Two-barrier problems in applied probability: Algorithms and analysis

Mats Pihlsgård

Centre for Mathematical Sciences
Mathematical Statistics
Lund University,
2005

ISBN 91-628-6671-0
LUNFMS--1016--2005

Abstract:

This thesis consists of five papers (A-E).

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 required for the queue length starting at x<n to reach level n. 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 set-up of (quasi) birth-death processes.

Paper B investigates reflection of a random walk at two barriers, 0 and K>0. We define the loss rate due to the reflection. The main result is sharp asymptotics for the loss rate 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 an explicit comparison between asymptotic and exact results for the loss rate.

Paper C deals with queues and insurance risk processes where a generic service time, respectively generic claim, has a truncated heavy-tailed distribution. We study the compound Poisson ruin probability (or, equivalently, the tail of the M/G/1 steady-state waiting time) numerically. Furthermore, we investigate the asymptotics of the asymptotic exponential decay rate as the truncation level tends to infinity in a more general truncated Lévy process set-up.

Paper D 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 is based on optional stopping of the Kella-Whitt martingale for the reflected process. Once the identification is performed, we derive asymptotics for the loss rate in the case of a light-tailed Lévy measure.

Paper E is also a sequel of Paper B. We present an algorithm for simulating the loss rate for a reflected random walk. The algorithm is efficient in the sense of bounded relative error.

Key words:

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, efficient simulation