Stochastic Networks: Control and Performance

本项目的目的是研究与随机网络的控制和性能分析相关的数学问题。正在考虑的网络模型是异构的，可能有复杂的反馈机制，并允许在到达，服务时间和路由的随机变化。网络中的缓冲区可以存储无法立即提供的作业。此外，具有控制的模型允许动态排序和作业的交替路由。由于这些网络模型的复杂性通常排除了精确的分析，重点是近似模型。两个层次的近似被认为是，即流体模型（一阶近似）和扩散模型（二阶近似）。调查这些模型之间的相互作用是研究的一个重要特点。两个主要的主题是（i）通过排序和交替路由的动态控制，以及（ii）处理器共享网络的性能分析。关于主题（i），一些作者已经成功地使用扩散控制问题作为形式化工具，为某些特定的网络模型生成良好的控制策略。然而，很少有严格的分析，这些政策的业绩。PI正在开发一种系统的方法来寻找和解释扩散控制问题的解决方案，并分析以这种方式产生的政策的性能。关于主题（ii），处理器共享规程是服务规程的一个示例，其中自然状态描述符涉及测量值过程（以跟踪网络中所有作业的剩余服务时间）。PI正在研究处理器共享网络的流体和扩散近似，目的是了解此类网络的动态，获得性能指标，并开发用于研究与网络模型相关的度量值过程的通用工具。 随机网络被用作复杂制造、电信和计算机系统的模型。这种网络的一个具有挑战性的问题是设计控制，是简单的实施，但在适当的意义上接近最优。受这些问题的启发，一些与控制和分析随机网络性能相关的数学问题正在研究中。由于随机网络模型的复杂性通常排除了精确的分析，重点是近似模型与层次结构。与最近的一些网络的性能分析工作一样，在这个层次结构中的层次之间的相互作用是研究的一个重要特征。

The aim of this project is to study mathematical problems associated with the control and performance analysis of stochastic networks. The network models being considered are heterogeneous, may have complex feedback mechanisms, and allow for stochastic variability in arrivals, service times and routing. Buffers in the network enable jobs to be stored that cannot be served immediately. In addition, the models with control allow for dynamic sequencing and alternate routing of jobs. Since the complexity of these network models usually precludes exact analysis, the focus is on approximate models. Two levels of approximation are being considered, namely fluid models (first order approximations) and diffusion models (second order approximations). Investigating the interplay between these models is an important feature of the research. The two main topics are (i) dynamic control through sequencing and alternate routing, and (ii) performance analysis for processor sharing networks. With regard to topic (i), some authors have successfully used diffusion control problems as formal tools for generating good control policies for some specific network models. However, there are few rigorous analyses of the performance of such policies. The PI is developing a systematic approach to finding and interpreting solutions of diffusion control problems and to analyzing the performance of the policies generated in this manner. Regarding topic (ii), the processor sharing discipline is an example of a service discipline for which a natural state descriptor involves a measure-valued process (to keep track of the residual service times of all jobs in the network). The PI is studying fluid and diffusion approximations of processor sharing networks with the aims of understanding the dynamics of such networks, obtaining measures of performance, and developing general tools for studying measure-valued processes associated with network models. Stochastic networks are used as models for complex manufacturing, telecommunications and computer systems. A challenging problem for such networks is to design controls that are simple to implement and yet are near optimal in an appropriate sense. Motivated by such problems, a number of mathematical questions associated with controlling and analyzing the performance of stochastic networks are being studied under this grant. Since the complexity of stochastic network models usually precludes exact analysis, the focus is on approximate models with a hierarchical structure. As with recent work on the performance analysis of some networks, the interplay between the levels in this hierarchy is an important feature of the research.