We provide contributions to two classical areas of queueing. The first part of this thesis focuses on finding new conditions for a
Markov chain on a general state space to be Harris recurrent,
positive Harris recurrent or geometrically ergodic. Most of our
results show that establishing each property listed above is
equivalent to finding a good enough feasible solution to a
particular optimal stopping problem, and they provide a more
complete understanding of the role Foster's criterion plays in the
theory of Markov chains.
The second and third parts of the thesis involve analyzing queues
from a transient, or time-dependent perspective. In part two, we
are interested in looking at a queueing system from the
perspective of a customer that arrives at a fixed time t. Doing
this requires us to use tools from Palm theory. From an intuitive
standpoint, Palm probabilities provide us with a way of computing
probabilities of events, while conditioning on sets of measure
zero. Many studies exist in the literature that deal with Palm
probabilities for stationary systems, but very few treat the
non-stationary case. As an application of our main results, we
show that many classical results from queueing (in particular ASTA and Little's law) can be generalized to a time-dependent
setting.
In part three, we establish a continuity result for what we refer
to as jump processes. From a queueing perspective, we basically
show that if the primitives and the initial conditions of a
sequence of queueing processes converge weakly, then the
corresponding queue-length processes converge weakly as well in
some sense. Here the notion of convergence used depends on
properties of the limiting process, therefore our results
generalize classical continuity results that exist in the
literature. The way our results can be used to approximate
queueing systems is analogous to the way phase-type random
variables can be used to approximate other types of random
variables.
| Identifer | oai:union.ndltd.org:GATECH/oai:smartech.gatech.edu:1853/14569 |
| Date | 10 January 2007 |
| Creators | Fralix, Brian Haskel |
| Publisher | Georgia Institute of Technology |
| Source Sets | Georgia Tech Electronic Thesis and Dissertation Archive |
| Detected Language | English |
| Type | Dissertation |
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