Validity of heavy traffic steady-state approximations in generalized Jackson networks

We consider a single class open queueing network, also known as a gen-eralized Jackson network (GJN). A classical result in heavy-traffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validat-ing a so-called “interchange-of-limits ” for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong ap-proximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN. 1. Introduction. Jackson networks are one of the most fundamental objects in the theory of stochastic processing networks, owing to the remarkable simplic