Diffusion Limit of Fair Resource Control-Stationarity and Interchange of Limits

• 成果类型: Article
• 署名作者: Ye, Heng-Qing; Yao, David D.
• 署名单位: Hong Kong Polytechnic University; Columbia University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2015.0773
• 发表日期: 2016
• 页码: 1161-1207
• 关键词: state-space collapse; multiclass queuing-networks; heavy-traffic optimality; Stochastic Network; fluid; STABILITY; approximations; queues; policies; halfin

摘要

We study a resource-sharing network where each job requires the concurrent occupancy of a subset of links (servers/resources), and each link's capacity is shared among job classes that require its service. The real-time allocation of the service capacity among job classes is determined by the so-called proportional fair scheme, which allocates the capacity among job classes taking into account the queue lengths and the shadow prices of link capacity. We show that the usual traffic condition is necessary and sufficient for the diffusion limit to have a stationary distribution. We also establish the uniform stability of the prelimit networks, and hence the existence of their stationary distributions. To justify the interchange of two limits, the limit in time and limit in diffusion scaling, we identify a bounded workload condition, and show it is a sufficient condition to justify the interchange for the stationary distributions and their moments. This last result is essential for the validity of the diffusion limit as an approximation to the stationary performance of the original network. We present a set of examples to illustrate justifying the validity of diffusion approximation in resource-sharing networks, and also discuss extensions to other multiclass networks via the well-known Kumar-Seidman/Rybko-Stolyar model.