Abstract
This thesis consists of five papers (AE).
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 KellaWhitt martingale and the second on more
traditional results on level crossing times of birthdeath
processes. Furthermore, we try to find an equivalent to the theory of
the natural scale for diffusion processes to fit into the setup of
(quasi) birthdeath 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 phasetype 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 heavytailed distribution. We study the compound Poisson ruin
probability (or, equivalently, the tail of the M/G/1 steadystate 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 setup.
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 nontrivial in the Lévy process case. The
technique we use is based on optional stopping of the KellaWhitt
martingale for the reflected process. Once the
identification is performed, we derive asymptotics for the loss rate in the case of a lighttailed 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:
manyserver queues, quasi birthdeath processes, KellaWhitt
martingale, optional stopping, heterogeneous servers, reflected random
walks, loss rate, Lundberg's equation, CramérLundberg approximation,
WienerHopf factorization, asymptotics, phasetype 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 KellaWhitt martingale and the second on more
traditional results on level crossing times of birthdeath
processes. Furthermore, we try to find an equivalent to the theory of
the natural scale for diffusion processes to fit into the setup of
(quasi) birthdeath 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 phasetype 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 heavytailed distribution. We study the compound Poisson ruin
probability (or, equivalently, the tail of the M/G/1 steadystate 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 setup.
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 nontrivial in the Lévy process case. The
technique we use is based on optional stopping of the KellaWhitt
martingale for the reflected process. Once the
identification is performed, we derive asymptotics for the loss rate in the case of a lighttailed 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:
manyserver queues, quasi birthdeath processes, KellaWhitt
martingale, optional stopping, heterogeneous servers, reflected random
walks, loss rate, Lundberg's equation, CramérLundberg approximation,
WienerHopf factorization, asymptotics, phasetype distributions,
reflected Lévy processes, light tails, efficient simulation.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  2005 Dec 2 
Publisher  
ISBN (Print)  9162866710 
Publication status  Published  2005 
Bibliographical note
Defence detailsDate: 20051202
Time: 09:15
Place: Matematikcentrum, Sölvegatan 18, sal MH:A
External reviewer(s)
Name: Zwart, Bert
Title: Professor
Affiliation: Tekniska Högskolan i Eindhoven

Subject classification (UKÄ)
 Probability Theory and Statistics
Free keywords
 Statistics
 operations research
 programming
 actuarial mathematics
 Statistik
 Matematik
 Mathematics
 Naturvetenskap
 Natural science
 Reflection
 Stochastic processes
 Applied probability
 Queueing
 operationsanalys
 programmering
 aktuariematematik