  In this paper we investigate the continuum limits of a class of Markov
chains. The investigation of such limits is motivated by the desire to model
very large networks. We show that under some conditions, a sequence of Markov
chains converges in some sense to the solution of a partial differential
equation. Based on such convergence we approximate Markov chains modeling
networks with a large number of components by partial differential equations.
While traditional Monte Carlo simulation for very large networks is practically
infeasible, partial differential equations can be solved with reasonable
computational overhead using well-established mathematical tools.
