Jumping Fluid Models and Delay Stability of Max-Weight Dynamics under Heavy-Tailed Traffic

TL;DR

This paper investigates delay stability in Max-Weight queueing networks with mixed heavy-tailed and light-tailed traffic, introducing jumping fluid models to characterize stability conditions and elucidate delay instability mechanisms.

Contribution

It develops a necessary and sufficient condition for delay stability using jumping fluid models, extending traditional fluid models to handle heavy-tailed traffic.

Findings

Delay stability depends on tail exponents and resource sharing.

Jumping fluid models accurately predict delay instability.

Lyapunov functions are effective in analyzing delay stability.

Abstract

We say that a random variable is $light$-$tailed$ if moments of order $2+\epsilon$ are finite for some $\epsilon>0$; otherwise, we say that it is $heavy$-$tailed$. We study queueing networks that operate under the Max-Weight scheduling policy, for the case where some queues receive heavy-tailed and some receive light-tailed traffic. Queues with light-tailed arrivals are often delay stable (that is, expected queue sizes are uniformly bounded over time) but can also become delay unstable because of resource-sharing with other queues that receive heavy-tailed arrivals. Within this context, and for any given "tail exponents" of the input traffic, we develop a necessary and sufficient condition under which a queue is robustly delay stable, in terms of $jumping$ $fluid$ models - an extension of traditional fluid models that allows for jumps along coordinates associated with heavy-tailed flows. Our result elucidates the precise mechanism that leads to delay instability, through a coordination of multiple abnormally large arrivals at possibly different times and queues and settles an earlier open question on the sufficiency of a particular fluid-based criterion. Finally, we explore the power of Lyapunov functions in the study of delay stability.