# Uniform polynomial rates of convergence for a class of L\'evy-driven controlled SDEs arising in multiclass many-server queues

## TL;DR

This paper investigates the ergodic behavior of L9-stable driven controlled SDEs from multiclass queueing models, establishing polynomial and exponential convergence rates under different conditions.

## Contribution

It provides new uniform ergodicity results with explicit convergence rates for a class of L9-stable driven SDEs in queueing theory, including lower bounds and extensions of functional CLTs.

## Key findings

- Polynomial convergence rate when safety staffing is positive
- Exponential ergodicity when abandonment rates are positive
- Uniform ergodic properties over all stationary Markov controls

## Abstract

We study the ergodic properties of a class of controlled stochastic differential equations (SDEs) driven by $\alpha$-stable processes which arise as the limiting equations of multiclass queueing models in the Halfin-Whitt regime that have heavy-tailed arrival processes. When the safety staffing parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a polynomial rate of convergence to the invariant probability measure in total variation, which is uniform over all stationary Markov controls resulting in a locally Lipschitz continuous drift. We also derive a matching lower bound on the rate of convergence (under no abandonment). On the other hand, when all abandonment rates are positive, we show that the SDEs are exponentially ergodic uniformly over the above-mentioned class of controls. Analogous results are obtained for L\'evy-driven SDEs arising from multiclass many-server queues under asymptotically negligible service interruptions. For these equations, we show that the aforementioned ergodic properties are uniform over all stationary Markov controls. We also extend a key functional central limit theorem concerning diffusion approximations so as to make it applicable to the models studied here.