Stochastic Modelling and Optimization With Applications in Queues and Insurance

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In a large number of scientific disciplines, stochastic models are used to represent systems which, more often than not, involve several random components interacting in a complicated fashion. For instance, queueing models find important uses in a variety of applications including inventory systems, vehicular traffic control, and health care scheduling. The study of queueing theory provides the necessary mathematical framework for how to effectively reduce congestion in waiting lines and optimize system performance. A second example involves the development and subsequent analysis of insurance risk models, as they are paramount in their ability to adequately address the insurance industry's need for a more sophisticated and integrated risk management framework which better assesses an insurer's solvency risk. In this research program, I plan to contribute to the design and analysis of queueing models and insurance risk processes which strive to incorporate more flexibility and realism than many of the traditional models found in the literature. The proposed research essentially involves a continuation of my present work, and through collaboration with students at the undergraduate, Master's, and doctoral levels, I hope to accomplish three primary objectives: (1) continue to develop dynamic priority queueing models which are able to assign priority to customers in ways that are somehow dependent on their time within the system, (2) expand the class of polling models (i.e., a system of queues served sequentially according to a defined set of rules) that can be analyzed via the use of matrix analytic methods, in particular so that models featuring customer jockeying/reneging and server switchover/vacation times can be successfully handled by these methods, and (3) make mathematical and computational advances to the fundamental theory of insurance risk processes, so as to subsequently allow for a quantitative measurement of the benefits of proper risk management practices. Both analytic and probabilistic techniques are expected to be used to conduct the proposed research. Graduate students will be an integral part of the research process on all levels. They will gain skills in mathematical analysis, stochastic modelling, and algorithmic/computer implementation. Furthermore, it is expected that the results from this proposed research will prove useful in adding to the current state of knowledge concerning these kinds of stochastic models as well as stimulate further ideas for investigation. In a broader sense, I anticipate the results of this work to have a meaningful impact on the applied probability discipline as a whole, and perhaps contribute more significantly in specific research areas such as priority queues and ruin theory.