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Annals of Operations Research

, Volume 8, Issue 1, pp 57–73 | Cite as

Asymptotic marginal independence in large networks of loss systems

  • D. P. Heyman
Part I Numerical Problems In Probability

Abstract

Network models in which each node is a loss system frequently arise in telephony. Models with several hundred nodes are common. Suppose a customer requires a server from each of several nodes. It would be convenient if the probability that the required servers are all free were approximately a product, where each term is the probability a required node has a free server. We present some theorems to support this approximation. Most of the theorems are restricted to nodes with one server. Some of the difficulties in analyzing nodes with multiple servers are described.

Keywords and phrases

Approximations product-form solutions limit theorems 

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References

  1. [1]
    J.M. Akinpelu, The overload performance of engineered networks with nonhierarchical and hierarchical routing, AT&T Tech. J. 63(1984)1261.Google Scholar
  2. [2]
    P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
  3. [3]
    D.Y. Burman, J.P. Lohoczky and Y. Lim, Insensitivity of blocking probabilities in a circuit switching network, J. Appl. Prob. 21(1984)850.Google Scholar
  4. [4]
    D.Y. Burman, Insensitivity in queueing systems, Adv. Appl. Prob. 13(1981)846Google Scholar
  5. [5]
    L. Green, A queueing system in which customers require a random number of servers, Oper. Res. 28(1980)1335.Google Scholar
  6. [6]
    D.P. Heyman and M.J. Sobel,Stochastic Models in Operations Research, Vol. I (McGraw-Hill, New York, 1982).Google Scholar
  7. [7]
    A.F. Karr, Weak convergence of a sequence of Markov chains, Z. Wahrscheinlichkeitstheorie verw. Gebiete 33(1975)41.Google Scholar
  8. [8]
    D. Mitra and P.J. Weinberger, Probabilistic models of database locking: Solutions, computational algorithms and asymptotics, J. Assoc. Comput. Mach. 31(1984)855.Google Scholar
  9. [9]
    A.G. Pakes, Some conditions for ergodicity and recurrence of Markov chains, Oper. Res. 17(1969)1058.Google Scholar
  10. [10]
    D. Sonderman, Comparing semi-Markov processes, Math. Oper. Res. 5(1980)110.Google Scholar
  11. [11]
    W. Whitt, Continuity of generalized semi-Markov processes, Math. Oper. Res. 5(1980)494.Google Scholar
  12. [12]
    W. Whitt, Blocking when service is required from several facilities simultaneously, AT&T Tech. J. (1985)1807.Google Scholar
  13. [13]
    E. Wolman, The camp-on problem for multiple-address traffic, Bell Syst. Tech. J. 51(1972)1363.Google Scholar

Copyright information

© J.C. Baltzer A.G., Scientific Publishing Company 1987

Authors and Affiliations

  • D. P. Heyman
    • 1
  1. 1.Bell Communications ResearchRed BankUSA

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