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

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 $\gamma \in (0,1)$.