Topic: Mathematical Methods for Stochastic Models of Complex Networks

Professor: Professor R. J. Williams, AP&M 7161.
Time: Tu, Th 5-6.30pm.
Place: AP&M 2402.

NOTE: There will be no class on Thu October 12, Tu October 24, Thu October 26. This class time is being made up with extra time for the other lectures and an additional make-up time.

Office Hour: Thursdays, AP&M 7161, 11-11.45 a.m. 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 biological 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, especially to biological 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, stationary distributions for Markov processes, fluid models and reflected diffusion processes.

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.

    Biochemical Reaction Networks

  • 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.

    References on Reflected Brownian Motions

  • Harrison, J. Michael; Reiman, Martin I. Reflected Brownian Motion on an Orthant. Ann. Probab. 9 (1981), no. 2, 302--308. doi:10.1214/aop/1176994471. Click here for the paper.
  • Reiman, M. I., and Williams, R. J., A boundary property of semimartingale reflecting Brownian motions, Probability Theory and Related Fields 77 (1988), 87-97. Link to paper. Correctional note link.
  • Taylor, L. M. and Williams, R. J., Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant, Probability Theory and Related Fields 96 (1993), 283-317, Click here to access the paper.
  • Dupuis, Paul; Ishii, Hitoshi. SDEs with Oblique Reflection on Nonsmooth Domains. Ann. Probab. 21 (1993), no. 1, 554--580. doi:10.1214/aop/1176989415. Click here for the paper. Correctional note: Ann. Probab. 36 (2008), no. 5, 1992--1997. doi:10.1214/07-AOP374. Click here for correctional note.
  • M. Shashiashvili, A lemma of variational distance between maximal functions with applications to the skorokhod problem in a nonnegative orthant with state-dependent reflection directions, Stochastics and Stochastics Reports, 48 (1994), 161-194. For the paper, click here.
  • R. J. Williams, Semimartingale reflecting Brownian motions in the orthant, Stochastic Networks, IMA Volumes in Mathematics and Its Applications, Volume 71, eds. F. P. Kelly and R. J. Williams, Springer-Verlag, New York, 1995, pp. 125-137. Survey up through 1995. For a copy click here.
  • S. Ramasubramanian, A subsidy-surplus model and the Skorokhod Problem in an orthant, Math of Operations Research, 25 (2000), 509-538.
  • Kang, W.; Williams, R. J. An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries. Ann. Appl. Probab. 17 (2007), no. 2, 741--779. doi:10.1214/105051606000000899. Click here for the paper.

    Stein's Method

  • Anton Braverman, J.G. Dai, Jiekun Feng, Stein's method for steady-state diffusion approximations: an introduction through the Erlang-A and Erlang-C models, Stochastic Systems, 2 (2016), 301-366. For a copy, click here.
  • Braverman, Anton; Dai, J. G. Stein's method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems. Ann. Appl. Probab. 27 (2017), 550--581. doi:10.1214/16-AAP1211. For a copy, click here.
  • Anton Braverman, Stein's method for steady-state diffusion approximations, Ph.D. thesis, 2017.
  • Nathan Ross, Fundamentals of Stein's method, Probability Surveys Vol. 8 (2011) 210-293. DOI: 10.1214/11-PS182. For a copy, click here.
  • A. D. Barbour and L. H. Y. Chen Eds: An introduction to Stein's method. IMS Lecture Note Series Volume 4, World Scientific Press, Singapore (2005) Publisher's website.
  • A. D. Barbour and L. H. Y. Chen Eds: Stein's method and applications IMS Lecture Note Series Volume 5, World Scientific Press, Singapore (2005) Publisher's website.
  • A. D. Barbour and L. H. Y. Chen, Stein's magic method.

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

    Last updated December 26, 2017.