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.
@article{1141654281,
author = {Gamarnik, David and Zeevi, Assaf},
title = {Validity of heavy traffic steady-state approximations in generalized Jackson networks},
journal = {Ann. Appl. Probab.},
volume = {16},
number = {1},
year = {2006},
pages = { 56-90},
language = {en},
url = {http://dml.mathdoc.fr/item/1141654281}
}
Gamarnik, David; Zeevi, Assaf. Validity of heavy traffic steady-state approximations in generalized Jackson networks. Ann. Appl. Probab., Tome 16 (2006) no. 1, pp. 56-90. http://gdmltest.u-ga.fr/item/1141654281/