A multiclass closed queueing network with unconventional heavy traffic behavior

A multiclass closed queueing network with unconventional heavy traffic behavior
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具有非常规大流量行为的多类封闭排队网络

DOI：
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发表时间：
1996
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影响因子：
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通讯作者：
Ruth J. Williams
Ruth J. Williams
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文献类型：
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作者：
J. Harrison;Ruth J. Williams

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我们考虑一个多类封闭的网络模型类似于开放的网络模型的Rybko和Stolyar和Lu和Kumar。封闭网络有两个单服务器站和一个大小为n的固定客户群。客户通过四个不同的类，其中两个是在每个站服务的循环方式路由，每个服务器使用一个抢占式恢复优先级纪律。每个客户类的服务时间分布是指数的，注意力集中在关键的情况下，所有四个类有相同的平均服务时间。当n趋于无穷大时，我们证明了一个在三个方面都是非常规的重载交通极限定理。首先，在我们的大流量缩放的时间长度的过程和累积空闲过程中，时间被压缩了一个因子n，而不是在传统理论中发生的因子n。第二，应用于时间长度和空闲过程的某些分量的空间尺度是与中心极限定理相关联的，但应用于其他分量的尺度是与大数定律相关联的。因此，在量子力学的语言中，我们的交通量大的极限定理涉及布朗标度和流体标度的混合。最后，我们得到的极限过程不是一个普通的反射布朗运动，如在传统的重交通定理，虽然它是相关的或从布朗运动。
We consider a multiclass closed queueing network model analogous to the open network models of Rybko and Stolyar and of Lu and Kumar. The closed network has two single-server stations and a fixed customer population of size n. Customers are routed in cyclic fashion through four distinct classes, two of which are served at each station, and each server uses a preemptive-resume priority discipline. The service time distribution for each customer class is exponential, and attention is focused on the critical case where all four classes have the same mean service time. Letting n approach infinity, we prove a heavy traffic limit theorem that is unconventional in three regards. First, in our heavy traffic scaling of both queue-length processes and cumulative idleness processes, time is compressed by a factor of n rather than the factor of n occurring in conventional theory. Second, the spatial scaling applied to some components of the queue-length and idleness processes is that associated with the central limit theorem, but the scaling applied to other components is that associated with the law of large numbers. Thus, in the language of queueing theory, our heavy traffic limit theorem involves a mixture of Brownian scaling and fluid scaling. Finally, the limit process that we obtain is not an ordinary reflected Brownian motion, as in conventional heavy traffic theorems, although it is related to or derived from Brownian motion.