Topic: Stochastic Networks

Professor: Professor R. J. Williams, AP&M 7161.
Time: Tu, Th 4-5.30pm
Place: AP&M 5402.

NOTE: There will be no class on Tuesday, April 29. This class time is being made up with extra time for the other lectures.

Office Hour: Thursday, 2-3pm and by appointment.

Background: Stochastic models of complex networks arise in a wide variety of applications in science and engineering. Specific instances include high-tech manufacturing, telecommunications, computer systems, service systems and biochemical reaction networks. There are challenging mathematical problems stemming from the need to analyse and control such networks.

Content: This course will describe some general mathematical techniques for modeling stochastic networks, for deriving approximations at various scales (especially deterministic differential equation and diffusion approximations), and for analyzing the behavior of these models. Applications to queueing networks and biochemical reaction networks will be used to illustrate the concepts developed.

The mathematical topics covered will include path spaces for stochastic processes, weak convergence of processes, fundamental building block processes and invariance principles, quasireversibility and stationary distributions for Markov processes, reflected Brownian motion and Skorokhod problems.

Prerequisites: This is an advanced graduate probability course featuring a topic of current research interest. Students enrolling in this course should have a background in probability at the level of Math 280AB (courses such as Math 285, Math 280C, Math 286 provide additional useful background).


Background References:

  • P. Billingsley, Convergence of Probability Measures, Wiley, 1999.
  • S. N. Ethier and T. Kurtz, Markov Processes, Wiley, 1986.
  • J. Jacod and A. Shiryaev, Limit theorems for stochastic processes, 2nd edition, Springer-Verlag, New York, 2003.
  • F. P. Kelly, Reversibility and Stochastic Networks, Wiley, Chichester, 1979, reprinted 1987, 1994.
  • W. Whitt, Stochastic Process Limits, Springer, 2002.

    Some Papers:

  • A. Mandelbaum, W. A. Massey, M. I. Reiman, Strong Approximations for Markovian Service Networks, Queueing Systems, 30 (1998), pp. 149-201.
  • T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic processes and their applications, 6 (1978), 223-240.
  • Karen Ball, Tom Kurtz, Lea Popovic, Greg Rempala, Asymptotic analysis of multiscale approximations to reaction networks, Ann. Appl. Probab. 16 (2006), 1925-1961.
  • D. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Physical Chemistry, Vol. 81, 1977, 2340--2361.
  • Martin Feinberg, Lecture notes on chemical reaction networks, 1979.
  • Jeremy Gunawardena, Chemical Reaction Network Theory for in-silico Biologists, lecture notes, 2003.
  • F. Horn and R. Jackson, General mass action kinetics, Arch. Rat. Mech. Anal., 47 (1972), 81-116.
  • M. Feinberg, Complex balancing in general kinetic systems, Arch. Rat. Mech. Anal., 49 (1972), 187-194.
  • D. Anderson, Global Asymptotic Stability for a Class of Nonlinear Chemical Equations, SIAM J. Appl. Math. Volume 68, Issue 5, pp. 1464-1476 (2008).
  • D. F. Anderson, G. Craciun, T. G. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, preprint.
  • G. Craciun, J. W. Helton and R. J. Williams, Homotopy methods for counting reaction network equilibria, to appear in Mathematical Biosciences. Also see references on the Software webpage linked from that for the CHW paper.
  • J. M. Harrison and M. I. Reiman, Reflected Brownian motion in an orthant, Annals of Probability, 9 (1981), 302-308.
  • D. F. Anderson and T. G. Kurtz, Continuous time Markov chain models for chemical reaction networks, chapter in Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology, H. Koeppl. et al. (eds.), Springer.
  • C. Gadgil, C. H. Lee and H. G. Othmer, A stochastic analysis of first-order reaction networks, Bull. Math. Biology, 2005 (67), 901--946, Elsevier.

    Questions: Please direct any questions to Professor Williams (williams at math dot ucsd dot edu).

    Last updated April 1, 2014.