Diffusion Limits for Shortest Remaining Processing Time Queues under Nonstandard Spatial Scaling
Diffusion Limits for Shortest Remaining Processing Time Queues under Nonstandard Spatial Scaling
复制标题
非标准空间尺度下最短剩余处理时间队列的扩散极限
DOI：
10.1214/14-aap1076
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Amber L. Puha
中科院分区：
文献类型：
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作者：
Amber L. Puha
我们针对单服务器队列中采用最短剩余处理时间（SRPT）的队列长度过程，开发了非标准空间缩放下的大流量扩散极限定理。对于具有无限支持的处理时间分布，已经证明标准扩散缩放产生相同的零极限。我们指定产生非零限制的替代空间缩放。我们的模型允许续签到达和 i.i.d.处理时间满足快速变化条件。我们向标准扩散缩放添加了校正空间缩放因子，并指定非常规缩放的队列长度过程序列在分布中收敛到与常规缩放的工作负载过程序列收敛到的相同非零反射布朗运动的条件。因此，该校正空间比例因子表征了大流量中 SRPT 队列的队列长度和工作负载过程之间的数量级差异。它由处理时间分布确定，使得其趋于无穷大的速率取决于处理时间分布的尾部趋于零的速率。对于威布尔处理时间分布，我们以一种使结果状态空间崩溃更加明显的方式重申了这个结果。
We develop a heavy traffic diffusion limit theorem under nonstandard spatial scaling for the queue length process in a single server queue employing shortest remaining processing time (SRPT). For processing time distributions with unbounded support, it has been shown that standard diffusion scaling yields an identically zero limit. We specify an alternative spatial scaling that produces a nonzero limit. Our model allows for renewal arrivals and i.i.d. processing times satisfying a rapid variation condition. We add a corrective spatial scale factor to standard diffusion scaling, and specify conditions under which the sequence of unconventionally scaled queue length processes converges in distribution to the same nonzero reflected Brownian motion to which the sequence of conventionally scaled workload processes converges. Consequently, this corrective spatial scale factor characterizes the order of magnitude difference between the queue length and workload processes of SRPT queues in heavy traffic. It is determined by the processing time distribution such that the rate at which it tends to infinity depends on the rate at which the tail of the processing time distribution tends to zero. For Weibull processing time distributions, we restate this result in a manner that makes the resulting state space collapse more apparent.