Continuum limits of Markov chains with application to wireless network modeling and control - Zhang, Yang, author; Chong, Edwin K. P., advisor; Estep, Donald, committee member; Luo, J. Rockey, committee member; Pezeshki, Ali, committee member (Colorado State University. Libraries, 2014)
We investigate the continuum limits of a class of Markov chains. The investigation of such limits is motivated by the desire to model networks with a very large number of nodes. We show that a sequence of such Markov chains indexed by N , the number of components in the system that they model, converges in a certain sense to its continuum limit, which is the solution of a partial differential equation (PDE), as N goes to infinity. We provide sufficient conditions for the convergence and characterize the rate of convergence. As an application we approximate Markov chains modeling large wireless networks by PDEs. We first describe PDE models for networks with uniformly located nodes, and then generalize to networks with nonuniformly located, and possibly mobile, nodes. While traditional Monte Carlo simulation for very large networks is practically infeasible, PDEs can be solved with reasonable computation overhead using well-established mathematical tools. Based on the PDE models, we develop a method to control the transmissions in nonuniform networks so that the continuum limit is invariant under perturbations in node locations. This enables the networks to maintain stable global characteristics in the presence of varying node locations.
Keywords: partial differential equations; network modeling; modeling; Markov processes