Twobarrier problems in applied probability: Algorithms and analysis
Mats Pihlsgård
Centre for Mathematical Sciences
Mathematical Statistics
Lund University,
2005
ISBN 9162866710
LUNFMS10162005

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