Research on Applied Probability Models

该研究项目旨在开发综合方法，包括数学分析、算法和涉及概率行为的系统的计算机实验。长期目标是量化嵌入点过程的属性。在这个项目中，将从这个角度研究各种重要的排队问题，目的是对这些系统进行物理理解。本研究将与一组杰出的国际同事合作进行，以调查以下问题：(1)在显著的增量或递减运行之间，系统的行为是什么？(2)在暂态阶段，输出流比输入流更规则还是更不规则？(3)在不同的交通强度下，投入产出流是否具有根本不同的行为特征？(4)到达新的最大队列长度的时间分布如何随着系统状态和时间的变化而变化？利用矩阵-几何方法获取这些系统矩量的研究成果将在本项目中得到推广和利用。所提出的方法具有广泛的适用性，特别是对电信、生物和制造系统。这一研究结果可以指导改进设计随机系统的方法。从所研究的问题中发展出来的直觉可以为系统设计提供指导，取代浪费的试验和错误以及昂贵的模拟。所开发的数学方法可能允许对所建议的系统设计进行准确的性能预测，并使分析人员能够对各种组合事件的发生进行概率估计。系统管理人员可以更有效地分配资源来运行这些系统，并避免拥塞和损失。其结果可以为扩展系统容量和早期发现系统性能变化提供明智的决策，确保在不过度投资的情况下提供满意的客户服务。

The research project is for the development of integrated methodology consisting of mathematical analysis, algorithms, and computer experimentation for systems involving probabilistic behavior. The long-term goal is to quantify properties of embedded point processes. In this project a variety of important queuing problems will be studied from this perspective with the intent of developing a physical understanding of these systems. This research will be performed in collaboration with a group of distinguished international colleagues to investigate questions such as the following: (1) What is the behavior of the system between significant runs of increments or decrements? (2) During transient phases, is the output stream more or less regular than the input stream? (3) Does the input-output stream have fundamentally different behavioral characteristics at different levels of traffic intensity? and (4) How does the distribution of time until a new maximum queue length evolve with the system state and time? Research results on the use of matrix-geometric methods to obtain the moments of these systems will be extended and utilized in this project.The proposed methodology has widespread applicability, particularly to telecommunication, biological, and manufacturing systems. The results of this research can lead to improved methods for designing stochastic systems. The intuition developed from the problems studied can lead to guidelines for system design that replace wasteful trial and error and costly simulation. The mathematical methods developed may allow accurate performance prediction of proposed system designs and can enable analysts to produce probabilistic estimates for the occurrence of various combined events. System managers may be able to more effectively allocate resources to operate these systems and to avoid congestion and losses. The results can lead to intelligent decisions for expanding system capacity and for early detection of changes in system performance ensuring satisfactory customer service without excessive investment.