Two-Barrier Problems in Applied Probability: Algorithms and Analysis
Author
Summary, in English
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.
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.
Department/s
Publishing year
2005
Language
English
Document type
Dissertation
Publisher
Centre for Mathematical Sciences, Lund University
Topic
- 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
Status
Published
Supervisor
- Sören Asmussen
ISBN/ISSN/Other
- ISBN: 91-628-6671-0
- ISRN: LUNFMS-1016-2005
Defence date
2 December 2005
Defence time
09:15
Defence place
Matematikcentrum, Sölvegatan 18, sal MH:A
Opponent
- Bert Zwart (Professor)