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We carry out a delay stability analysis (i.e., determine conditions underwhich expected steady-state delays at a queue are finite) for a simple 3-queuesystem operated under the Max-Weight scheduling policy, for the case where oneof the queues is fed by heavy-tailed traffic (i.e, when the number of arrivalsat each time slot has infinite second moment). This particular systemexemplifies an intricate phenomenon whereby heavy-tailed traffic at one queuemay or may not result in the delay instability of another queue, depending onthe arrival rates.  While the ordinary stability region (in the sense of convergence to asteady-state distribution) is straightforward to determine, the determinationof the delay stability region is more involved: (i) we use "fluid-type" samplepath arguments, combined with renewal theory, to prove delay instabilityoutside a certain region; (ii) we use a piecewise linear Lyapunov function toprove delay stability in the interior of that same region; (iii) as anintermediate step in establishing delay stability, we show that the expectedworkload 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 $\gamma \in (0,1)$.

Delay Stability Regions of the Max-Weight Policy under Heavy-Tailed Traffic