The Annals of Applied Probability

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

Ari Arapostathis, Guodong Pang, and Nikola Sandrić

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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.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 1070-1126.

Dates
Received: July 2017
Revised: May 2018
First available in Project Euclid: 24 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1548298937

Digital Object Identifier
doi:10.1214/18-AAP1430

Mathematical Reviews number (MathSciNet)
MR3910024

Zentralblatt MATH identifier
07047445

Subjects
Primary: 60J75: Jump processes 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60G17: Sample path properties 60J25: Continuous-time Markov processes on general state spaces 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Multidimensional piecewise Ornstein–Uhlenbeck processes with jumps pure-jump Lévy process (an)isotropic Lévy process (sub)exponential ergodicity multiclass many-server queues Halfin–Whitt regime heavy-tailed arrivals service interruptions

Citation

Arapostathis, Ari; Pang, Guodong; Sandrić, Nikola. Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues. Ann. Appl. Probab. 29 (2019), no. 2, 1070--1126. doi:10.1214/18-AAP1430. https://projecteuclid.org/euclid.aoap/1548298937


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