Analysis of Large Scale Interacting Systems by Mean Field Method

Modeling and analysing very large stochastic systems composed of interacting entities is a very challenging and complex task. The usual approach, relying on the generation of the whole state space, is bounded by the state space explosion, even if symmetry properties, often included in the model, allow to apply lumping techniques and building the overall model by means of tensor algebra operations. In this paper we resort to the mean field theory. The main idea of the mean field theory is to focus on one particular tagged entity and to replace all interactions with the other entities with an average or effective interaction. The reduction of a multibody problem into an effective one-body problem makes the solution easier while at the same time taking into account the contribution of an averaged interdependence of the whole system on the specific entity. We apply the mean field approach to very large systems of interacting continuous time Markov chains, in which the averaged interaction depends on the distribution of the entity population in each state. We report several examples of interacting Markovian queues, showing the potentialities of the proposed technique.