When considering a Markov process with two distinct component, it can happen that they evolve on different time scales (one is fast and the other is slow). In some cases, we can hope that the fast one "averages" so that it only influences the slow one through its steady state mean. When that is the case we say there is a separation of time scales (or homogenization).
During this talk, I will present a work in progress, developing a novel method to obtain condition on the joint dynamic to to establish a time scale separation using functional inequalities (Log-Sobolev and Gronwall). As an application example we'll expose different scheduling algorithm for wireless networks with a focus on CSMA-Queue based algorithms. We will also see how to derive stability using homogenization in this case.