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

• 成果类型: Article
• 署名作者: Gamarnik, D; Zeevi, A
• 署名单位: Massachusetts Institute of Technology (MIT); Columbia University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051605000000638
• 发表日期: 2006
• 页码: 56-90
• 关键词: multiclass queuing-networks; achievable region approach; changeover times; partial sums; STABILITY; optimization; queues; performance; bounds

摘要

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.