零等待负载均衡算法的大系统不敏感性
Large-System Insensitivity of Zero-Waiting Load Balancing Algorithms
摘要：
本文研究了一类在次Halfin-Whitt区实现渐近零等待的负载平衡算法LB-零的灵敏度（或不灵敏度）。现有的零等待负载均衡算法的结果大多假设服务时间是指数分布的。本文建立了服务时间服从Coxian分布且具有有限个阶段的工件的LB-零{ \em大系统不灵敏度}。这一结果表明，LB-零实现渐近零等待的一大类服务时间分布，这是在我们的模拟证实。为了证明这一结果，本文提出了一种新的技术，称为“迭代状态空间剥离”（或简称ISSP）。ISSP首先确定了排队状态上界和下界之间的迭代关系，然后证明了系统以高概率生活在迭代界的不动点附近。在此基础上，利用Stein方法在不动点附近进一步分析了系统的稳态分布。ISSP，像状态空间崩溃在繁忙的交通分析，是一种通用的方法，可用于研究其他复杂的随机系统。
英文摘要：
This paper studies the sensitivity (or insensitivity) of a class of load balancing algorithms that achieve asymptotic zero-waiting in the sub-Halfin-Whitt regime, named LB-zero. Most existing results on zero-waiting load balancing algorithms assume the service time distribution is exponential. This paper establishes the {\em large-system insensitivity} of LB-zero for jobs whose service time follows a Coxian distribution with a finite number of phases. This result suggests that LB-zero achieves asymptotic zero-waiting for a large class of service time distributions, which is confirmed in our simulations. To prove this result, this paper develops a new technique, called "Iterative State-Space Peeling" (or ISSP for short). ISSP first identifies an iterative relation between the upper and lower bounds on the queue states and then proves that the system lives near the fixed point of the iterative bounds with a high probability. Based on ISSP, the steady-state distribution of the system is further analyzed by applying Stein's method in the neighborhood of the fixed point. ISSP, like state-space collapse in heavy-traffic analysis, is a general approach that may be used to study other complex stochastic systems.
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