Speed of convergence to the quasi-stationary distribution for L\'evy input fluid queues

TL;DR

This paper proves that the workload in a Lévy-driven queue converges to its quasi-stationary distribution at a rate proportional to 1/t, providing explicit Laplace transforms and examples.

Contribution

It establishes the convergence rate to the quasi-stationary distribution for Lévy input queues and derives the Laplace transform of the convergence measure.

Findings

Convergence rate is of order 1/t.

Laplace transform of the convergence measure is identified.

Provides examples illustrating the results.

Abstract

In this note we prove that the speed of convergence of the workload of a L\'evy-driven queue to the quasi-stationary distribution is of order $1/t$. We identify also the Laplace transform of the measure giving this speed and provide some examples.