  We study a single server queue operating under the shortest remaining
processing time (SRPT) scheduling policy; that is, the server preemptively
serves the job with the shortest remaining processing time first. In this work
we are interested in studying the asymptotic behavior of suitably scaled
measure-valued state descriptors that describe the evolution of a sequence of
SRPT queuing systems. Gromoll, Kruk, and Puha (2011) have studied this problem
under diffusive scaling. In the setting where the processing time distributions
have unbounded support, under suitable conditions, they show that the diffusion
scaled measures converge in distribution to the process that is identically
zero. In Puha (2015) for the setting where the processing time distributions
have unbounded support and light tails, a non-standard scaling of the queue
length process is shown to give rise to a form of state space collapse that
results in a nonzero limit. In the current work we consider the case where
processing time distributions have finite second moments and regularly varying
tails. We show that the measure valued process, under a non-standard scaling,
converges in distribution in the space of paths of measures. In sharp contrast
with previous results, there is no state space collapse. Nevertheless, the
description of the limit is simple and given explicitly in terms of a certain
$\mathbb{R}_+$ valued random field which is determined from a single Brownian
motion. Along the way we establish convergence of suitably scaled workload and
queue length processes. We also show that as the tail of the distribution of
job processing times becomes lighter in an appropriate fashion, the difference
between the limiting queue length process and the limiting workload process
converges to zero, thereby approaching the behavior of state space collapse.

Probability