many important models of stochastic networks exhibit congestion and delay. this implies that the total time a job spends in the system (the "sojourn time") is typically longer than the actual amount of service time needed by the job. the sojourn time is a classical measure of the performance of a queueing system. recently, various authors have begun to consider more general measures of the delay experienced in a queueing system. one such measure is the "lead time," a dynamic quantity describing the time until expiration of some deadline which the job may have. the initial lead time of a job could be random and different from the service time of the job. in this talk, we will discuss recent results for the gi/gi/1 processor sharing queue when jobs have timing requirements represented as lead times. our primary tools are fluid model and state space collapse techniques involving a measure valued process that jointly keeps track of residual service times and lead times of individual jobs in the system. the main result is a heavy traffic diffusion approximation for the (appropriately rescaled) measure valued process. this is joint work with lukasz kruk.