We obtain the classical exponential limit law for steady-state FIFO customer delay in heavy traffic for single-server queues fed by stationary input; the G/G/1 case. By postulating a given fixed stationary input (satisfying reasonable general assumptions), and modifying it (via scaling for example) to reach heavy traffic, our method considerably weakens previously known sufficient conditions. We consider two different setups, each alone sufficient to yield the exponential limit result:
(1) We assume that the stationary input satisfies a functional central limit theorem to Brownian motion, and a minor uniform integrability condition.

(2) We assume that the input satisfies a “strong approximation principle” (to Brownian motion).
This is joint work with Peter Glynn (Stanford University).