Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter
Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter
复制标题
Halfin–Whitt 机制中的加入最短队列扩散极限：对大流量参数的敏感性
DOI：
10.1214/19-aap1496
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Debankur Mukherjee
中科院分区：
文献类型：
--
作者：
Sayantan Banerjee;Debankur Mukherjee
考虑一个$N$并行单服务器队列系统，该系统具有单位指数服务时间分布和单个调度程序，其中任务作为速率为$\lambda(N)$的泊松过程到达。当任务到达时，调度程序根据最短队列加入（join -the- short Queue， JSQ）策略将其分配给其中一台服务器。Eschenfeldt和Gamarnik（2015）发现了一种新的限制扩散过程，该过程出现在Halfin-Whitt制度下JSQ政策下系统适当比例占用度量的弱极限，其中$(N - \lambda(N)) / \sqrt{N} \to \beta > 0$为$N \to \infty$。对这种扩散的分析超越了技术的水平，甚至证明了它的遍历性是非平凡的，这是一个悬而未决的问题。最近，Braverman（2018）利用Stein's方法利用发电机展开框架建立了其指数遍历性，Banerjee和Mukherjee（2018）采用再生方法分析了扩散平稳分布和路径波动的尾部特性。
然而，在这项工作之前，对平稳分布的总体行为（即力矩）的分析仍然很棘手。在本文中，我们对扩散过程平稳分布的总体行为进行了深入的分析，并发现它表现出惊人的不同定性行为，这取决于重流量参数$\beta$的值。此外，当$\beta$趋于0和$\infty$时，我们得到了中心和缩放稳态分布的精确渐近定律。
Consider a system of $N$ parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate $\lambda(N)$. When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (2015) identified a novel limiting diffusion process that arises as the weak-limit of the appropriately scaled occupancy measure of the system under the JSQ policy in the Halfin-Whitt regime, where $(N - \lambda(N)) / \sqrt{N} \to \beta > 0$ as $N \to \infty$. The analysis of this diffusion goes beyond the state of the art techniques, and even proving its ergodicity is non-trivial, and was left as an open question. Recently, exploiting a generator expansion framework via the Stein's method, Braverman (2018) established its exponential ergodicity, and adapting a regenerative approach, Banerjee and Mukherjee (2018) analyzed the tail properties of the stationary distribution and path fluctuations of the diffusion.
However, the analysis of the bulk behavior of the stationary distribution, viz., the moments, remained intractable until this work. In this paper, we perform a thorough analysis of the bulk behavior of the stationary distribution of the diffusion process, and discover that it exhibits surprisingly different qualitative behavior, depending on the value of the heavy-traffic parameter $\beta$. Moreover, we obtain precise asymptotic laws of the centered and scaled steady state distribution, as $\beta$ tends to 0 and $\infty$.
DOI：
10.48550/arxiv.1510.02328
发表时间：
2015
期刊：
arXiv e-prints
影响因子：
--
作者：
Banerjee Sayan
通讯作者：
Banerjee Sayan