Stochastic Methods and Models in Operations Research and Related Areas

运筹学及相关领域的随机方法和模型

• 批准号: RGPIN-2014-05697
• 负责人: Brill, Percy
• 金额: $ 1.46万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611789
• 关键词: Stochastic; Methods; Models; Operations; Research

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.

要完成的工作的性质涉及现实世界系统的数学建模，例如队列，生产库存，精算风险过程，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顺序），水坝（对铁路桥梁设计，洪水控制等很重要），药代动力学（多次给药时血流中药物浓度随时间的动态）。这些模型具有共同的重要特性。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），这些变量在本质上是随机的。这些关键的随机变量称为状态变量。样本路径包含有关过程随时间演变的状态变量的信息。状态变量可以取的值的范围称为状态空间。状态变量的一个重要的基本性质是它在从时间零点开始测量的有限时间点上的概率分布，或者随着时间趋于无穷大。一个有用的分析工具，获得这样的概率分布是随机模型的水平交叉方法，这是起源于本研究人员在1974年首次应用。这将是拟议研究中使用的主要分析方法。我们描述的方法，通过考虑一个典型的样本路径的状态变量的利益，如工作量在队列中，或库存的库存，或保险公司的货币盈余的精算风险模型。这种模型的运行特性通常取决于状态变量的概率分布。在水平交叉方法中，用于获得概率分布的过程从样本路径开始，并且考虑从时间零直到感兴趣的时间点的状态空间水平的水平交叉率。水平交叉率原来是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用来构造积分方程（有积分作为项）通过检查！通过解析或数值求解这些方程，得到了概率分布的显式公式。水平交叉法通常比其他分析方法更快，更容易和更直观地推导出所需的概率分布。拟议的工作还将解决更复杂的模型，并使用更通用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，微分和积分方程，计算机编程，模拟，更新理论，再生过程，马尔可夫过程。它将提供替代的解决方法和观点，并提出新的研究方向，这将有助于研究人员，从业人员和学生更好地理解模型。拟议的研究是重要的，因为它将有助于增加知识的一大类随机模型，同时继续发展水平交叉方法。它将为科学文献增加和应用一套有用的新工具。它将得出有趣的，实际的关系完全不同的模型，获得关键的预期首次通过时间，并得到新的结果库存，风险模型等预期的结果是一组新的，有用的分析随机模型在一个大的科学范畴。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进行研究而使加拿大受益。