Sebastian Allmeier

Abstract:

Mean Field methods have consistently been of interest to the scientific community due to their capacity to approximate a wide array of stochastic population systems. This methodology has proven to be a cornerstone in understanding and predicting the behavior of large-scale, complex systems. The key idea of the mean field approximation is to replace the complex stochastic behavior of systems with a simpler deterministic counterpart. The approximation therefore assumes that individuals become increasingly independent for large system sizes. The behavior of the system is thus obtained by replacing individual interactions with the average state of individuals. Despite its longstanding application and the advancements made in various fields, numerous questions and challenges remain open. In the scope of this thesis, we present our contributions and advancements for two general types of population models (1) heterogeneous mean field models and (2) mean field models with a fast-changing environment. In the first part of the thesis, we focus on stochastic systems with heterogeneous components. We consider two types of heterogeneity, individual-level diversity as well as graph-structured interactions. For both cases, we provide accuracy bounds for the expected state of finite-sized systems. The results are supported through practical examples, including cache replacement policies, supermarket models, load balancing, and bike-sharing systems, highlighting their computational tractability and precision. In the case of individual-level diversity, we further adapt the refined mean field idea and show that the refined approximation significantly reduces the error and provides accurate predictions for small to moderate-sized systems. In the second part of the thesis, we turn our focus to mean field approximation techniques for stochastic systems with a coupled, fast-changing environment. By studying the ‘average’ mean field approximation, we obtain accuracy bounds for the expected system state. Furthermore, we derive a bias refinement term, which increases the accuracy of the approximation. Expanding on these results, we extend the methodology to stochastic approximation with constant step size and state-dependent Markovian noise. We show how to adapt the ideas to obtain accuracy results and a computable bias extension.