Two-Barrier Problems in Applied Probability: Algorithms and Analysis

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.

Abstract (Swedish)

Popular Abstract in Swedish

Avhandlingen består av fem upsatser (A-E):



I uppsats A undersöks tidsberoende egenskaper hos kölängdsprocessen hos ett kösystem med flera betjänter under mycket allmänna Markovska ankomstprocesser.



I uppsats B undersöks en endimensionell slumpvandring som reflekteras i två barriärer.



I uppsats C undersöks ruinsannolikheten i en Cramér-Lundberg-modell där utbetalningarna har en trunkerad tungsvansad fördelning.



I uppsats D undersöks en Lévyprocess med två reflekterande barriärer.



I uppsats E presenteras en algoritm för att simulera den förlust som uppkommer då en slumpvandring reflekteras i två barriärer (se uppsats B).

• author: Pihlsgård, Mats
• supervisor: Sören Asmussen
• opponent: Professor Zwart, Bert, Tekniska Högskolan i Eindhoven
• organization: Mathematical Statistics
• publishing date: 2005
• type: Thesis
• publication status: published
• subject: Probability Theory and Statistics
• keywords: Statistics, operations research, programming, actuarial mathematics, Statistik, Matematik, Mathematics, Naturvetenskap, Natural science, Reflection, Stochastic processes, Applied probability, Queueing, operationsanalys, programmering, aktuariematematik
• pages: 150 pages
• defense location: Matematikcentrum, Sölvegatan 18, sal MH:A
• defense date: 2005-12-02 09:15:00
• external identifiers: other:ISRN: LUNFMS-1016-2005
• ISBN: 91-628-6671-0
• language: English
• LU publication?: yes
• id: d6320f64-9673-4608-9eb8-d2b9dfa4e9b6 (old id 545975)
• date added to LUP: 2016-04-01 16:48:24
• date last changed: 2018-11-21 20:44:20