Stochastic Modelling and Optimization With Applications in Queues and Insurance

在许多科学学科中，随机模型被用来表示系统，这些系统通常涉及几个以复杂方式相互作用的随机组件。例如，排队模型在各种应用中都有重要的用途，包括库存系统、车辆交通控制和医疗保健调度。排队论的研究为如何有效地减少排队等待中的拥堵，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，因为这些模型最重要的是它们能够充分满足保险业对更复杂和更综合的风险管理框架的需求，以便更好地评估承保人的偿付能力风险。 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找到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我目前工作的继续，并通过与本科生、硕士和博士生的合作，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)扩展可通过使用矩阵分析方法分析的轮询模型的类别(即，根据一组定义的规则顺序服务的队列系统)，特别是使得这些方法可以成功地处理以客户操纵/放弃和服务器切换/休假时间为特征的模型，以及(3)对保险风险过程的基本理论进行数学和计算上的改进，以便随后允许对适当的风险管理实践的益处进行定量测量。 预计将使用分析和概率技术来进行拟议的研究。研究生将成为所有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/计算机实施方面的技能。此外，预计这项拟议研究的结果将有助于增加关于这类随机模型的现有知识状况，并激发进一步研究的想法。在更广泛的意义上，我预计这项工作的结果将对应用概率学科作为一个整体产生有意义的影响，并可能在特定的研究领域做出更大的贡献，如优先排队和破产理论。

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.