Stochastic models for risk and queueing applications

在试图分析与健康保险以及涉及房主和汽车保险的财产和意外伤害保险情况有关的保险损失时，必须了解保险业务的投资组合在其过去和未来基于风险的行为的性质方面是非常复杂的。 有各种确定性和随机性的影响在发挥作用，和未来索赔经验的准确预测，有必要确定所有这些影响和他们的影响。 概率和统计的作用在评估一个定量的（即，理论）模型能够准确地预测保险业务组合的索赔经验。 概率和统计学在分析排队等候或排队时的拥堵方面也起着至关重要的作用，这在日常生活的许多方面都经常发生。 例如，我们通常都会在超市、银行和医生办公室等普通场所经历一定程度的等待。 应用概率论的一个分支--决策理论的研究为决策者提供了必要的数学框架，以便有效地简化流程，消除/减少医院急诊室和计算机网络系统等环境中可能出现的任何瓶颈。 在这个建议中，它是我的目标，开发数学上易于处理的结果，结合有效的计算程序，有关的各种随机模型中产生的保险风险和保险应用。预计这项研究的结果将有助于增加有关这些随机模型的现有知识，也许更重要的是，激发进一步的研究思路，以期更准确地反映现实世界的条件。

In attempting to analyze insurance losses arising in connection with health coverages as well as property and casualty insurance situations involving homeowner and automobile coverages, it is imperative to understand that a portfolio of insurance business is very complicated in terms of the nature of its past and future risk-based behaviour. There are various deterministic and stochastic influences at play, and the precise prediction of the future claims experience necessitates that all such influences and their effects be identified. The role of probability and statistics is vitally important in assessing whether a quantitative (i.e., theoretical) model is able to accurately predict the claims experience of a portfolio of insurance business. Probability and statistics also play a crucial role in terms of analyzing congestion in waiting lines or queues, which frequently occur in many facets of everyday life. For example, we all typically experience some amount of waiting in common places such as the supermarket, the bank, and the doctor's office. The study of queueing theory, a branch of applied probability, provides the essential mathematical framework that decision makers need in order to efficiently streamline processes to remove/reduce any bottlenecks which may arise in settings such as hospital emergency rooms and computer network systems. In this proposal, it is my goal to develop mathematically tractable results, in conjunction with effective computational procedures, pertaining to a variety of stochastic models arising in both insurance risk and queueing applications. 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, and perhaps more importantly, stimulate further ideas for investigation with a view towards more precisely reflecting real-world conditions.