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Loss networks under diverse routing: the symmetric star network

Published online by Cambridge University Press:  01 July 2016

P. J. Hunt*
Affiliation:
University of Cambridge
*
*Present address: NatWest Capital Markets, 135 Bishopsgate, London EC2M 3UR, UK.

Abstract

A highly symmetric loss network is considered, the symmetric star network previously considered by Whitt and Ziedins and Kelly. As the number of links, K, becomes large, the state space for this process also grows, so we consider a functional of the network, one which contains all information relevant to blocking probabilities within the network but which is easier to analyse. We show that this reduced process obeys a functional law of large numbers and a functional central limit theorem, the limit in this latter case being an Ornstein-Uhlenbeck diffusion process. Finally, by considering the network in equilibrium, we are able to prove that the well-known Erlang fixed point approximation for blocking probabilities is correct to within o(K-1/2) as K → ∞.

MSC classification

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Supported by Christ's College, Cambridge.

References

[1] Arnold, L. (1974) Stochastic Differential Equations: Theory and Applications. Wiley, New York.Google Scholar
[2] Arnold, V. I. (1973) Ordinary Differential Equations. MIT Colonial Press, Cambridge, MA.Google Scholar
[3] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[4] Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.CrossRefGoogle Scholar
[5] Hunt, P. J. (1989) Implied costs in loss networks. Adv. Appl. Prob. 21, 661680.CrossRefGoogle Scholar
[6] Hunt, P. J. (1990) Limit Theorems for Stochastic Loss Networks. PhD thesis, University of Cambridge.Google Scholar
[7] Hunt, P. J. (1992) Loss networks under diverse routing, II: L-symmetric networks. Technical Report, University of Cambridge.Google Scholar
[8] Kelly, F. P. (1986) Blocking probabilities in large circuit-switched networks. Adv. Appl. Prob. 18, 473505.Google Scholar
[9] Kelly, F. P. (1991) Loss networks. Ann. Appl. Prob. To appear.CrossRefGoogle Scholar
[10] Louth, G. M. (1990) Stochastic networks: complexity, dependence and routing. Unpublished.Google Scholar
[11] Whitt, W. (1980) Some useful functions for functional limit theorems. Math. Operat. Res. 5, 6785.Google Scholar
[12] Whitt, W. (1985) Blocking when service is required from several facilities simultaneously. AT. & T Tech. J. 64, 18071856.CrossRefGoogle Scholar
[13] Ziedins, I. B. (1987) Stochastic Models of Traffic in Star and Line Networks. PhD thesis, University of Cambridge.Google Scholar
[14] Ziedins, I. B. and Kelly, F. P. (1989) Limit theorems for loss networks with diverse routing. Adv. Appl. Prob. 21, 804830.Google Scholar