Validity of heavy-Traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines

Itai Gurvich*

*Corresponding author for this work

Research output: Contribution to journalArticle

9 Scopus citations

Abstract

A class of stochastic processes known as semi-martingale reflecting Brownian motions (SRBMs) is often used to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson [Bramson M (1998) State space collapse with applications to heavy-traffic limits for multiclass queueing networks. Queueing Systems 30:89-148] and Williams [Williams RJ (1998b) Diffusion approximations for open multiclass queueuing networks: Sufficient conditions involving state space collapse. Queueing Systems 30:27-88] laid out a general and structured approach for proving the validity of such heavy-traffic approximations, in which an SRBM is obtained as a diffusion limit from a sequence of suitably normalized workload processes. However, for multiclass networks it is still not known in general whether the steady-state distribution of the SRBM provides a valid approximation for the steady-state distribution of the original network. In this paper we study the case of queue-ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative. In addition to standard assumptions made in the literature towards the stability of the pre- and post-limit processes and the existence of diffusion limits, we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate. For the special case of static-priority networks such linear attraction is known to hold under certain conditions on the network primitives. The analysis elucidates interesting connections between stability of the pre- and post-limit processes, their respective fluid models and state-space collapse, and identifies the respective roles played by all of the above in establishing validity of heavy-traffic steady-state approximations.

Original languageEnglish (US)
Pages (from-to)121-162
Number of pages42
JournalMathematics of Operations Research
Volume39
Issue number1
DOIs
StatePublished - Feb 2014

Keywords

  • Heavy traffic
  • Multiclass
  • Network
  • Queue-ratio
  • Steady-state

ASJC Scopus subject areas

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

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