Fluid Limits for Longest Remaining Time First Queues

A single-server queue with renewal arrivals and generally distributed independent and identically distributed service times is considered. Customers are served using the longest remaining time first scheduling algorithm. In case of a tie, processor sharing is utilized. We introduce a fluid model for the evolution of a measure-valued state descriptor of this queue, and we investigate its properties. We also prove a fluid limit theorem justifying our fluid model as the first-order approximation of the queueing system under consideration.