State space collapse with application to heavy traffic limits for multiclass queueing networks

Maury Bramson

Research output: Contribution to journalArticlepeer-review

152 Scopus citations

Abstract

Heavy traffic limits for multiclass queueing networks are a topic of continuing interest. Presently, the class of networks for which these limits have been rigorously derived is restricted. An important ingredient in such work is the demonstration of state space collapse. Here, we demonstrate state space collapse for two families of networks, first-in first-out (FIFO) queueing networks of Kelly type and head-of-the-line proportional processor sharing (HLPPS) queueing networks. We then apply our techniques to more general networks. To demonstrate state space collapse for FIFO networks of Kelly type and HLPPS networks, we employ law of large number estimates to show a form of compactness for appropriately scaled solutions. The limits of these solutions are next shown to satisfy fluid model equations corresponding to the above queueing networks. Results from Bramson [4,5] on the asymptotic behavior of these limits then imply state space collapse. The desired heavy traffic limits for FIFO networks of Kelly type and HLPPS networks follow from this and the general criteria set forth in the companion paper Williams [41]. State space collapse and the ensuing heavy traffic limits also hold for more general queueing networks, provided the solutions of their fluid model equations converge. Partial results are given for such networks, which include the static priority disciplines.

Original languageEnglish (US)
Pages (from-to)89-140
Number of pages52
JournalQueueing Systems
Volume30
Issue number1-2
DOIs
StatePublished - Nov 1998

Keywords

  • First-in first-out
  • Fluid models
  • Heavy traffic limits
  • Processor sharing
  • Queueing networks

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