ABSTRACT: 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.