Abstract and Applied Analysis
Volume 2013 (2013), Article ID 262581, 16 pages
http://dx.doi.org/10.1155/2013/262581
Research Article

Continuum Modeling and Control of Large Nonuniform Wireless Networks via Nonlinear Partial Differential Equations

Yang Zhang,1 Edwin K. P. Chong,1 Jan Hannig,2 and Donald Estep3

1Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523-1373, USA
2Department of Statistics and Operation Research, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3260, USA
3Department of Statistics, Colorado State University, Fort Collins, CO 80523-1373, USA

Received 4 January 2013; Revised 27 February 2013; Accepted 8 March 2013

Academic Editor: Lan Xu

Copyright © 2013 Yang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a continuum modeling method to approximate a class of large wireless networks by nonlinear partial differential equations (PDEs). This method is based on the convergence of a sequence of underlying Markov chains of the network indexed by , the number of nodes in the network. As goes to infinity, the sequence converges to a continuum limit, which is the solution of a certain nonlinear PDE. We first describe PDE models for networks with uniformly located nodes and then generalize to networks with nonuniformly located, and possibly mobile, nodes. 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.