Abstract: 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.

Adviser: Edwin K. P. Chong
Co-Adviser: N/A
Non-ECE Member: Don Estep, Stat
Member 3: J. Rockey Luo, ECE
Addional Members: Ali Pezeshki, ECE

Publications:
Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, "Approximating Extremely Large Networks via Continuum Limits, IEEE Access, Accepted, 2013
Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, "Continuum modeling and control of large nonuniform wireless networks via nonlinear partial differential equations," Abstract and Applied Analysis, vol. 2013, Article ID 262581, 16 pages, 2013. doi:10.1155/2013/262581.
Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, "Continuum modeling and control of large nonuniform networks," in the 49th Annual Allerton Conference on Communication, Control, and Computing, pp. 1670-1677, 28-30 Sept. 2011.
Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, "On continuum limits of Markov chains and network modeling," in Decision and Control (CDC), 2010 49th IEEE Conference on, pp.6779-6784, 15-17 Dec. 2010.

Program of Study:
ECE 514
ECE 516
ECE 520
ECE 752
ECE 799
MATH 545
STAT 720
N/A