Mathematics – Probability
Scientific paper
2004-10-04
Annals of Applied Probability 2006, Vol. 16, No. 1, 56-90
Mathematics
Probability
Published at http://dx.doi.org/10.1214/105051605000000638 in the Annals of Applied Probability (http://www.imstat.org/aap/) by
Scientific paper
10.1214/105051605000000638
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.
Gamarnik David
Zeevi Assaf
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