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

We say that a random variable is light-tailed if moments of order 2 + are finite for some > 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.