## The Annals of Applied Probability

### Diffusion limits for shortest remaining processing time queues under nonstandard spatial scaling

Amber L. Puha

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3381-3404.

Dates
Revised: July 2014
First available in Project Euclid: 1 October 2015

https://projecteuclid.org/euclid.aoap/1443703777

Digital Object Identifier
doi:10.1214/14-AAP1076

Mathematical Reviews number (MathSciNet)
MR3404639

Zentralblatt MATH identifier
1328.60205

#### Citation

Puha, Amber L. Diffusion limits for shortest remaining processing time queues under nonstandard spatial scaling. Ann. Appl. Probab. 25 (2015), no. 6, 3381--3404. doi:10.1214/14-AAP1076. https://projecteuclid.org/euclid.aoap/1443703777

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