Stochastic Methods and Models in Operations Research and Related Areas

要做的工作的性质涉及对现实世界系统的数学建模，例如排队、生产库存、精算风险流程、当周期完成时重新启动的更换模型(如随着时间的推移更换电池的序列)、大坝(对铁路桥设计、防洪等很重要)、药代动力学(一种药物在血液中的浓度随时间变化的动力学)。这些模型具有重要的共同属性。它们受到不确定性的影响，它们随时间的演变可以记录为关键变量的跟踪或实现(也称为样本路径)，这些变量本质上是随机的。这些关键随机变量称为状态变量。样例路径包含有关流程随时间演变的状态变量的信息。状态变量可以采用的值范围称为状态空间。状态变量的一个重要的基本属性是它在从时间零开始测量的有限时间点上的概率分布，或者当时间趋于无穷大时。获得这种概率分布的一个有用的分析工具是随机模型的水平交叉法，它是由本研究人员于1974年提出并首次应用的。这将是拟议研究中使用的主要分析方法。在精算风险模型中，我们通过考虑感兴趣的状态变量的典型样本路径来描述该方法，例如队列中的工作量、库存中的库存或保险公司的货币盈余。这种模型的运行特性通常取决于状态变量的概率分布。在水平交叉法中，获得概率分布的过程从样本路径开始，并考虑从时间零到感兴趣的时间点的状态空间水平的水平交叉率。水平交叉率变成了简单的数学函数或状态变量的概率密度函数的积分变换。因此，交叉率和一个简单、直观的守恒定律可以通过检验来构造积分方程组(以积分为项)！通过对这些方程的解析或数值求解，得到了概率分布的显式公式。与其他分析方法相比，水平交叉法通常更快、更容易、更直观地得出所需的概率分布。拟议的工作还将处理更复杂的模型，并使用更一般的水平交叉方法来分析它们。它还将应用其他定量技术，包括：概率论、应用概率、应用数学、微分方程式和积分方程式、计算机编程、模拟、更新理论、再生过程、马尔可夫过程。它将提供可供选择的解决方法和观点，并提出新的研究方向，这将有助于研究人员、实践者和学生更好地理解模型。这项拟议的研究很重要，因为它将有助于增加对一大类随机模型的知识，同时继续发展水平交叉方法。它将在科学文献中添加和应用一套有用的新工具。它将在完全不同的模型之间推导出有趣的、实用的关系，获得关键的预期首次通过时间，并获得库存、风险模型等的新结果。预期结果是在一个大的科学类别中对随机模型进行一组新颖、有用的分析。这项工作将通过培训新的研究人员和在一个重要研究领域的前沿产生研究成果，使加拿大受益。

The nature of the work to be done involves mathematical modeling of real-world systems, such as queues, production-inventories, actuarial risk processes, replacement models that restart when cycles are completed (like a sequence of battery replacements over time), dams (important for railway bridge design, flood control, etc.), pharmacokinetics (the dynamics of the concentration of a drug in the blood stream with multiple dosing over time). These models have important properties in common. They are subject to uncertainty, and their evolution over time can be recorded as a tracing or realization (also called sample path) of key variables, which are random in nature. These key random variables are called state variables. The sample path contains information about the state variable as the process evolves over time. The range of values that the state variable can assume is called the state space. An important, basic property of the state variable is its probability distribution at finite time-points measured from time zero, or as time tends to infinity. A useful analytical tool for obtaining such probability distributions is the level crossing method for stochastic models, which was originated and first applied by the present researcher in 1974. This will be a dominant method of analysis used in the proposed research. We describe the method by considering a typical sample path of the state variable of interest, such as the workload in a queue, or the stock on hand in an inventory, or the monetary surplus of an insurance company in an actuarial risk model. Operating characteristics of such models often depend on the probability distribution of the state variable. In the level crossing method, the procedure for obtaining the probability distribution starts with the sample path, and considers level crossing rates of state-space levels, from time zero until the time point of interest. Level crossing rates turn out to be simple mathematical functions or integral transforms of the probability density function of the state variable. Thus the crossing rates and a simple, intuitive conservation law can be used to construct integral equations (having integrals as terms) by inspection! Explicit formulas for the probability distributions are obtained by solving these equations analytically or numerically. The level crossing method is often faster, easier and more intuitive for deriving the desired probability distributions, than other methods of analysis. The proposed work will also address more complex models and use more general level crossing methods to analyze them. It will also apply other quantitative techniques including: probability theory, applied probability, applied mathematics, differential and integral equations, computer programming, simulation, renewal theory, regenerative processes, Markov processes. It will offer alternative solution methods and points of view, and suggest new directions of research, which will aid researchers, practitioners and students to understand the models better. The proposed research is important because it will help to increase knowledge about a large class of stochastic models, while continuing to develop the level crossing methodology. It will add and apply a set of useful new tools to the scientific literature. It will derive interesting, practical relationships between completely different models, obtain key expected first passage times, and get new results for inventories, risk models, etc. The anticipated outcome is a set of novel, useful analyses of stochastic models in a large scientific category. The work will benefit Canada by training new researchers and producing studies at the forefront of an important field of research.