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April 2019 Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues
Ari Arapostathis, Guodong Pang, Nikola Sandrić
Ann. Appl. Probab. 29(2): 1070-1126 (April 2019). DOI: 10.1214/18-AAP1430

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 $\alpha $-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) $\alpha $-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.

Citation

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Ari Arapostathis. Guodong Pang. Nikola Sandrić. "Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues." Ann. Appl. Probab. 29 (2) 1070 - 1126, April 2019. https://doi.org/10.1214/18-AAP1430

Information

Received: 1 July 2017; Revised: 1 May 2018; Published: April 2019
First available in Project Euclid: 24 January 2019

zbMATH: 07047445
MathSciNet: MR3910024
Digital Object Identifier: 10.1214/18-AAP1430

Subjects:
Primary: 60H10, 60J75
Secondary: 60G17, 60J25, 60K25

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 2 • April 2019
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