Hostname: page-component-6766d58669-kl59c Total loading time: 0 Render date: 2026-05-24T18:27:33.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Implied costs in loss networks

Published online by Cambridge University Press:  01 July 2016

P. J. Hunt
P. J. Hunt*
Affiliation:
University of Cambridge
*
∗ Postal address: Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

Implied costs in loss networks are measures of the rate of change of an objective function with respect to the parameters of the network. This paper considers these costs and the costs predicted by the Erlang fixed-point approximation. We derive exact expressions for the implied costs and consider the asymptotic accuracy of the approximation. We show that the approximation is asymptotically valid in some cases but is not valid in one important limiting regime. We also show that a linearity approximation for the implied costs is asymptotically correct when taken over suitable subsets of links.

Keywords

OPTIMISATIONERLANG'S FORMULA

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 3 , September 1989 , pp. 661 - 680
Copyright
Copyright © Applied Probability Trust 1989 

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.)

Article purchase

Temporarily unavailable

Footnotes

Supported by SERC grant No. 87001346.

References

[1] Arnold, L. (1974) Stochastic Differential Equations: Theory and Applications. Wiley, New York.Google Scholar
[2] Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
[3] Gibbens, R. J., Kelly, F. P. and Key, P. B. (1988) Dynamic alternative routing—modelling and behaviour. International Teletraffic Congress 12, Turin 1988.Google Scholar
[4] Girard, A. and Ouimet, Y. (1983) End-to-end blocking for circuit-switched networks: polynomial algorithms for some special cases. IEEE Trans. Comm. 31, 1269–1273.Google Scholar
[5] Holtzman, J. M. (1971) Analysis of dependence effects in telephone trunking networks. Bell System Tech. J. 50, 2647–2663.Google Scholar
[6] Hunt, P. J. and Kelly, F. P. (1988) On critically loaded loss networks. To appear.Google Scholar
[7] Jagerman, D. L. (1974) Some properties of the Erlang loss function. Bell System Tech. J. 53, 525–551.Google Scholar
[8] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
[9] Kelly, F. P. (1986) Blocking probabilities in large circuit-switched networks. Adv. Appl. Prob. 18, 473–505.Google Scholar
[10] Kelly, F. P. (1988) Routing in circuit-switched networks optimization, shadow prices and decentralization. Adv. Appl. Prob. 20, 112–144.Google Scholar
[11] Key, P. B. and Whitehead, M. J. (1988) Cost-effective use of networks employing dynamic alternative routing. International Teletraffic Congress 12, Turin 1988.Google Scholar
[12] Rao, C. R. (1965) Linear Statistical Inference and its Applications. Wiley, New York.Google Scholar
[13] Rogers, L. C. G. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales, Volume 2: Itô Calculus. Wiley, Chichester.Google Scholar
[14] Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
[15] Whitt, W. (1985) Blocking when service is required from several facilities simultaneously. A. T. & T. Tech. J. 64, 1807–1856.Google Scholar
[16] Whittle, P. (1988) Approximation in large-scale circuit-switched networks. Prob. Eng. Inf. Sci. To appear.Google Scholar
[17] Ziedins, I. B. (1985) Blocking in queueing and loss systems. Knight Prize Essay, University of Cambridge.Google Scholar