Two-Barrier Problems in Applied Probability: Algorithms and Analysis

• 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 $x0. 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.

• 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).

Ämnesord

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

Nyckelord

Statistics

operations research

programming

actuarial mathematics

Statistik

Matematik

Mathematics

Naturvetenskap

Natural science

Reflection

Stochastic processes

Applied probability

Queueing

operationsanalys

programmering

aktuariematematik

Publikations- och innehållstyp

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