  This paper studies the rate of convergence of a family of continuous-time
Markov chains (CTMC) to a mean-field model. When the mean-field model is a
finite-dimensional dynamical system with a unique equilibrium point, an
analysis based on Stein's method and the perturbation theory shows that under
some mild conditions, the stationary distributions of CTMCs converge (in the
mean-square sense) to the equilibrium point of the mean-field model if the
mean-field model is globally asymptotically stable and locally exponentially
stable. In particular, the mean square difference between the $M$th CTMC in the
steady state and the equilibrium point of the mean-field system is $O(1/M),$
where $M$ is the size of the $M$th CTMC. This approach based on Stein's method
provides a new framework for studying the convergence of CTMCs to their
mean-field limit by mainly looking into the stability of the mean-field model,
which is a deterministic system and is often easier to analyze than the CTMCs.
More importantly, this approach quantifies the rate of convergence, which
reveals the approximation error of using mean-field models for approximating
finite-size systems.
