  We study a G/GI/1 single-server queuing model with i.i.d.\ service times that
are independent of a stationary process of inter-arrival times. We show that
the distribution of the waiting time converges to a stationary law as time
tends to infinity provided that inter-arrival times satisfy a G\"artner-Ellis
type condition. A convergence rate is given and a law of large numbers
established. These results provide tools for the statistical analysis of such
systems, transcending the standard case with independent inter-arrival times.
