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

By David Gamarnik, Assaf J. Zeevi. 2004

197 citations recorded by the source corpus.

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.

Subfields: Control and Systems Engineering, Management Information Systems, Transportation

View Paper: https://www.semanticscholar.org/paper/86a138b5164b60a42d471b36b326239b12b75261

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Citations in the corpus

Heavy traffic analysis for EDF queues with reneging (2011, result): “a result that grows out of the work of Kingman [19], [20]; see [10] for a general result that specializes to the case under consideration.”
Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. (2014, background): “A relatively general framework toward proving the required uniform integrability has been developed in [12] and applied there to generalized Jackson networks; see also [6].”
Bounding Stationary Expectations of Markov Processes (2008, background): “…that setting, the goal is typically to bound the steady-state queue lengths or workload [see, e.g., Bertsimas, Paschalidis and Tsitsikilis (1994), Kumar and Kumar (1994), Sigman and Yao (1997), Bertsimas, Gamarnik and Tsitsikilis (2001) and Gamarnik and Zeevi (2006), as well as references therein].”
Asymptotic approximations for stationary distributions of many-server queues with abandonment (2010, background): “, Gamarnik and Zeevi [9] for an interchange of limits result in the context of generalized Jackson networks).”
Heavy traffic approximation for the stationary distribution of a generalized Jackson network: The BAR approach (2015, background): “2014/10/16 file: GJN-HTP_ssyfinal_arxivcorrection.tex date: June 28, 2017 that if a sequence of characteristic functions (CF) converges, and the limit is continuous at zero, then the corresponding sequence of probability measures converges weakly to a limiting probability measure.”