  This paper considers a GI/GI/1 processor sharing queue in which jobs have
soft deadlines. At each point in time, the collection of residual service times
and deadlines is modeled using a random counting measure on the right
half-plane. The limit of this measure valued process is obtained under
diffusion scaling and heavy traffic conditions and is characterized as a
deterministic function of the limiting queue length process. As special cases,
one obtains diffusion approximations for the lead time profile and the profile
of times in queue. One also obtains a snapshot principle for sojourn times.
