Time and Place: 4:10-5:25pm on Mondays and Wednesdays in 603 Hamilton Hall (Call Number 80782)

• Office hours: after class and by appointment; email: ww2040@columbia.edu

Focus

• The course will focus on stochastic-process limits and their applications to queueing models. We will explore ways in which large scale in complex stochastic systems can be converted into an advantage instead of a disadvantage via asymptotics exploiting that scale. We will consider how relatively simple approximations for the performance of complex systems can be developed by establishing appropriate stochastic-process limits.

• The course is an advanced doctoral course, being a sequel to the first-year doctoral courses, IEOR 6711 and IEOR 6712, Stochastic Models I and II. These core courses provide an introduction to stochastic processes, covering Markov chains, the Poisson process, renewal theory, martingales and Brownian motion, at the level of the 1996 book, Stochastic Processes, by Sheldon Ross and the 1975 book, A first Course in Stochastic Processes, by Samuel Karlin and Howard Taylor. It would also be good to have taken or be currently taking a course on measure-theoretic probability, such as Math G4151, Analysis and Probability I, taught this year by Professor Julien Dubedat on Tuesdays and Thursdays at 4:10-5:25pm.

Textbooks

• W. Whitt, Stochastic-Process Limits, Springer, 2002.

• P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968; second edition 1999.

Lecture Notes

Syllabus

• The course will have three components: First, there will be an introduction to stochastic-process limits, covering topics such as Donsker's functional central limit theorem, the martingale central limit theorem, and the continuous mapping theorem. Second, there will be an introduction to applications to queueing models. Specifically, we will consider heavy-traffic limit theorems, including the three principle kinds: (1) a fixed number of servers, (2) an infinite number of servers and (3) many-server limits, in which a finite number of servers is allowed to increase as the traffic intensity increases, so that the probability all servers are busy approaches a nondegenerate limit strictly between 0 and 1. Finally, the third component will be students reporting on course projects, conducted individually or in pairs.

• This course is intended to be a research course, giving students the opportunity to conduct independent research. Each student will be required to conduct a research project, on which they will give an oral presentation and submit a written report. A wide range of research topics is acceptable, including asymptotic methods in probability with other applications as well as other approaches to queueing models, i.e., the research topic should be in the union of (i) stochastic-process limits and (ii) queueing theory. Generally, the course is intended to enhance the student's ability to work with stochastic models. More specifically, it is intended to help students be able to effectively conduct research on stochastic models. There will be lectures and homework in the first parts of the course, and then student presentations thereafter. There will be no exams.

Possible Research Topics

Background Introductory Queueing Textbooks

• S. Asmussen, Applied Probability and Queues, Second Edition, Springer, 2003. (introductory, but at a high mathematical level; concise, but difficult for beginners)

• R. B. Cooper, Introduction to Queueing Theory, second edition, North Holland, 1981. On reserve. Out of print, but available in .pdf format (about 13mb zipped) from Michael Taaffe at VPI, with approval from the author. (truly introductory, especially good at the telecommunications perspective, drawing on early experience at Bell Labs)

• R. W. Hall, Queueing Methods for Service and Manufacturing, Prentice Hall, 1991. (very nice applied perspective)

• L. Kleinrock, Queueing Systems, vols. I and II, Wiley, New York, 1975-6. (introductory with computer science and networking perspective)

• R. Wolff, Stochastic Modeling and the Theory of Queues, Prentice-Hall, Englewood Cliffs, NJ, 1989. (truly introductory)

Overview Papers

• Stochastic Models of Call Centers

  • Z. Arkin, M. Armony and V. Mehrotra, "The modern call center: a multidisciplinary perspective on operations management," Production and Operations Management, vol. 16, 2007, 665-688.

  • N. Gans, G. Koole and A. Mandelbaum, "Telephone call centers: tutorial, review and research prospects," Manufacturing and Service Operations Management, vol. 5, 2003, 79--141. PDF

  • W. Whitt, "Stochastic models for the design and management of customer contact centers: some research directions," March 2002. PDF

• Many-Server Heavy-Traffic Limits

  • G. Pang, R. Talreja and W. Whitt, "Martingale Proofs of Many-Server Heavy-Traffic Limits for Markovian Queues." Probability Surveys, vol. 4, 2007, 193-267. [PDF].

My Recent Research Papers on Stochastic-Process Limits for Many-Server Queueing Models.

Other Researchers Working on Asymptotics for Many-Server Queues.