Ruth Williams' current research concerns mathematical problems
stemming from the challenges of analyzing and controlling the dynamics of stochastic models of complex networks.
Such "stochastic networks" arise in a variety of applications in science and engineering, e.g.,
in systems biology, high-tech manufacturing, computer systems, telecommunications, transportation, and business service systems. These networks typically have entities, such as molecules, jobs, packets, vehicles, or customers, that move along paths or routes in a network, wait to receive processing from various resources, and that are subject to the effects of stochastic variability through such variables as arrival times, processing times, and routing protocols. Modern networks, such as the Internet, are often highly complex and heterogeneous, presenting interesting mathematical challenges for their analysis and control. Some aspects of Williams' work involve the development of general theory for broad classes of networks, while others focus on mathematical problems directly motivated by specific applications. As examples of the latter, Williams has recently analyzed models of the Internet to understand the effects of using fair bandwidth-sharing policies, and has developed theory to study coupled enzymatic processing in protein networks, in collaboration with researchers in synthetic biology.
Some stochastic process aspects of her research include justifying the approximation of density dependent Markov chains by reflected diffusion processes, analyzing measure-valued processes used to track residual job sizes or ages of jobs in stochastic network models with resource sharing, solving singular diffusion control problems, and addressing foundational questions for reflected processes.
To get a flavor of some of the problems Ruth Williams works on, you may wish to consult this survey article on Stochastic Processing Networks. This mathematical research involves advanced techniques in analysis and probability. Students wishing to work in this area are advised to obtain a strong background in real analysis and measure-theoretic probability, including stochastic processes/stochastic differential equations. An overv ew of Ruth Williams' research on Stochastic Networks and Reflecting Brownian Motion written by Ioana Dumitriu, Todd Kemp and Kavita Ramanan for the American Mathematical Society Notices is available here.