Asymptotic analysis of Levy-driven tandem queues

We analyze tail asymptotics of a two-node tandem queue with spectrally-positive L\'evy input. A first focus lies on tail probabilities of the type ${\mathbb P}(Q_1> \alpha x, Q_2>(1-\alpha)x)$, for $\alpha\in(0,1)$ and $x$ large, and $Q_i$ denoting the steady-state workload in the $i$th queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize to the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed L\'evy input. It is also indicated how the results can be extended to tandem queues with more than two nodes.