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December 2015 Diffusion limits for shortest remaining processing time queues under nonstandard spatial scaling
Amber L. Puha
Ann. Appl. Probab. 25(6): 3381-3404 (December 2015). DOI: 10.1214/14-AAP1076

Abstract

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.

Citation

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Amber L. Puha. "Diffusion limits for shortest remaining processing time queues under nonstandard spatial scaling." Ann. Appl. Probab. 25 (6) 3381 - 3404, December 2015. https://doi.org/10.1214/14-AAP1076

Information

Received: 1 January 2014; Revised: 1 July 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1328.60205
MathSciNet: MR3404639
Digital Object Identifier: 10.1214/14-AAP1076

Subjects:
Primary: 60F17, 60K25
Secondary: 60G57, 68M20, 90B22

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 6 • December 2015
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