Stationary M/G/1 excursions in the presence of heavy tails

It is shown that the stationary excursions above level x for the stationary M/G/1 queue with the service time distribution belonging to a certain class ℐ* of subexponential distributions are asymptotically of two types as x → ∞: either the excursion starts with a jump from a level which is O(1) and the initial excess over x converges to ∞ or it starts from a level of the form x - O(1) and the excess has a proper limit distribution. The two types occur with probabilities ρ, resp. 1 - ρ.