Mean Field Approximation of Uncertain Stochastic Models

Abstract

We consider stochastic models in presence of uncertainty, originating from lack of knowledge of parameters or by unpredictable effects of the environment. We focus on population processes, encompassing a large class of systems, from queueing networks to epidemic spreading. We set up a formal framework for imprecise stochastic processes, where some parameters are allowed to vary in time within a given domain, but with no further constraint. We then consider the limit behaviour of these systems as the population size goes to infinity. We prove that this limit is given by a differential inclusion that can be constructed from the (imprecise) drift. We provide results both for the transient and the steady state behaviour. Finally, we discuss different approaches to compute bounds of the so-obtained differential inclusions, proposing an effective control-theoretic method based on Pontryagin principle for transient bounds. This provides an efficient approach for the analysis and design of large-scale uncertain and imprecise stochastic models. The theoretical results are accompanied by an in-depth analysis of an epidemic model and a queueing network. These examples demonstrate the applicability of the numerical methods and the tightness of the approximation.