The Annals of Applied Probability

Validity of heavy traffic steady-state approximations in generalized Jackson networks

David Gamarnik and Assaf Zeevi

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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.

Article information

Source
Ann. Appl. Probab. Volume 16, Number 1 (2006), 56-90.

Dates
First available in Project Euclid: 6 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1141654281

Digital Object Identifier
doi:10.1214/105051605000000638

Mathematical Reviews number (MathSciNet)
MR2209336

Zentralblatt MATH identifier
1094.60052

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60J65: Brownian motion [See also 58J65] 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Diffusion approximations stationary distribution weak convergence Lyapunov functions Markov processes reflected Brownian motion

Citation

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.


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