Abstract

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queue M/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues lambda is an element of (0, lambda(c)], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.