  We carry out a delay stability analysis (i.e., determine conditions under
which expected steady-state delays at a queue are finite) for a simple 3-queue
system operated under the Max-Weight scheduling policy, for the case where one
of the queues is fed by heavy-tailed traffic (i.e, when the number of arrivals
at each time slot has infinite second moment). This particular system
exemplifies an intricate phenomenon whereby heavy-tailed traffic at one queue
may or may not result in the delay instability of another queue, depending on
the arrival rates.
  While the ordinary stability region (in the sense of convergence to a
steady-state distribution) is straightforward to determine, the determination
of the delay stability region is more involved: (i) we use "fluid-type" sample
path arguments, combined with renewal theory, to prove delay instability
outside a certain region; (ii) we use a piecewise linear Lyapunov function to
prove delay stability in the interior of that same region; (iii) as an
intermediate step in establishing delay stability, we show that the expected
workload of a stable M/GI/1 queue scales with time as
$\mathcal{O}(t^{1/(1+\gamma)})$, assuming that service times have a finite
$1+\gamma$ moment, where $\gamma \in (0,1)$.
