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

We consider a single class open queueing network, also known as a generalized 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 Validating a so-called "interchange-of-limits" for this class of networks, Our method of proof involves a combination of Lyapunov function techniques. strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.