MATH 289A: TOPICS IN PROBABILITY (FALL 2017)
  • Lecture 1: Introduction to Stochastic Processing Networks; Simple example- M/M/1 queue, GI/GI/1 queue - fluid model.
    A reference on Little's Law.
  • Lectures 2-3: GI/GI/1 diffusion approximation; simple biochemical reaction network example, M/M/infinity queue, fluid model. Summary of results on convergence of stochastic processes (for notes on this, click here).
    Extra reading: notes by David Aldous on weak convergence and general theory of processes.
  • Lecture 4: Sample Use of Summary Results - Fluid limit for GI/GI/1 queue
  • Lecture 5: Fluid model for GI/GI/1 queue, stability.
    Reference notes on Stability of Queueing Networks via Fluid Models, by Maury Bramson. Note equation (1.17) has a typo.
  • Lecture 6: Reference for proof of the Skorokhod representation theorem -- from Ethier and Kurtz, Markov Processes.
  • Lecture 7: Stability via fluid limits (Dai's thm for GI/GI/1 queue) and diffusion approximation for GI/GI/1 queue.
  • Lecture 8: One-Dimensional Skorokhod Problem.
  • Lecture 9: Multi-Dimensional Skorokhod Problem for a generalized M-matrix.
  • Lecture 10: Multi-Dimensional Skorokhod Problem for a completely-S matrix, counterexample to uniqueness for a P-matrix.
  • Lecture 11: Weak existence and uniqueness of SRBM. Fluid limit for a single class open queueing network.
  • Lecture 12: Stability and diffusion approximation for a single class open queueing network.
  • Lecture 13 and 14: Stability and diffusion approximations for multiclass HL open queueing networks. Click here for papers by Bramson and Williams related to proving a diffusion approximation for a multiclass HL queueing network. For another illustration of using multiplicative state space collapse to prove a diffusion approximation, see the paper by Kang, Kelly, Lee and Williams (2009), by clicking here. This is perhaps a good place to start in reading a multiplicative state space collapse arguement for the first time. A paper that uses this approach is Bramson and Dai, Heavy Traffic Limits for Some Queueing Networks, Annals of Applied Probability, 11 (2001), 49-90.
  • Lecture 15: Presentation on SRBM analysis. Biochemical Reaction Networks: setup, stochastic model. For a summary of Markov chain models for biochemical reaction networks, see 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.
  • Lecture 16: Presentation of paper by Pal and Pitman. More on biochemical reaction networks - propensities.
  • Lecture 17: Fluid limit of a biochemical reaction network model.
  • Lecture 18: van Kampen diffusion approximation for a biochemical reaction network model.
    Possible papers to present.
  • 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.
  • S. Pal and J. Pitman, One-dimensional Brownian particle systems with rank-dependent drifts, Annals of Applied Probability, 18 (2008), 2179-2207.