Exact and Asymptotic Solutions to Applied Stochastic Models Arising in Queueing Theory and Other Areas

我们研究的目的是发展和应用新的构造差分、微分-差分、积分和积分-微分方程解的方法。 这些渐近和奇异摄动技术是基于WKB方法，边界层分析和匹配渐近展开。 新的nontrivialextensions这些方法以前已经开发解决问题，在各种各样的应用，这导致了相当准确的结果。 我们计划继续开发这些方法，并将其应用于特别是在性能评估的模型中的嵌入式系统。 该项目的几个方面与高速通信系统分析中出现的问题直接相关。 我们还建议研究与扩散-对流、分叉和隧穿势垒有关的物理学问题。我们研究计划的一个重要方面是队列或线的研究。 在最简单的形式中，排队系统由服务器和等候室或等候室组成。 客户可能在随机时间到达，并向服务器请求随机长度的服务。 如果系统是空的，到达的客户被直接发送到服务器。 否则，客户将被降级到队列中等待。 一是建立排队系统的数学模型，计算排队系统的平均等待顾客数、平均等待时间、利用率等性能指标，to help understand理解thebehavior行为of the system系统. 对于复杂的嵌入式系统，其性能度量是很难计算的，我们研究的一个主要目标是构造有用的近似。 我们工作的主要应用是高速通信网络，如互联网。 在这里，客户是信息包（比特），如数据文件、语音消息或视频。 服务器是一种硬件，如路由器或交换机，它沿着Internet传输数据包。 在每台交换机上，如果需要，数据包在传输之前存储在队列中。 这些缓冲区只能存储固定数量的数据包. 到达时发现缓冲区已满会丢失，因此系统性能会降低。 在高速通信网络的设计和调试中，性能指标，如平均缓冲区大小、平均包延迟和丢失信息的概率是非常重要的。 我们的研究为网络设计者提供了数学工具来近似这些性能测量。

The purpose of our research is to develop and apply new mathematicalmethods for constructing solutions to difference, differential-difference,integral, and integro- differential equations. These asymptotic andsingular perturbation techniques are based on the WKB method, boundarylayer analysis and matched asymptotic expansions. New nontrivialextensions of these methods have previously been developed to solveproblems in a wide variety of applications, which led to remarkablyaccurate results. We plan to continue the development of these methods,and to apply them in particular to models arising in the performanceevaluation of queueing systems. Several aspects of the project aredirectly related to problems arising in the analysis of high-speedcommunication systems. We also propose to study problems in mathematicalphysics, related to diffusion-convection, bifurcation, and tunnelingthrough potential barriers.An important aspect of our research program is the study of queues orlines. In its simplest form, a queueing system consists of a server and aqueue or waiting room. Customers arrive, possibly at random times, andrequest service of random length from the server. If the system is empty,an arriving customer is sent directly to the server. Otherwise, thecustomer is relegated to wait in the queue. One is interested indeveloping mathematical models of the queueing system and computingperformance measures, such as the average number of customers waiting, theaverage waiting time, the utilization, etc., to help understand thebehavior of the system. For complicated queueing systems, the performancemeasures are difficult to compute and a major goal of our research is toconstruct useful approximations to them. The main application of our workin queueing is to high-speed communication networks such as the Internet. Here the customers are packets of information (bits), such as data files,voice messages, or video. The server is hardware such as a router orswitch, which transmits the packets along the Internet. At each switch,the packets are stored in a queue, if needed, before transmission. Thesebuffers can only a fixed number of packets. Arrivals finding a fullbuffer are lost and hence the system performance is degraded. Theperformance measures, average buffer size, average packet delays, and theprobability of losing information, is important in the design and tuningof high-speed communication networks. Our research provides networkdesigners with mathematical tools for approximating these performancemeasures.