Queueing systems with nonstandard input processes

This dissertation is mainly about prestation-evaluation of queueing systems with unusual arrival processes. The arrival processes considered in this thesis range from Hawkes processes and doubly-stochastic Poisson processes driven by shot noise to inhomogeneous Poisson and Lévy processes. Homogeneous Poisson arrival processes are standard in queueing theory. Such processes generate arrivals that are uniformly distributed over time: a property that does not always hold in practice. With Hawkes or shot-noise Poisson processes we are able to generate more bursty arrival processes. In case of Hawkes processes, these bursts are the result of endogeneity: more arrivals will lead to even more arrivals. For shot noise Poisson processes, the effect is exogenous: an independent stochastic processes in the background generates fluctuating demand. Prestation evaluation of queueing systems with inhomogeneous Poisson processes has been performed extensively in the literature. In this thesis we consider a new angle: suppose that the model is known and the number of customers are observed (but the customers are not identifiable). We have solved the statistical inversion problem of inferring the distribution of service times. Furthermore, we consider some heavy traffic regimes in Lévy-driven tandem fluid queues (i.e., a system where fluid flows continuously through two sequential buffers). There are two different asymptotical regimes: one in which only the last buffer is seen as a bottleneck, and one in which both buffers are seen as a bottleneck. We characterize the behavior of the system in both regimes and note that they are entirely different. Next, we study a related fluid model with applications to blockchains. In particular, the model is able to estimate the waiting time distribution as a function of the fee that is given to miners.