Kinetic Limits for Large Communication Networks

Communication networks are modeled using Markov processes, which have huge state spaces even in very simple cases. Relevant quantities are then given in terms of intractable linear systems, and practical and precise approximations must be found. We let the network size go to infinity as the offered load at each site is kept fixed, and seek to obtain a tractable limit, for which the sites become independent, with evolutions satisfying decoupled nonlinear problems. We study such chaoticity results on path space, for chaotic initial conditions as well as in equilibrium, which imply laws of large numbers. The main tools are martingale problems and characterizations, compactness-uniqueness arguments, random graph and tree constructions, couplings, and the study of the long-time behavior of kinetic equations. Phase transition may occur at the limit, indicating metastability of the finite systems. Rates of convergence are given both directly, and through functional central limit and large deviation theorems.

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