On pathwise analysis and existence of empirical distributions for G/G/1 queues

In this paper we study the existence of empirical distributions of G/G/1 queues via a sample-path approach. We show the convergence along a given trajectory of empirical distributions of the workload process of a G/G/1 queue under the condition that the work brought into the system has strictly stationary increments and the time average of the queue load converges along the trajectory to a quantity ϱ < 1. In particular, we identify the limit as the expectation with respect to the Palm distribution associated with the beginning of busy cycles. The approach is via the use of a sample-path version of Beneš result describing the time evolution of the workload process. It turns out that the Beneš equation leads to consideration of the renovation arguments similar to those used in the framework of Borovkovʹs renovating events.