  We present a heavy traffic analysis for a single server queue with renewal
arrivals and generally distributed i.i.d. service times, in which the server
employs the Shortest Remaining Processing Time (SRPT) policy. Under typical
heavy traffic assumptions, we prove a diffusion limit theorem for a
measure-valued state descriptor, from which we conclude a similar theorem for
the queue length process. These results allow us to make some observations on
the queue length optimality of SRPT. In particular, they provide the sharpest
illustration of the well-known tension between queue length optimality and
quality of service for this policy.
