Some Two-Barrier Problems in Applied Probability

Mats Pihlsgård

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2003

ISSN 1404-028X

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Abstract:

This thesis is concerned with first time passage problems in queueing and loss rates in reflected random walks and Lévy processes. It is divided into three papers.

In Paper A, we study transient properties of the queue length process in various queueing settings. We focus on computing the mean and the Laplace transform of the time it takes for the queue length to reach level n (we assume that we start at some queue length less than n). To this end, we use two different techniques. The first one is based on optional stopping of the Kella-Whitt martingale and the second on more traditional results on level crossing times of birth-death processes. Furthermore, we try to find an equivalent to the theory of the natural scale for diffusion processes to fit into the setting of (quasi) birth-death processes.

Paper B investigates two different ways of reflecting a random walk at two barriers, 0 and K>0, with corresponding definitions of the loss rate, i.e. the expected overflow of level K per time unit in

stationarity. The main results are asymptotics for the two alternative loss rates as K tends to infinity. As a major example, we consider the case where the increments of the random walk may be written as the difference between two phase-type distributed random variables. In this example we perform explicit comparison between asymptotic and exact results for one of the two loss rate alternatives.

Paper C is a sequel of Paper B. We consider a Lévy process reflected at 0 and K>0 and define the loss rate. The first step is to identify the loss rate, which is non-trivial in the Lévy process case. The technique we use here is based on observation of the Kella-Whitt martingale for the reflected process at time 1. Once the identification is performed, we derive asymptotics for the Lévy process loss rate.

Keywords:

many-server queues, quasi birth-death processes, Kella-Whitt martingale, optional stopping, heterogeneous servers, reflected random walks, loss rate, Lundberg's equation, Cramér-Lundberg approximation, Wiener-Hopf factorization, asymptotics, phase-type distributions, reflected Lévy processes, light tails.