Some Two-Barrier Problems in Applied Probability
Centre for Mathematical Sciences
Lund Institute of Technology,
This thesis is concerned with first time passage problems in queueing and
loss rates in reflected random walks and Lévy processes. It is divided
into three papers.
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 it takes for the queue length to reach level n
(we assume that we start at some queue length less than n). To this
end, 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 setting of (quasi) birth-death processes.
Paper B investigates two different ways of reflecting a random walk at two
barriers, 0 and K>0, with corresponding definitions of the loss
rate, i.e. the expected overflow of level K per time unit in
stationarity. The main results are asymptotics for the two alternative loss
rates 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
explicit comparison between asymptotic and exact results for one of the two
loss rate alternatives.
Paper C 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 here is based on observation of the Kella-Whitt martingale
for the reflected process at time 1. Once the identification is performed,
we derive asymptotics for the Lévy process loss rate.
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.