  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.
