# The Fluid Limit of a Shortest Remaining Processing Time Queue.

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Consider a GI/GI/1 queue operating under shortest remaining processing time with preemption. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. Of particular interest is the waiting time for large jobs, which can be tracked using the frontier process, the largest service time of any job that has ever been in service. We propose a fluid model and present a functional limit theorem justifying it as an approximation of this system. The fluid model state descriptor is a measure valued function for which the left edge of the support is the fluid analog for the frontier process.

Under mild assumptions, we prove existence and uniqueness of fluid model solutions.

Furthermore, we are able to characterize the left edge of fluid model

solutions as the right continuous inverse of a simple functional of the initial condition,

arrival rate, and service time distribution. When applied to various examples, this

characterization reveals the dependence on service time distribution of the rate at which the

left edge of the fluid model increases.