  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.
