| Random walks and queues are active research topics in applied probability and stochastic operations research, with applications in computer-communications, manufacturing and many other fields, while Lévy processes form an important topic in probability theory. Although there is a huge and mature literature on random walks and queues, as well as on Lévy processes, there is only a relatively small collection of studies devoted to their interface. The goal of the project is to systematically investigate problems at the interface of random walks, queues and Lévy processes. In Part I we shall study one-dimensional queueing models with as input a Lévy process. This generalizes the classical M/G/1-type queues, in which the input process is a compound Poisson process - which is a special case of a Lévy process. We mainly focus on queues with the additional feature of server vacations, which may depend on the previous busy period. We also consider Lévy processes, reflected at the origin, for which the Laplace exponent changes when a certain workload threshold is exceeded. In Part II we study multi-dimensional models: Polling models with Lévy input, networks of queues with Lévy input, and two-queue models with a coupled input. We aim for both exact and asymptotic workload results. These asymptotic results could follow from Laplace transforms obtained, but also for instance by applying large-deviations techniques or change-of-measure arguments. |
