Rigorous Approximations of Stochastic Network Dynamics, with Applications to Real-World Networks

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Stochastic networks are comprised of "jobs" in the form of packets or customers that arrive to a network and wait in buffers at different nodes of the networks until their processing requirements are fulfilled. Stochastic variability arises from randomness in arrival and processing times, as well as from routing and scheduling decisions. Such networks are ubiquitous and arise as models in diverse fields ranging from telecommunications and service systems to biological systems. A better understanding of these networks has the potential to lead to new algorithms that dramatically improve performance and enable the support of novel network applications. This award supports the development of a general mathematical framework for the analysis of two broad classes of stochastic networks: large-scale load-balancing networks that arise, for example, in web-server farms, and queueing networks that use scheduling policies involving prioritization, which are relevant for real-time scheduling in computer networks and health care systems. The goal is to identify tractable approximations of both transient dynamics and equilibrium behavior, rigorously establish their accuracy for suitable values of network parameters, and to use them to gain insight into network design. There will be mentorship of graduate students, a multidisciplinary project for an undergraduate student and opportunity for outreach. The project also involves interactions with industry, which increases the potential of impacting the design of real networks.Stochastic networks are typically too complex to admit an exact analysis. The goal then is to obtain tractable approximations, whose accuracy can be rigorously justified in a suitable (asymptotic) regime of network parameters via limit theorems for suitably scaled state processes. While there is a well developed mathematical theory of "scaling limits" for certain classes of networks, it does not cover the networks considered in this project, which involve many servers at a single node, general service distributions and non-head of the line scheduling policies. One of the challenges is that Markovian scaling limits of these networks are typically not finite-dimensional. The research will develop tractable infinite-dimensional Markovian representations of these systems, involving interacting measure-valued processes and infinite-dimensional Skorokhod maps, and rigorously establish scaling limit theorems. The analysis will combine methods from different fields, including probability, stochastic analysis, dynamical systems, partial differential equations and optimization. The tools developed will potentially be applicable more broadly for the study of analogous stochastic models arising in other applications. Simulations will also be carried out to ascertain the validity of these approximations for finite systems.