The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 16, Number 1 (2006), 56-90.
Validity of heavy traffic steady-state approximations in generalized Jackson networks
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.
Ann. Appl. Probab., Volume 16, Number 1 (2006), 56-90.
First available in Project Euclid: 6 March 2006
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Gamarnik, David; Zeevi, Assaf. Validity of heavy traffic steady-state approximations in generalized Jackson networks. Ann. Appl. Probab. 16 (2006), no. 1, 56--90. doi:10.1214/105051605000000638. https://projecteuclid.org/euclid.aoap/1141654281