We study one-dimensional continuous loss networks with length distribution G and cable capacity C. We prove that the unique stationary distribution ηL 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 ηL exists and is unique. The limiting distribution turns out to be invariant for the infinite loss network. This was conjectured by Kelly (1991).354963975Daley, D.J., Verb-Jones, D., (1988) An Introduction to the Theory of Point Processes, , Springer, New YorkDarroch, J.N., Seneta, E., On quasi-stationary distributions in absorbing continuous time finite markov chains (1967) J. Appl. Prob., 4, pp. 192-196Ethier, S.N., Kurtz, T.G., (1986) Markov Processes: Characterization and Convergence, , Wiley, New YorkGarcia, N.L., Birth and death processes as projections of higher dimensional poisson processes (1995) Adv. Appl. Prob., 27, pp. 911-930Hall, P., On continuum percolation (1985) Ann. Prob., 13, pp. 1250-1266Hall, P., (1988) Introduction to the Theory of Coverage Processes, , Wiley, New YorkKelly, F.P., One dimensional circuit-switched networks (1987) Ann. Prob., 15, pp. 1166-1179Kelly, F.P., Loss networks (1991) Ann. Appl. Prob., 1, pp. 319-378Kurtz, T.G., Gaussian approximations for markov chains and counting processes (1983) Bull. Internat. Statist. Inst., Proceedings of the 44th Session, Invited Papers, 1, pp. 361-376Kurtz, T.G., Stochastic processes as projections of poisson random measures (1989) IMS Meeting, , Special invited paper Washington, DC. UnpublishedLotwick, H.W., Silverman, B.W., Convergence of spatial birth-and-death processes (1981) Math. Proc. Camb. Phil. Soc., 90, pp. 155-165Neuts, M.F., (1981) Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach, , Johns Hopkins University Press, Baltimore, MDPollett, P., Reversibility, invariance and μ-invariance (1988) Adv. Appl. Prob., 20, pp. 600-621Ycart, B., The philosophers' process: An ergodic reversible nearest particle system (1993) Ann. Appl. Prob., 3, pp. 356-363Ziedins, I., Quasi-stationary distributions and one-dimensional circuit-switched networks (1987) J. Appl. Prob., 24, pp. 965-977