A Tandem Fluid Network with L\'evy Input in Heavy Traffic

TL;DR

This paper analyzes the stationary workload distribution of a fluid tandem queue with Lévy input in heavy traffic, considering different Lévy processes and regimes, revealing new asymptotic behaviors and improved approximations.

Contribution

It introduces a comprehensive analysis of heavy-traffic regimes for fluid tandem queues with Lévy input, including simultaneous load tending to one, which was not previously studied.

Findings

Different heavy-traffic asymptotics for finite and infinite variance Lévy inputs.

Simultaneous heavy-traffic regime leads to distinct asymptotic behavior.

Numerical results show improved approximation accuracy under certain conditions.

Abstract

In this paper we study the stationary workload distribution of a fluid tandem queue in heavy traffic. We consider different types of L\'evy input, covering compound Poisson, $\alpha$-stable L\'evy motion (with $1<\alpha<2$), and Brownian motion. In our analysis we separately deal with L\'evy input processes with increments that have finite and infinite variance. A distinguishing feature of this paper is that we do not only consider the usual heavy-traffic regime, in which the load at one of the nodes goes to unity, but also a regime in which we simultaneously let the load of both servers tend to one, which, as it turns out, leads to entirely different heavy-traffic asymptotics. Numerical experiments indicate that under specific conditions the resulting simultaneous heavy-traffic approximation significantly outperforms the usual heavy-traffic approximation.