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.
"Heavy traffic limit for a processor sharing queue with soft deadlines." Ann. Appl. Probab. 17 (3) 1049 - 1101, June 2007. https://doi.org/10.1214/105051607000000014