Asymptotic Analysis and Control of Stochastic Networks

本研究的目的是开发数学工具，以分析和设计电信、计算机和服务系统中出现的复杂随机网络。这些网络通常过于复杂，无法进行精确的分析。本研究的主要目标是开发新的技术来获得这些系统的各种渐近逼近。具体来说，这些包括描述系统平均行为的所谓流体或一阶近似，捕获平均值周围波动的扩散近似，以及提供对系统工作至关重要的罕见事件概率估计的大偏差近似。这些技术将被应用于深入了解几个具体类型的网络的行为。特别是，将为所谓的“实时”开发新的准入控制算法。处理有最后期限的任务的系统，例如，携带数字化语音和跟踪系统的电信系统。此外，还将获得呼叫中心中出现的多服务器系统的性能度量估计。我们还将研究阻塞网络（用于模拟移动无线网络）的平衡特性，以及利用所谓的“砰砰”控制的网络的稳定性。分析随机网络的数学挑战之一是，它们通常由表现出不连续转换的约束过程来描述，因此许多经典理论不再适用。如果成功，这项研究将为分析这种受限过程提供一套新的数学工具，这可能具有更广泛的适用性。这项研究也有可能有助于改进实时排队网络、呼叫中心和无线网络的设计和控制，这可能会导致这些系统运行的更大经济效益。

The objective of this proposed research is to develop mathematical tools for the analysis and design of complex stochastic networks arising in telecommunications, computer and service systems. These networks are typically too complex to lend themselves to an exact analysis. The primary goal of this research is to develop new techniques for obtaining a variety of asymptotic approximations for these systems. Specifically, these include so-called fluid or first-order approximations that describe the mean behavior of the system, diffusion approximations that capture fluctuations around the mean, and large deviations approximations that provide estimates for the probabilities of rare events that are critical to the working of the system. These techniques will be applied to gain insight into the behavior of several concrete classes of networks. In particular, new admission control algorithms will be developed for so-called ?real-time? systems that process tasks with deadlines such as, for example, telecom systems carrying digitized voice and tracking systems. In addition, estimates of performance measures will be obtained for multi-server systems that arise in call centers. We will also investigate the equilibrium properties of networks with blocking (used to model mobile wireless networks), as well as the stability of networks utilizing so-called bang-bang controls. One of the mathematical challenges in analyzing stochastic networks is that they are often described by constrained processes that exhibit discontinuous transitions, and so much of the classical theory can no longer be applied. If successful, this research would provide a new set of mathematical tools for the analysis of such constrained processes, which could be of broader applicability. The research also has the potential to contribute to improving the design and control of real-time queueing networks, call centers and wireless networks, which could lead to greater economy in the running of these systems.