Steady-State Analysis of the Join-The-Shortest-Queue Model in the Halfin Whitt Regime

Anton Braverman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

This paper studies the steady-state properties of the join-The-shortest-queue model in the Halfin Whitt regime. We focus on the process tracking the number of idle servers and the number of servers with nonempty buffers. Recently, Eschenfeldt and Gamarnik proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper, we prove that the diffusion limit is exponentially ergodic and that the diffusion scaled sequence of the steady-state number of idle servers and nonempty buffers is tight. Combined with the process-level convergence proved by Eschenfeldt and Gamarnik, our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein s method, also referred to as the drift-based fluid limit Lyapunov function approach in Stolyar. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity.

Original languageEnglish (US)
Pages (from-to)1069-1103
Number of pages35
JournalMathematics of Operations Research
Volume45
Issue number3
DOIs
StatePublished - Aug 2020

Keywords

  • drift-based fluid model
  • exponential ergodicity
  • fluid model
  • generator expansion
  • Halfin Whitt
  • join shortest queue
  • limit interchange
  • steady state
  • Stein method

ASJC Scopus subject areas

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

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