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

Abstract

We say that a random variable is light-tailed if moments of order [Formula: see text] are finite for some [Formula: see text]; otherwise, we say that it is heavy-tailed. We study queueing networks that operate under the max-weight scheduling policy for the case in which 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.