L\'evy-driven GPS queues with heavy-tailed input

In this paper we derive exact large-buffer asymptotics for a two-class Generalized Processor Sharing (GPS) model, under the assumption that the input traffic streams generated by both classes correspond to heavy-tailed L\'evy processes. Four scenarios need to be distinguished, which differ in terms of (i)~the level of heavy-tailedness of the driving L\'evy processes as well as (ii)~the values of the corresponding mean rates relative to the GPS weights. The derived results are illustrated by two important special cases, in which the queues' inputs are modeled by heavy-tailed compound Poisson processes and by $\alpha$-stable L\'evy motions.