Ergodicity of L\'evy-driven SDEs arising from multiclass many-server queues

TL;DR

This paper investigates the ergodic behavior of multidimensional L\'evy-driven Ornstein-Uhlenbeck processes, including queueing models with heavy-tailed arrivals, identifying conditions for exponential and subexponential ergodicity and characterizing convergence rates.

Contribution

It provides sharp conditions for ergodicity of L\'evy-driven SDEs from multiclass many-server queues, including necessary conditions and convergence rate analysis.

Findings

Conditions for exponential ergodicity are identified.

Conditions for subexponential ergodicity are established.

Polynomial convergence rates are characterized with bounds.

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 It\^o equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump L\'evy process, or (2) an anisotropic L\'evy process with independent one-dimensional symmetric $\alpha$-stable components, or (3) an anisotropic L\'evy process as in (2) and a pure-jump L\'evy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) $\alpha$-stable L\'evy process as a special case. We identify conditions on the parameters in the drift, the L\'evy 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.