Fluid approximations for a processor-sharing queue

This paper studies the uid approximation, also known as the functional strong law-of-large-numbers, for a GI/G/1 queue under a processor-sharing service discipline. The uid (approximation) limit in general depends on the service time distribution, and the convergence is in general in the Skorohod J 1 topology. This is in contrast to the known result for the GI/G/1 queue under a FIFO service discipline, where the uid limit depends on the service time distribution only through its mean, and the convergence is in the uniform topology. Speciically, we prove a standard uid approximation theorem for the case with no arrivals, and prove a weaker version of the uid approximation theorem for the more general case. We also characterize the uid limit.