Stochastic Modelling and Optimization With Applications in Queues and Insurance

在许多科学学科中，随机模型通常用于表示系统，这些系统通常包含几个随机组件以复杂的方式相互作用。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程序中都有重要的用途。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后续分析，因为它们在充分满足保险业对更复杂和综合风险管理框架的需求方面至关重要，该框架可以更好地评估保险公司的偿付能力风险。***在这个研究项目中，我计划对排队模型和保险风险过程的设计和分析做出贡献，这些模型力求比文献中的许多传统模型更具灵活性和现实性。拟议的研究基本上涉及到我目前工作的延续，并通过与本科生，硕士和博士水平的学生合作，我希望实现三个主要目标：(1)继续开发动态优先级排队模型，该模型能够以某种程度上依赖于客户在系统中的时间的方式为客户分配优先级；(2)扩展轮询模型的类别（即，根据一组定义的规则顺序服务的队列系统），可以通过使用矩阵分析方法进行分析；特别是这样，以客户欺骗/违约和服务器切换/休假时间为特征的模型可以通过这些方法成功处理，并且(3)对保险风险过程的基本理论进行数学和计算上的进步，从而随后允许对适当的风险管理实践的好处进行定量测量。***预计将使用分析和概率技术来进行拟议的研究。研究生将是各个层次研究过程中不可或缺的一部分。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，预计本研究的结果将有助于增加有关这些类型的随机模型的现有知识，并激发进一步的研究思路。从更广泛的意义上讲，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，并可能在优先队列和破产理论等特定研究领域做出更重大的贡献

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.**