Two-barrier problems in applied probability: Algorithms and analysis
Centre for Mathematical Sciences
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.
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