Ergodicity Conditions For A Continuous One-dimensional Loss Network

One dimensional continuous loss networks are spatial birth-and-death processes which can be dominated by a multitype branching process. Using the Peron-Frobenius theory for sub-criticality of branching process we obtain a sufficient condition for ergodicity of one-dimensional loss networks on ℝ with arbitrary length distribution π and cable capacity C.343349360Fernández, R., Ferrari, P.A., Garcia, N.L., Loss network representation of Pierls contours (2001) Annals of Probability, 29 (2), pp. 902-937Fernández, R., Ferrari, P.A., Garcia, N.L., Perfect simulation for interacting point processes, loss networks and Ising models (2002) Los Alamos Archive: Mathematics. Abstract Math., PR-9911162T. , PreprintFerrari, P.A., Garcia, N.L., One-dimensional loss networks and conditioned M/G/∞ queues (1998) J. Appl. Probab., 35 (4), pp. 963-975Garcia, N.L., Perfect simulation of spatial processes (2000) Resenhas IME-USP, 4 (3), pp. 283-325Kelly, F.P., Loss networks (1991) Ann. Appl. Probab., 1 (3), pp. 319-37