Stochastic Modelling and Optimization With Applications in Queues and Insurance

在大量的科学学科中，随机模型被用来表示系统，通常涉及以复杂方式相互作用的几个随机组件。例如，排队模型在包括库存系统、车辆交通控制和医疗保健调度在内的各种应用中找到重要用途。排队理论的研究为如何有效地减少排队拥挤和优化系统性能提供了必要的数学框架。 第二个例子涉及保险风险模型的开发和随后的分析，因为它们能够充分满足保险业对更复杂和综合的风险管理框架的需求，从而更好地评估保险公司的偿付能力风险。在这项研究计划中，我计划有助于设计和分析的保险模型和保险风险的过程，努力纳入更多的灵活性和现实主义比许多传统的模型在文献中找到。拟议的研究基本上涉及我目前的工作的延续，并通过与本科生，硕士和博士水平的学生合作，我希望完成三个主要目标：（1）继续开发动态优先级分配模型，该模型能够以某种方式依赖于客户在系统内的时间来向客户分配优先级，（2）扩展轮询模型的类别（即，根据定义的一组规则顺序服务的队列系统），其可以通过使用矩阵分析方法进行分析，特别是使得以客户操纵/违约和服务器延迟/休假时间为特征的模型可以通过这些方法成功地处理，以及（3）对保险风险过程的基本理论进行数学和计算改进，以便随后能够对适当风险管理做法的好处进行量化衡量。预计分析和概率技术将用于进行拟议的研究。研究生将是各级研究过程中不可分割的一部分。他们将获得数学分析，随机建模和算法/计算机实现方面的技能。 此外，预计这项研究的结果将被证明是有用的，在增加有关这些类型的随机模型的知识的现状，以及激发进一步的想法进行调查。从更广泛的意义上说，我预计这项工作的结果将对整个应用概率学科产生有意义的影响，也许在特定的研究领域，如优先队列和破产理论，贡献更大。

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.**