RESEARCH IN PROBABILITY: R. J. WILLIAMS
For a brief description of Ruth Williams' current research interests, click here.
Google Scholar Citations for Ruth J Williams are here.
LIST OF PUBLICATIONS
ABSTRACTS AND PREPRINTS OF RECENT RESEARCH
  • Error bounds for one-dimensional constrained Langevin approximations for nearly density dependent Markov chains
  • Comparison Theorems for Stochastic Chemical Reaction Networks
  • Epigenetic cell memory: The gene's inner chromatin modification circuit, PLoS Computational Biology, April 2022, 18(4): e1009961.
  • Asymptotic Behavior of a Critical Fluid Model for Bandwidth Sharing with General File Size Distributions
  • Limit Theorems and Ergodicity for General Bootstrap Random Walks
  • Stability of a Subcritical Fluid Model for Fair Bandwidth Sharing with General File Size Distributions
  • Fluid Model of a Traffic Network with Information Feedback and Onramp Controls
  • Asymptotic Behavior of a Critical Fluid Model for a Multiclass Processor Sharing Queue via Relative Entropy
  • On Constrained Langevin Equations and (Bio)Chemical Reaction Networks (introductory paper)
  • A constrained Langevin approximation for chemical reaction networks
  • Asymptotic behavior of a critical fluid model for a processor sharing queue via relative entropy
  • Criticality and adaptivity in enzymatic networks
  • Existence, uniqueness and stability of slowly oscillating periodic solutions for delay differential equations with non-negativity constraints
  • Stochastic Processing Networks
  • Dynamic Scheduling for Parallel Server Systems in Heavy Traffic: Graphical Structure, Decoupled Workload Matrix and Some Sufficient Conditions for Solvability of the Brownian Control Problem
  • Translational cross talk in gene networks.
  • Diffusion approximation for an input-queued switch operating under a maximum weight matching policy
  • Queueing up for enzymatic processing: correlated signaling through coupled degradation
  • Factorized time-dependent distributions for certain multiclass queueing networks and an application to enzymatic processing networks
  • Correlation resonance generated by coupled enzymatic processing
  • Heavy traffic on a controlled motorway
  • On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints
  • Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation
  • Homotopy methods for counting reaction network equilibria
  • State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy
  • HJB equations for certain singularly controlled diffusions
  • Fluid model for a data network with alpha fair bandwidth sharing and general document size distributions: two examples of stability.
  • On the performance of a two-user downlink system in heavy traffic.
  • Fluid limits for networks with bandwidth sharing and general document size distributions.
  • An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries.
  • Workload interpretation for Brownian models of stochastic processing networks.
  • The fluid limit of an overloaded processor sharing queue.
  • Product form stationary distributions for diffusion approximations to a flow level model operating under a proportional fair sharing policy (MAMA 07 - to appear in Performance Evaluation Review)
  • On fluid and Brownian approximations for an Internet congestion control model.
  • Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: asymptotic optimality of a threshold policy.
  • Workload reduction of a generalized Brownian network.
  • Fluid model for a network operating under a fair bandwidth-sharing policy.
  • Two workload properties for Brownian networks.
  • Diffusion approximation for a processor sharing queue in heavy traffic by H. C. Gromoll.
  • Invariant states and rates of convergence for a critical fluid model of a processor sharing queue.
  • The fluid limit of a heavily loaded processor sharing queue
  • On dynamic scheduling of a parallel server system with complete resource pooling
  • Dynamic scheduling of a system with two parallel servers in heavy traffic with complete resource pooling: asymptotic optimality of a continuous review threshold policy
  • Stabilization of stochastic nonlinear systems driven by noise of unknown covariance
  • Reflecting diffusions and queueing networks
  • Heavy traffic limits for open multiclass queueing networks
  • On K. L. Chung's Research on Stopped Feynman-Kac Functionals and the Schrodinger Equation.
    SNAPSHOTS: Some selected snapshots of recent research
  • Queueing Up for Enzymatic Processing
  • Dynamic Scheduling for Parallel Server Systems
  • American Mathematical Society Notices article on Stochastic Networks and Reflecting Brownian Motion: The Mathematics of Ruth Williams.
    TALKS
  • 40 years of reflected Brownian motion and related topics, April 2023.
    1. RBM and Queueing Theory: An Early History, Part 2.
    2. Biochemical reaction networks and reflected diffusions.
  • Australian and New Zealand Industrial and Applied Mathematics Conference (ANZIAM), February 2022, plenary lecture, Stochastic Networks: Bottlenecks, Entrainment and Reflection.
    For a video of this talk, please click here.
  • Stochastic Networks, Applied Probability and Performance Seminar (SNAPP), May 24, 2021. For slides from the talk, click here.
  • Wasserstrom Distringuished Lecture, Northwestern University, May 2021.
    Slides are available here. here. >
  • Frontier Probability Days, Oregon State University, March 2018.
  • Two talks given at the 39th Midwest Probability Colloquium, Northwestern University, October 2017. Overall title: Reflected diffusions, stochastic networks and biology.
    First talk: Reflected Diffusions and (Bio)Chemical Reaction Networks.
    Second talk: Criticality and Adaptivity in Enzymatic Networks.
  • Queueing up for enzymatic processing: correlations through coupled degradation, lecture at the Synthetic Biology Summer School held at UCSD as part of the Sixth q-bio Summer School, August 2012.
  • The Mathematics of Queueing. Talk given at 54th Annual High School Honors Mathematics Contest Awards Banquet, sponsored by the Greater San Diego Mathematics Council and UCSD, April 2011.
  • A Random Walk Through Analysis, Networks and Biology, 3 Lectures given at Gonville and Caius College, Cambridge as G. C. Steward Visiting Fellow, April 2010.
  • Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation, Stochastic Networks meeting, Paris, June 2008.
  • Markov Lecture, Annual INFORMS meeting, Seattle, November 2007. For a pdf copy of this talk, click here. For a recent paper related to this talk, click here.
  • 3rd Cornell Probability Summer School, June 2007: Reflecting Brownian motions.
    Lecture 1: Multiclass Queueing Networks and Semimartingale Reflecting Brownian Motions in the Orthant.
    Click here for a copy with white background and 6 to a page for easy printing.
    Lecture 2: Stochastic Processing Networks and Semimartingale Reflecting Brownian Motions in Piecewise Smooth Domains.
    Click here for a copy with 6 to a page for easy printing.
    (Copyright Ruth Williams 2007.)
  • Stochastic Processing Networks, a lecture series at the University of Melbourne, September 4-6, 2006. Jointly sponsored by the Maurice H. Belz trust and MASCOS. (Copyright Ruth Williams 2006. All print and electronic rights and use rights reserved. Personal, non-co mmerical use only, for individuals with permission from author --- write to williams@math.ucsd.edu for this.)
  • A measure-valued process in a bandwidth sharing model with general document size distributions, Madison, WI, July 2006. (Copyright Ruth Williams 2006. All print and electronic rights and use rights reserved. Personal, non-commerical use only, for ind ividuals with permission from author --- write to williams@math.ucsd.edu for this.)
  • Input Queued Crossbar Switch Operating under a Maximum Weight MatchingPolicy in Heavy Traffic, Stochastic Networks meeting, Urbana, June 2006. (Copyright Ruth Williams 2006. All print and electronic rights and use rights reserved. Personal, non-commerical use only, for individuals with permission from author --- write to williams@math.ucsd.edu for this.)
  • Hotelling Lecture Series, University of North Carolina, Chapel Hill, November 1-5, 2004. (Copyright Ruth Williams 2004. All print and electronic rights and use rights reserved. Personal, non-commerical use only, for individuals with permission from author --- write to williams@math.ucsd.edu for this.)
  • Transparencies from talk given at IMA Workshop on Control and Pricing in Communication and Power Networks, March 8-13, 2004. (Copyright Ruth Williams 2004. All print and electronic rights and use rights reserved. Personal, non-commerical use only, for individuals with permission from author --- write to williams@math.ucsd.edu for this.)
    Fluid and Brownian Models of Congestion at Flow Level.

  • Transparencies from tutorial given at SAMSI Workshop, October 31, 2003. (Copyright Ruth Williams 2003. All print and electronic rights and use rights reserved. Personal, non-commerical use only, for individuals with permission from author --- write to williams@math.ucsd.edu for this.)
    Fluid and diffusion approximations for stochastic processing networks, and a flow level model of congestion control.

  • Transparencies from lectures given at the Eight Summer Workshop of the New Zealand Mathematics Research Institute, Napier, January 6-11, 2002. (Copyright Ruth Williams 2002. All print and electronic rights and use rights reserved. Personal, non-commerical use only, for individuals with permission from author --- write to williams@math.ucsd.edu for this.)
    -Performance analysis for multiclass queueing networks via reflecting diffusions.
    -Dynamic scheduling for stochastic networks in heavy traffic.

  • La Trobe University, Bendigo, G.S. Watson Annual Lecture, 1999
    "From Queueing Networks to Reflecting Diffusions", by Professor Ruth     Williams,
    July 19, 1999, 4pm to 5pm, Education Lecture Theatre, La Trobe University, Bendigo, Australia.

  • Transparencies from talk at the Applied Probability Day, Columbia University, April 23, 1999. Multiclass queueing networks and reflecting diffusion processes.
    (Copyright Ruth Williams 1999. All print and electronic rights and use rights reserved. Personal, non-commerical use only, for individuals with permission from author --- write to williams@math.ucsd.edu for this.)
    ACKNOWLEDGEMENT
    The research of Ruth Williams is supported in part by the National Science Foundation and the National Institutes of Health. Any opinions, findings and conclusions or recommendations expressed in the material found on these web pages are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF) or the National Institutes of Health (NIH). Some of the research of Ruth Williams has also been partially supported by a gift from the David and Holly Mendel Fund, a gift from AT&T Bell Labs, a John Simon Guggenheim Fellowship and the Charles Lee Powell Chair in Mathematics I.
    PHD STUDENTS: MATHEMATICS GENEALOGY PROJECT
    Includes a list of Ph.D. students of R. J. Williams.


    Professor Williams discussing research with Ph.D. students Yingjia Fu and Felipe Campos-vergara (images courtesy of UC San Diego Publications).
    NEWS ITEMS RELATED TO RESEARCH IN PROBABILITY AND STOCHASTIC PROCESSES
  • Expository article describing the role of mathematics in modeling networks, especially communication and transport networks (produced by the Institute of Mathematics and the EPSRC, United Kingdom, circa 2008).
  • You Tube video on the use of queueing theory for management of checkout service (2010).
  • Impact of variable speed limits on traffic congestion, The Jam Busters: Can maths keep us moving?, article in the UK Independent.
    USEFUL INFORMATIONAL LINKS
  • Maury Bramson's homepage with access to his 2006 Saint Flour lecture notes on stability of queueing networks. For a direct link to the notes, click here.
  • Avi Mandelbaum's site on Service Engineering, with particular emphasis on call centers.
  • Ward Whitt's page with links to Call Center related websites, including Avi Mandelbaum's Research Bibliography on Call Centers.
  • Nosokinetics - The mathematics of patient flow through health and social care systems.
  • Student's 1927 paper entitled "Errors of Routine Analysis" (see especially page 10 for a memory aid concerning kurtosis).
  • Systems and Synthetic Biology - links to some researcher web pages.
  • Brownian motion: a description of aspects of this process and some of its applications
  • Links to some web sites of researchers in Stochastic Networks
  • Ernst Hairer's website with links to code for numerical approximation of ordinary differential equations with delay.
  • Matthew Peet's website with links to matlab code for numerical approximation of differential equations with delays.