随机负载平衡网络的流体动力学极限
The hydrodynamic limit of a randomized load balancing network
摘要：
随机负载均衡网络出现在各种应用中，并且允许资源的有效共享，同时相对容易实现。我们考虑一个网络的并行队列中，传入的工作与独立和相同分布的服务时间被分配到最短的队列中随机选择的子集的$d$队列，并离开网络完成服务。以前的工作，这个模型的动力学性质集中在指数服务分布的情况下。在这项工作中，我们分析了更现实的情况下，一般服务分布。首先，我们引入了一种新的粒子表示的网络状态，并通过一系列相互作用的测度值随机过程的状态动力学特征。在温和的假设下，我们表明，缩放状态过程的序列收敛，作为服务器的数量趋于无穷大，到一个流体动力学的限制，其特征在于作为唯一的解决方案，耦合确定性测度值方程的可数系统。我们还建立了一个混沌传播的结果，表明有限的队列集合是渐近独立的。在这里开发的一般框架是潜在的有用的分析在不同领域，包括生物学和材料科学中产生的模型的更大的类。
英文摘要：
Randomized load balancing networks arise in a variety of applications, and allow for efficient sharing of resources, while being relatively easy to implement. We consider a network of parallel queues in which incoming jobs with independent and identically distributed service times are assigned to the shortest queue among a randomly chosen subset of $d$ queues, and leave the network on completion of service. Prior work on dynamical properties of this model has focused on the case of exponential service distributions. In this work, we analyze the more realistic case of general service distributions. We first introduce a novel particle representation of the state of the network, and characterize the state dynamics via a sequence of interacting measure-valued stochastic processes. Under mild assumptions, we show that the sequence of scaled state processes converges, as the number of servers goes to infinity, to a hydrodynamic limit that is characterized as the unique solution to a countable system of coupled deterministic measure-valued equations. We also establish a propagation of chaos result that shows that finite collections of queues are asymptotically independent. The general framework developed here is potentially useful for analyzing a larger class of models arising in diverse fields including biology and materials science.
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