On Queues in Heavy Traffic

We say that a single server queue is in "heavy traffic" when the traffic intensity ρ is less than, but very near, unity. Then the equilibrium waiting time w will be large, and it is proved that, under certain conditions, the distribution of (1 - ρ)w tends, as ρ tends to 1, to a negative exponential distribution. The conditions under which this result holds are fairly weak, and, in particular, it is not necessary to make the assumption of independent inter-arrival times.