BOOK LIST.
You can borrow most of these books from our offices.
- 4 homeworks -- worth 50% total.
- Midterm 1 -- Written -- 20%.
- Midterm 2 -- Oral -- 20%.
- Class participation -- 10%.
ANTICIPATED OUTLINE OF TOPICS FOR THIS CLASS:
Stochastic Orderings (2 lectures Alan)
Heavy Tailed Workloads (2 lectures Alan)
- Definitions and motivation behind the definitions
- Several lectures on examples of proofs and applications
- Inventory applications
- Scheduling apps from Shanthikumar
- FIFO minimizes variance application
Queueing Theory Laws (1 lecture Mor)
- Alan's thesis
- Applications -- Load balancing
- Applications -- Opt number of servers.
Introduction to Transforms (1 lecture Mor)
- Quick review of Little's Law Derivation
- Generalization to H = LG
- Rate Conservation Law (RCL)
- Simple applications of these more advanced laws
- Lecture 5 notes -- H=LG and RCL : in postscript and in pdf
Transform analysis of M/G/1 (1 lecture Mor)
- Definiton of Laplace Transform
- Definition of z-transform
- Practice with transforms
- Sums of Random Variables
- Sums of Random Number of Random Variables
- Busy Periods
- Lecture 6 notes -- Intro to transforms : in postscript and in pdf .
Scheduling via Transforms (1-2 lectures Mor)
- Transform of Age/Excess
- Kleinrock method for M/G/1
- RCL method for M/G/1
- Transform analysis of server with setup cost
- Lecture 7 notes -- M/G/1 transform : in postscript and in pdf .
Possible additional applications of transforms (1 lecture Mor, if time)
- Busy periods with exceptional first service, Other busy period properties.
- Go through all scheduling policies -- derive Laplace transform
- Comparison on policies
- Lecture 8 notes -- Scheduling transform : in postscript and in pdf .
Fluid analysis (2 lectures Alan)
- RCL method for server with vacations
- Lecture 9 notes -- M/G/1 with Vacations : in postscript and in pdf .
Delay Tail (1 lecture Alan)
- Application: c-mu rule/reentrant lines.
- Stolyar recent papers
- Application involving work coming in a fluid sense
- Workload analysis
Matrix-Analytic Methods (1 or 2 lectures Mor)
- Ward Whitt closed form
- Onno & Bert -- scheduling tail results -- closed form
- Defining the technique
- Phase-type distributions
- Application: Analysis with phase-type service distributions
- Interpretations of the R-matrix
- Application Time-Varying-Load
- Application Join-the-Shortest-Queue
- Application: Dimensionality Reduction
- Lecture 12 notes -- Matrix-Analytic Methods : handout in pdf and answers to handout in pdf .
- Lecture 13 notes -- Interpretations and Applications of Matrix-Analytic Methods : notes in pdf