Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T02:58:19.191Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected].
∗∗Postal address: Universidade Estadual de Campinas, IMECC, UNICAMP, Caixa Postal 6065, 13081–970 – Campinas SP, Brazil.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Daley, D. J., and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Darroch, J. N., and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous time finite Markov chains. J. Appl. Prob. 4, 192–196.Google Scholar
Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Garcia, N. L. (1995). Birth and death processes as projections of higher dimensional Poisson processes. Adv. Appl. Prob. 27, 911–930.Google Scholar
Hall, P. (1985). On continuum percolation. Ann. Prob. 13, 1250–1266.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
Kelly, F. P. (1987). One dimensional circuit-switched networks. Ann. Prob. 15, 1166–1179.Google Scholar
Kelly, F. P. (1991). Loss networks. Ann. Appl. Prob. 1, 319–378.Google Scholar
Kurtz, T. G. (1983). Gaussian approximations for Markov chains and counting processes. Bull. Internat. Statist. Inst., Proceedings of the 44th Session, Invited Papers, 1, 361–376.Google Scholar
Kurtz, T. G. (1989). Stochastic processes as projections of Poisson random measures. Special invited paper at IMS meeting, Washington, DC. Unpublished.Google Scholar
Lotwick, H. W., and Silverman, B. W. (1981). Convergence of spatial birth-and-death processes. Math. Proc. Camb. Phil. Soc. 90, 155–165.Google Scholar
Neuts, M. F. (1981). Matrix-geometric Solutions in Stochastic Models. An algorithmic approach. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Pollett, P. (1988). Reversibility, invariance and μ-invariance. Adv. Appl. Prob. 20, 600–621.Google Scholar
Ycart, B. (1993). The philosophers' process: an ergodic reversible nearest particle system. Ann. Appl. Prob. 3, 356–363.Google Scholar
Ziedins, I. (1987). Quasi-stationary distributions and one-dimensional circuit-switched networks. J. Appl. Prob. 24, 965–977.Google Scholar