Scaling limits of random walks: combining analytic and probabilistic approaches

Summary This project is motivated by two related, areas of applied probability: queueing theory and insurance risk. In both areas, one-dimensional stochastic processes are well understood. However, in queueing theory one is usually faced with a network of interconnected and interacting resources, or with several classes of interacting customers; and thus one needs to study multidimensional stochastic processes. Similarly, in insurance risk one often needs to study several, related, books of a company, and the capitals of different companies are often linked via reinsurance contracts – again giving rise to multidimensional stochastic processes. In both areas, only few two- or higher-dimensional systems have allowed an exact analysis.

Our main goal is to develop mathematical tools for the analysis of two-dimensional stochastic processes, and thus get a much deeper insight into the intricate interactions which occur in, e.g., queueing systems and insurance risk processes. The project is methodologically oriented. Specifically we shall be looking at scaling limits of multidimensional reflected random walks in cases where the conventional approach in Stochastic network theory, using Skorokhod problems, breaks down. While one typically expects reflected Brownian motion (RBM) in the limit, the RBM process is not well defined - it may happen that RBM gets absorbed in the origin, for example. 

 The Dutch school on queueing theory is well positioned to make major advances in this area by combining probabilistic and asymptotic techniques. Specifically, it is possible to analyze two-dimensional random walks with analytic methods(boundary value problems), and to combine these with probabilistic techniques, as solutions to boundary value problems can be often interpreted probabilistically. Specifically, we investigate the invariant laws of two-dimensional coupled processor and polling models, both with light-tailed and heavy tailed input; these models are rich enough to lead to new qualitative phenomena. Both stationary and time-dependent behavior will be analyzed, by computing the time-dependent transform using conformal mapping approaches. Apart from scaling limits, which in queueing theory  determines how the system operates in heavy traffic, we are also interested in rare event. 

The research has possible algorithmic implications, in particular the dynamic control of interacting servers in heavy traffic.

Supervisors Michel Mandjes (UvA), Onno Boxma (TU/e)
PhD Student David Koops
Location University of Amsterdam (UvA)