Mathematical analysis of models from communications engineering

Abstract

This thesis deals with a mathematical analysis of models from communications engineering, which is thematically located in the field of applied probability and stochastic processes. The content of this work is divided into two chapters that address two different and independent models. The first chapter treats polling models that are multiple queue, cyclic service systems. The feature of the single server is that it may be forced to wait idly for new jobs at an empty queue instead of switching to the next station. We consider different wait-and-see strategies that govern these forced idle times. We assume that arrivals of new jobs occur according to Poisson processes and we allow general service and switchover time distributions. The results are formulas for the mean average queueing delay of a job, characterisations of the cases for a polling model with two stations where the wait-and-see strategies yield a lower delay compared to the exhaustive strategy, and a comparison of the strategies among each other. In the second chapter, we consider random rectangles that are distributed according to a Poisson random measure, i.e., independently and uniformly scattered in the plane. The distributions of the length and the width of the rectangles are heavy-tailed with different parameters. We investigate the scaling behaviour of the related random fields as the intensity of the random measure grows to infinity while the expected edge lengths tend to zero. We characterise the arising scaling regimes, identify the limiting random fields, and give statistical properties of these limits.