Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues1

Ari Arapostathis, Guodong Pang, Nikola Sandrić

Research output: Contribution to journalReview article

1 Citation (Scopus)

Abstract

We study the ergodic properties of a class of multidimensional piecewise Ornstein–Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin–Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric α-stable components or (3) an anisotropic Lévy process as in (2) and a pure-jump Lévy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) α-stable Lévy process as a special case. We identify conditions on the parameters in the drift, the Lévy measure and/or covariance function which result in subexponential and/or exponential ergodicity. We show that these assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence is polynomial, and we provide a sharp quantitative characterization of the rate via matching upper and lower bounds.

Original languageEnglish (US)
Pages (from-to)1070-1126
Number of pages57
JournalAnnals of Applied Probability
Volume29
Issue number2
DOIs
StatePublished - Apr 2019

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Multi-class
Ergodicity
Server
Jump Process
Queueing Model
Brownian motion
Exponential Ergodicity
Queueing Process
Covariance Function
Stable Process
Piecewise Linear
Upper and Lower Bounds
Jump
Rate of Convergence
Necessary Conditions
Polynomial
Invariant
Stochastic differential equations
Class
Model

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Arapostathis, Ari ; Pang, Guodong ; Sandrić, Nikola. / Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues1 In: Annals of Applied Probability. 2019 ; Vol. 29, No. 2. pp. 1070-1126.
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Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues1 . / Arapostathis, Ari; Pang, Guodong; Sandrić, Nikola.

In: Annals of Applied Probability, Vol. 29, No. 2, 04.2019, p. 1070-1126.

Research output: Contribution to journalReview article

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