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Queueing Systems

, Volume 62, Issue 4, pp 311–344 | Cite as

Exact asymptotics for the stationary distribution of a Markov chain: a production model

  • Ivo Adan
  • Robert D. Foley
  • David R. McDonald
Open Access
Article

Abstract

We derive rough and exact asymptotic expressions for the stationary distribution π of a Markov chain arising in a queueing/production context. The approach we develop can also handle “cascades,” which are situations where the fluid limit of the large deviation path from the origin to the increasingly rare event is nonlinear. Our approach considers a process that starts at the rare event. In our production example, we can have two sequences of states that asymptotically lie on the same line, yet π has different asymptotics on the two sequences.

Keywords

Rare events Large deviations Exact asymptotics Change of measure h transform Time reversal Markov additive process Markov chain R-transient 

Mathematics Subject Classification (2000)

60K25 60K20 

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Ivo Adan
    • 1
  • Robert D. Foley
    • 2
  • David R. McDonald
    • 3
  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Department of Industrial & Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of Mathematics and StatisticsThe University of OttawaOttawaCanada

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