2015/10/03 by Lei Ying, Ying, Lei
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #FOS: Computer and information sciences #FOS: Electrical engineering #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Performance (cs.PF) #Probability (math.PR) #Random Matrices and Applications #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1510.00761
openalex publication_date 2015/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 Mth 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 Mth 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.