Abstract: In this research 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 some 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. Our approach directly approximates
the state of Markov chains at specific temporal
and spatial points by continuum limits that are
deterministic functions, in contrast to
approximation methods that study the proportion of
components in certain states, and to those that
study the distribution of continuum limits that
are random processes. As an application we
approximate Markov chains modeling large networks
by PDEs, thus capturing the dynamic time-space
characteristics of the networks. 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.
Adviser: Edwin K. P. Chong Co-Adviser: N/A Non-ECE Member: Donald Estep Member 3: J. Rockey Luo Addional Members: Ali Pezeshki
Publications: 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.
Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, "Continuum limits of Markov chains and network modeling," presented at The Mathematical Association of America Rocky Mountain Section Meeting, 2010.
Y. Zhang and E. K. P. Chong, "Continuum models of large networks," presented in the poster session of the Graduate Student Visit Day of the ECE Dept., Colorado State University, 2010.
Y. Zhang, E. K. P. Chong, J. Hannig, and D. J. Estep, "Continuum limits of Markov chains with application to network modeling," arXiv:1106.4288v1, 2011.
Program of Study: ECE 514 ECE 516 ECE 520 ECE 752 ECE 799 MATH 545 STAT 720 N/A