Diffusion limit of fair resource control-stationarity and interchange of limits

Hengqing Ye, David D. Yao

Research output: Journal article publicationJournal articleAcademic researchpeer-review

9 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)1161-1207
Number of pages47
JournalMathematics of Operations Research
Issue number4
Publication statusPublished - 1 Nov 2016


  • Diffusion Limit
  • Interchange Of Limits
  • Proportional Fair Allocation
  • Stationary Distribution
  • Stochastic Processing Network
  • Uniform Stability

ASJC Scopus subject areas

  • Mathematics(all)
  • Computer Science Applications
  • Management Science and Operations Research


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