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

我们的研究目的是发展和应用新的数学方法来构造差分、微分-差分、积分和积分-微分方程的解。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。这些方法的新的非平凡扩展已经被开发出来，以解决各种各样的应用中的问题，这导致了非常准确的结果。我们计划继续开发这些方法，并将它们特别应用于排队系统性能评估中产生的模型。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。我们还建议研究数学物理中与扩散对流、分岔和穿越势垒隧道有关的问题。我们研究项目的一个重要方面是对在线排队的研究。在最简单的形式中，排队系统由服务器和队列或等候室组成。客户可能在随机时间到达，并从服务器请求随机长度的服务。如果系统为空，则直接将到达的客户发送到服务器。否则，客户将被降级到队列中等待。一个是对开发排队系统的数学模型和计算性能度量感兴趣，例如平均等待的客户数量、平均等待时间、利用率等，以帮助理解系统的行为。对于复杂的排队系统，性能度量很难计算，我们研究的一个主要目标是构建有用的近似值。我们的工作队列的主要应用是高速通信网络，如Internet。这里的客户是信息包（比特），如数据文件、语音信息或视频。服务器是硬件，如路由器或交换机，它在互联网上传输数据包。在每个交换机上，如果需要，在传输之前，数据包被存储在队列中。这些缓冲区只能容纳固定数量的数据包。到达发现一个满缓冲区丢失，因此系统性能下降。性能指标，平均缓冲区大小，平均数据包延迟和丢失信息的概率，在高速通信网络的设计和调优中是重要的。我们的研究为网络设计者提供了近似这些性能度量的数学工具。

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.