Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic

In a recent paper, Gamarnik and Zeevi [Gamarnik, D., A. Zeevi. 2006. Validity of heavy traffic steady-state approximations in open queueing networks. Ann. Appl. Probab.16(1) 56--90], it was shown that under suitable conditions stationary distributions of the (scaled) queue-lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certain exponential integrability assumptions on the primitives of the network. In this note we show that the above result holds under much weaker integrability conditions. We provide an alternative proof of this result assuming (in addition to natural heavy traffic and stability assumptions) only standard independence and square integrability conditions on the network primitives that are commonly used in heavy traffic analysis. Furthermore, under additional integrability conditions we establish convergence of moments of stationary distributions.