Validity of heavy traffic steady-state approximations in generalized Jackson networks
Open Access
- 1 February 2006
- journal article
- Published by Institute of Mathematical Statistics in The Annals of Applied Probability
- Vol. 16 (1), 56-90
- https://doi.org/10.1214/105051605000000638
Abstract
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.Other Versions
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