Asymptotics for Queueing Networks, Branching Random Walks, and Interacting Particle Systems

主要研究人员对随机过程的主要研究继续在排队网络领域，以及与分支随机游动和相互作用的粒子系统相关的主题。这些区域处理大型对象系统，如客户、作业或粒子，这些对象通过某种随机交互规则连接在一起。他最近还对某些有界域上的布朗运动的耦合问题感兴趣。在排队网络领域，过去二十年的一个重要问题是这样一个系统什么时候是稳定的，以及当它稳定时，如何分析它的行为；目前，仍然缺乏一个普遍的理论。PI打算研究几个网络系列的行为，例如客户总是加入最短队列的网络和连接到互联网流量的网络。在相互作用粒子系统领域，人们通常研究大型格值随机系统的演化，它模拟各种复杂的随机系统并显示出广泛的空间行为；它们经常发生在数学物理和数学生物学的背景下。这些模型有时与涉及分支随机行走的模型重叠，或与数学物理中的其他模型重叠。PI打算研究许多这样的模型的行为，包括分支随机游动、高斯自由场和排除过程。PI还打算研究布朗运动的某些性质，特别是在什么条件下，某些有界域上的布朗运动对可以以正概率始终保持分离。主要研究者的研究集中在概率论的几个领域。他在排队网络领域工作，研究排队网络的行为如何演变。例子包括用于制造过程的部件(例如，半导体晶片的制造)的人工生产线(客户)或装配线。当为客户或工作提供服务的规则涉及多个步骤时，这些队列的时间演变可能会产生意想不到的行为；了解这些行为对于为这些系统设计高效且具有成本效益的规则非常重要。PI在排队网络方面的大部分工作都是朝着这个方向进行的，包括客户或工作人员选择加入最短排队的系统，例如，当购物者在超市付款时。首席研究员还在相互作用的粒子系统领域进行研究，其中研究复杂随机系统的演化。这样的系统出现在物理学和生物学中，它们的组件可以对应于粒子、细胞或单个生物体，并可以模拟例如肿瘤细胞的扩散。由于不同系统表现出非常不同的行为，目前人们对许多方面只有部分了解，而且由于此类系统通常太复杂，无法进行显式计算，因此理解这些系统的数学理论很重要。这个项目将研究一些这样的模型的行为。

The principal investigator's main research on stochastic processes continues to be in the area of queueing networks, and on topics related to branching random walks and interacting particle systems. These areas deal with large systems of objects, such as customers, jobs, or particles, that are connected by some random interaction rule. He has recently also become interested in coupling problems for Brownian motion on certain bounded domains. In the area of queueing networks, an important question over the past two decades had been when such a system is stable and, when it is stable, to analyze its behavior; presently, a general theory is still lacking. The PI intends to study the behavior of several families of networks, such as those where the customer always joins the shortest queue and those connected with Internet traffic. In the area of interacting particle systems, one typically studies the evolution of large lattice-valued random systems, which model various complex random systems and exhibit a wide range of spatial behavior; they frequently occur in the context of mathematical physics and mathematical biology. These models sometimes overlap with models involving branching random walks or with other models from mathematical physics. The PI intends to study the behavior of a number of such models, including those for branching random walks, Gaussian free fields, and the exclusion process. The PI also intends to study certain properties of Brownian motion, in particular, under what conditions pairs of Brownian motions on certain bounded domains can, with positive probability, remain separated for all time.The principal investigator's research is concentrated in several areas of probability theory. He works in the area of queueing networks, where the behavior of how lines evolve is studied. Examples include human lines ("customers") or assembly lines for component parts ("jobs") of a manufacturing process, for instance, the manufacture of semiconductor wafers. When the rule by which customers or jobs are served involves multiple steps, the evolution in time of such lines can have unexpected behavior; understanding such behavior is important for designing efficient and cost-effective rules for these systems. Much of PI's work in queueing networks is in this direction, with continuing work including systems where customers or jobs choose to join the shortest line, for instance, when shoppers pay for purchases at a supermarket. The principal investigator also performs research in the area of interacting particle systems, where the evolution of complicated random systems is studied. Such systems arise in physics and biology, and their components can correspond to particles, cells, or individual organisms, and may model, for instance, the spread of tumor cells. Because of the very different behavior exhibited by different systems, many aspects are currently only partially understood, and since such systems are typically too complicated to do explicit computations, a mathematical theory to understand these systems is important. This project will study the behavior of a number of such models.