The heavy traffic limit of an unbalanced generalized processor sharing model

• 成果类型: Article
• 署名作者: Ramanan, Kavita; Reiman, Martin I.
• 署名单位: Carnegie Mellon University; Alcatel-Lucent
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP438
• 发表日期: 2008
• 页码: 22-58
• 关键词: multiclass queuing-networks; state-space collapse; skorokhod problem; Sufficient conditions; orthant; fluid; map

摘要

This work considers a server that processes J classes using the generalized processor sharing discipline with base weight vector alpha = (alpha(1), . . . , alpha(J)) and redistribution weight vector beta = (beta(1), . . . , beta(j)). The invariant manifold M of the so-called fluid limit associated with this model is shown to have the form M = [x is an element of R-+(J) : x(j) = 0 for J is an element of s), where s is the set of strictly sub-critical classes, which is identified explicitly in terms of the vectors a and P and the long-run average work arrival rates gamma(j) of each class j. In addition, under general assumptions, it is shown that when the heavy traffic condition Sigma(J)(j=1) gamma(j) = Sigma(J)(j=1) alpha(j) holds, the functional central limit of the scaled unfinished work process is a reflected diffusion process that lies in A. The reflected diffusion limit is characterized by the so-called extended Skorokhod map and may fail to be a semimartingale. This generalizes earlier results obtained for the simpler, balanced case where gamma(j) = alpha(j) for j = 1, . . . , J, in which case M = R-+(J) and there is no state-space collapse. Standard techniques for obtaining diffusion approximations cannot be applied in the unbalanced case due to the particular structure of the GPS model. Along the way, this work also establishes a comparison principle for solutions to the extended Skorokhod map associated with this model, which may be of independent interest.