Kavita Ramanan is a Professor at the Division of Applied Mathematics at Brown University. She is a fellow of the IMS and a recipient of the Erlang Prize of the INFORMS Applied Probability Society. She received her PhD from Brown University in 1998, was a post-doctoral fellow at the Technion, Israel, and has had appointments as a Member of Technical Staff at Bell Labs, a Professor at the Department of Mathematical Sciences at Carnegie Mellon University, and an Adjunct Professor at the Chennai Mathematical Institute. She has served on the editorial boards of several journals, including the Annals of Probability and the Annals of Applied Probability, and has been granted several patents.

Kavita Ramanan’s current research interests include stochastic analysis, large deviations, Gibbs measures and applications to stochastic networks.

Kavita will present her IMS Medallion lecture at the INFORMS Applied Probability Society conference to be held from July 5–8, 2015, at Koç University in Istanbul, Turkey. See http://home.ku.edu.tr/~aps2015/

Infinite-dimensional scaling limits of stochastic networks

Models of stochastic networks arise in a wide variety of applications, ranging from telecommunications and computer systems to manufacturing and service centers. These networks typically consist of jobs, e.g., in the form of customers or packets, that require processing from multiple servers in the network, and are routed through the network and stored in queues while awaiting service. Different measures of network performance, such as stability, mean queue lengths and probability of large delays, are relevant for different applications. The goal of the mathematical theory of stochastic networks is to develop general techniques for the analysis of the performance, design and control of broad classes of networks.

Given the complexity of these networks, an exact analysis is often infeasible. Instead, valuable insight can often be gained from the study of approximate models that are more tractable and can be rigorously justified via a limit theorem to be exact in a suitable asymptotic regime. These “scaling limits” and the methods developed to analyze them are often of independent interest and useful in other areas of probability. While much of the research in the last two decades was devoted to the study of so-called multi-class queueing networks, whose scaling limits are characterized by reflected diffusions in non-smooth domains, more recently the focus has shifted to the study of more complex networks.

In my lecture, I will describe mathematical techniques developed to study two classes of networks that have been the focus of my recent work: single-server networks that use a certain class of scheduling policies that employ a continuous parameter to prioritize the service of different jobs, and a class of many-server networks that arise as models of randomized load-balancing. A common feature that makes the analysis of both classes of models challenging is that the natural Markovian state space for establishing scaling limits of these models is infinite-dimensional.

In the context of single-server networks, it turns out that a certain infinite-dimensional Skorokhod map enables a unified analysis of several different scheduling policies in this class under quite general assumptions. For the class of many-server models, the analysis is much easier when the service distribution is exponentially distributed. However, the fact that service distributions are rarely exponentially distributed in practice leads to a natural set of questions: what is a suitable representation for the dynamics when the service distribution is general? What new tools are require to analyze scaling limits? Does the qualitative behavior of the network significantly differ from the exponential case?

My lecture will provide some answers: representations in terms of a system of interacting measure-valued stochastic processes appear to be fruitful; “hydrodynamic limits”, which capture mean behavior, can be characterized by (a countable system of) partial differential equations; “diffusion approximations” that characterize fluctuations around the mean, may be described by certain (non-standard) coupled stochastic partial differential equations.

Moreover, I will demonstrate via concrete examples that, despite being infinite-dimensional, the scaling limits can be analyzed to provide interesting (and sometimes counter-intuitive) insight into the behavior of the original network.

Finally, I will also describe some remaining challenges and open problems.