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

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Pablo A. Ferrari  and
Nancy Lopes Garcia
Pablo A. Ferrari*
Affiliation:
Universidade de São Paulo
Nancy Lopes Garcia*
Affiliation:
Universidade Estadual de Campinas
*
∗Postal address: Universidade de São Paulo, IME USP, Caixa Postal 66281, 05315–970 – São Paulo, Brazil. Email address: pablo@ime.usp.br.
∗∗Postal address: Universidade Estadual de Campinas, IMECC, UNICAMP, Caixa Postal 6065, 13081–970 – Campinas SP, Brazil.
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Abstract

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).

Keywords

Loss networksPoisson processprojection methodstationary distributionquasi-stationary distributions

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 963 - 975
Copyright
Copyright © Applied Probability Trust 1998 

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