Mathematical Sciences: Interacting Particle Systems and Queueing Networks

研究者的研究方向是随机过程，主要是相互作用粒子系统和排队网络。尽管典型地吸引着不同的研究群体，但这两个领域都处理由物体组成的大系统，如粒子或顾客，它们通过某种相互作用规则连接在一起。相互作用粒子系统的研究通常涉及格值随机系统，这些系统根据某些局部规则进行更新。这些系统模拟各种复杂的随机系统，并表现出广泛的空间行为；它们经常出现在数学物理和数学生物学的背景下。人们感兴趣的是这种系统的长期行为，比如相关平衡的行为。研究方向包括多类型粒子系统的结构、具有湮灭和创造的系统、某些半无限系统的平衡和界面动力学等。排队网络研究在不同的路由规则和分配优先级下个体队列的演化。随着时间的推移，个人进入系统，从一个队列移动到下一个队列，并在服务完成后退出系统。根据具体情况，这样的系统可能是稳定的，也可能不是稳定的；最近许多研究的目标都是分析这种行为。最近解决这类问题的一个重要工具是使用流体极限，有了它，关于网络稳定性的问题可以简化为相关流体模型的稳定性。研究者打算研究一些主要类别的流体模型的性质，并将这种分析应用于网络稳定性问题的研究以及网络大流量限制的相关主题。研究者的研究主要集中在概率论的两个领域，相互作用粒子系统和排队网络。相互作用粒子系统领域通常处理大型随机系统。这些系统的组成部分可以代表，例如，粒子、细胞或各种生物体。这样的系统模拟各种复杂的随机系统，并表现出广泛的行为；它们经常出现在数学物理和数学生物学的背景下。人们通常对这类系统的长期行为感兴趣。研究者打算研究的课题包括不同物种多样性和稳定性的生物学模型，以及化学反应进化的物理模型。排队网络研究一般情况下个体排队的行为。例如，这些个体可以是等待服务的客户，或者是某些制造过程中涉及的组件。随着时间的推移，个人进入系统，从一条线移动到另一条线，并在服务完成后退出系统。一般未解决的问题是找出哪些系统是稳定的，也就是说，它们的线的长度不会随着时间的增加而增长。然后，这些信息可以用来指示为客户服务的更有效的算法，以节省时间和材料。流体模型是分析排队网络的一个重要工具，它是原始网络的非随机简化。研究者打算通过研究和应用流体模型的性质来继续他在该领域的工作。

Bramson The investigator's research is in stochastic processes, primarily in the areas of interacting particle systems and queueing networks. Although typically appealing to different groups of researchers, both areas deal with large systems of objects, such as particles or customers, which are connected together by some interaction rule. Interacting particle systems research usually deals with lattice-valued random systems which update according to some local rule. Such systems 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. One is interested in the long-time behavior of such systems, such as the behavior of the associated equilibria. The investigator intends to work on topics such as the structure of multitype particle systems, systems with annihilation and creation, the equilibria of certain semi-infinite systems, and interface dynamics. Queueing networks research studies the evolution of queues of individuals under different rules for routing and assigned priorities. As time evolves, individuals enter the system, move from one queue to the next, and, upon completion of service, exit from the system. Depending on specifics, such a system may or may not be stable; the goal of much recent research has been to analyze such behavior. A recent important tool for such problems is the use of fluid limits, with which problems on stability of networks can be reduced to the stability of the associated fluid models. The investigator intends to investigate the properties of some of the main classes of fluid models, and apply this analysis to the study of stability questions for networks and to the related topic of heavy traffic limits for networks. The investigator's research lies primarily in two areas of probability theory, interacting particle systems and queueing networks. The field of interacting particle systems typically deals with large random systems. The compon ents of these systems can represent, for example, particles, cells, or various organisms. Such systems model various complex random systems and exhibit a wide range of behavior; they frequently occur in the context of mathematical physics and mathematical biology. One is typically interested in the long-time behavior of such systems. The topics the investigator intends to work on include biological models for the diversity and stability of different species, and physical models for the evolution of chemical reactions. The field of queueing networks studies the behavior of lines of individuals in general settings. These individuals can, for example, be customers waiting to be served, or components involved in some manufacturing process. As time evolves, individuals enter the system, move from one line to the next, and, upon completion of service, exit from the system. A general unsolved problem is to find which systems are stable, that is, the lengths of their lines do not tend to grow as time increases. This information can then be used to indicate more efficient algorithms for the service of customers, with the goal of saving time and materials. A recent important tool for the analysis of queueing networks is the use of fluid models, which are nonrandom simplifications of the original networks. The investigator intends to continue his work in the area by studying and applying the properties of fluid models.