Instability in stochastic and fluid queueing networks

• 成果类型: Article
• 署名作者: Gamarnik, D; Hasenbein, JJ
• 署名单位: International Business Machines (IBM); IBM USA; University of Texas System; University of Texas Austin
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051605000000179
• 发表日期: 2005
• 页码: 1652-1690
• 关键词: stability

摘要

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all nonidling policies. However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations, we prove that if the fluid model is not weakly stable under the class of all nonidling policies, then a corresponding queueing network is not rate stable under the class of all nonidling policies. We establish the result by building a particular nonidling scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition rho* <= 1, which was proven in [Oper Res. 48 (2000) 721-744] to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here rho* is a certain computable parameter of the network involving virtual station and push start conditions.