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

Abstract

We study one-dimensional continuous loss networks with length distribution G and cable capacity C. We prove that the unique stationary distribution η_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 distribution of a stationary M/G/∞ 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 → ∞ of η_L exists and is unique. The limiting distribution turns out to be invariant for the infinite loss network. This was conjectured by Kelly (1991).