Professor: Professor R. J. Williams, AP&M 6121.
Time: Tu, Th 2.30-3.50 p.m.
Place: AP&M 6218.

Office Hours: TBA

DESCRIPTION: Stochastic networks are used as models for complex manufacturing, telecommunications and computer systems. Since the complexity and heterogeneity of these networks usually preclude exact analysis, approximate models are frequently used. This course will discuss the mathematics associated with two classes of approximate models, namely, first order (functional law of large numbers) approximations called fluid models, and second order (functional central limit theorem) approximations which are frequently diffusion models. The interplay between these two levels will be an important subtheme throughout. The use of these approximate models for analysis and control of stochastic networks will also be presented. Associated mathematical topics that will be covered include weak convergence of stochastic processes, existence and uniqueness theory for reflected Brownian motions, positive recurrence of related processes.

PREREQUISITES: This is not a beginning graduate course. Knowledge gleaned from a graduate probability course such as Math 280, and some knowledge of stochastic processes, is advised. Although this course is labelled (B), there is no (A) course that is a prerequisite.


  • W. Whitt, Stochastic Process Limits, Springer, 2002.
  • H. J. Kushner, Heavy traffic analysis of controlled queueing and communication networks, Springer, 2001.
  • In addition to the above references, some notes will be made available during the course.

    Background Reference:

  • P. Billingsley, Convergence of Probability Measures, Wiley, 1999.
  • S. N. Ethier and T. Kurtz, Markov Processes, Wiley, 1986.

    Journal Articles:

  • S. Resnick and G. Samorodnitsky, A heavy traffic limit theorem for workload processes with heavy tailed service requirements, Management Science, 46 (2000), 1236--1248. For the abstract, click here.
  • M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits, Queueing Systems, 28 (1998), 7--31.
  • A. Mandelbaum, W. A. Massey and M. I. Reiman, Strong Approximations for Markovian Service Networks, Queueing Systems, 30 (1998), pp. 149-201.

    Links to Related Sites:

  • Stochastic Petri Nets and their use in Modeling Molecular Interactions.

    QUESTIONS: Please direct any questions to Professor Williams (