Two-Barrier Problems in Applied Probability: Algorithms and Analysis
(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... (More) - 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. (Less) - 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... (More) - 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). (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/545975
- author
- Pihlsgård, Mats LU
- supervisor
- opponent
-
- Professor Zwart, Bert, Tekniska Högskolan i Eindhoven
- organization
- publishing date
- 2005
- type
- Thesis
- publication status
- published
- subject
- 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
- publisher
- Centre for Mathematical Sciences, Lund University
- 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
@phdthesis{d6320f64-9673-4608-9eb8-d2b9dfa4e9b6, abstract = {{This thesis consists of five papers (A-E).<br/><br> <br/><br> In Paper A, we study transient properties of the queue length process<br/><br> <br/><br> in various queueing settings. We focus on computing the mean and the Laplace<br/><br> <br/><br> 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<br/><br> <br/><br> optional stopping of the Kella-Whitt martingale and the second on more<br/><br> <br/><br> traditional results on level crossing times of birth-death<br/><br> <br/><br> processes. Furthermore, we try to find an equivalent to the theory of<br/><br> <br/><br> the natural scale for diffusion processes to fit into the set-up of<br/><br> <br/><br> (quasi) birth-death processes.<br/><br> <br/><br> Paper B investigates reflection of a random walk at<br/><br> <br/><br> 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<br/><br> <br/><br> this example we perform an explicit comparison between asymptotic<br/><br> <br/><br> and exact results for the loss rate.<br/><br> <br/><br> Paper C deals with queues and insurance risk processes<br/><br> <br/><br> where a generic service time, respectively generic claim, has a truncated heavy-tailed distribution. We study the compound Poisson ruin<br/><br> <br/><br> probability (or, equivalently, the tail of the M/G/1 steady-state waiting time) numerically. Furthermore, we<br/><br> <br/><br> investigate the asymptotics of the asymptotic<br/><br> <br/><br> exponential decay rate as the truncation level tends to infinity in a more general truncated Lévy process set-up.<br/><br> <br/><br> Paper D is a sequel of Paper B. We consider a Lévy process reflected<br/><br> <br/><br> at 0 and $K$>0 and define the loss rate. The first step is to identify<br/><br> <br/><br> the loss rate, which is non-trivial in the Lévy process case. The<br/><br> <br/><br> technique we use is based on optional stopping of the Kella-Whitt<br/><br> <br/><br> martingale for the reflected process. Once the<br/><br> <br/><br> identification is performed, we derive asymptotics for the loss rate in the case of a light-tailed Lévy measure.<br/><br> <br/><br> 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.<br/><br> <br/><br> Key words:<br/><br> <br/><br> many-server queues, quasi birth-death processes, Kella-Whitt<br/><br> <br/><br> martingale, optional stopping, heterogeneous servers, reflected random<br/><br> <br/><br> walks, loss rate, Lundberg's equation, Cramér-Lundberg approximation,<br/><br> <br/><br> Wiener-Hopf factorization, asymptotics, phase-type distributions,<br/><br> <br/><br> reflected Lévy processes, light tails, efficient simulation.}}, author = {{Pihlsgård, Mats}}, isbn = {{91-628-6671-0}}, keywords = {{Statistics; operations research; programming; actuarial mathematics; Statistik; Matematik; Mathematics; Naturvetenskap; Natural science; Reflection; Stochastic processes; Applied probability; Queueing; operationsanalys; programmering; aktuariematematik}}, language = {{eng}}, publisher = {{Centre for Mathematical Sciences, Lund University}}, school = {{Lund University}}, title = {{Two-Barrier Problems in Applied Probability: Algorithms and Analysis}}, year = {{2005}}, }