One-dimensional loss networks and conditioned M/G/infinity queues

We study one-dimensional continuous loss networks with length distribution G and cable capacity C. We prove that the unique stationary distribution et a(L) of the network for which the restriction on the number of calls to be less than C is imposed only in the segment [-L, L] is the same as the distr ibution of a stationary M/G/infinity queue conditioned to be less than C in the time interval [-L, L]. For distributions G which are of phase type (= absorbing times of finite state Markov processes) we show that the limit as L --> infinity of eta(L) exists and is unique. The limiting distribution t urns out to be invariant for the infinite loss network. This was conjecture d by Kelly (1991).