Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic

Abstract

In a recent paper [5] 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 making (in addition to natural heavy traffic and stability assumptions) only standard independence and square integrability assumptions 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.

Heavy-Traffic Limit of the GI/GI/1 Stationary Departure Process and Its Variance Function

“…To do so, we apply the recent HT results for the stationary queue length (number in system) in Gamarnik and Zeevi (2006) and Budhiraja and Lee (2009) together with the HT limits for the general singleserver queue in Whitt (2002, Section 9.3) and the general reflection mapping with nonzero initial conditions in Whitt (2002, Section 13.5). As in Whitt (2002), a major component of the proof is the continuous mapping theorem.…”

“…In order to establish a tractable HT limit for the stationary departure process, we apply the recent HT limits for the stationary queue length in Gamarnik and Zeevi (2006) and Budhiraja and Lee (2009) …”

“…First, recall that the HT limit as ρ → 1 starting empty is the RBM that converges as t → ∞ to the exponential distribution in (60). We will be applying Gamarnik and Zeevi (2006) and Budhiraja and Lee (2009) to show that the two iterated limits involving ρ → 1 and t → ∞ are equal. Toward that end, we observe that, by Asmussen (2003, Sections X.3-X.4), the queue-length process has a proper steady-state distribution for each ρ.…”

“…What is needed is the extension of Gamarnik and Zeevi (2006) and Budhiraja and Lee (2009) to more general models. We conjecture that can be done for GI/GI/s and other models with regenerative structure in the arrival and service processes.…”

“…The key to the process limit here is the recent HT limits for the stationary vector of queue lengths in a generalized Jackson network in Gamarnik and Zeevi (2006) and the extension in Budhiraja and Lee (2009). The existence of the HT limit for the scaled variance function is a relatively direct consequence.…”