Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T04:30:47.911Z Has data issue: false hasContentIssue false
  • English
  • Français

Blocking probabilities in large circuit-switched networks

Published online by Cambridge University Press:  01 July 2016

F. P. Kelly
F. P. Kelly*
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

This paper is concerned with blocking and loss probabilities in circuit-switched networks. We show that when the capacity of links and the offered traffic are increased together, a limiting regime emerges in which loss probabilities are as if links block independently, with blocking probabilities given by the solution of a simple convex programming problem. We then show that an approximate procedure, based on solving Erlang&s formula under the assumption of independent blocking, produces a unique solution when routes are fixed, and that under the limiting regime the estimates of loss probabilities obtained from the procedure converge to the correct values.

Keywords

LOSS PROBABILITIESNETWORK FLOWLOCAL LIMIT THEOREMSERLANGS FORMULAFIXED ROUTINGDYNAMIC ROUTINGCELLULAR RADIO
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 473 - 505
Copyright
Copyright © Applied Probability Trust 1986 

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

1. Akinpelu, J. M. (1983) The overload performance of engineered networks with non-hierarchical routing. 10th Internat. Teletraffic Congress.Google Scholar
2. Appleby, M. (1983) Towards cellular radio. British Telecom J. Winter 1983/84, 4–6.Google Scholar
3. Brockmeyer, E., Halstrom, H. L. and Jensen, A. (1948) The Life and Works of A. K. Erlang. Academy of Technical Sciences, Copenhagen.Google Scholar
4. Burman, D. Y., Lehoczky, J. P. and Lim, Y. (1984) Insensitivity of blocking probabilities in a circuit-switching network. J. Appl. Prob. 21, 850–859.Google Scholar
5. De Souza E Silva, E., Lavenberg, S. S. and Muntz, R. R. (1984) A perspective on iterative methods for the approximate analysis of closed queueing networks. In Iazeolla, et al. [17], 225–244.Google Scholar
6. Disney, R. L. and König, D. (1985) Queueing networks: a survey of their random processes. SIAM Rev. 27, 335–403.CrossRefGoogle Scholar
7. Everitt, D. E. and Macfadyen, N. W. (1983) Analysis of multicellular mobile radiotelephone systems with loss. British Telecom. Technol. J. 1, 37–45.Google Scholar
8. Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
9. 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.CrossRefGoogle Scholar
10. Girard, A. and Hurtubise, S. (1983) Dynamic routing and call repacking in circuit-switched networks. IEEE Trans. Comm. 31, 1290–1294.Google Scholar
11. Gnedenko, B. V. (1968) Theory of Probability, 4th edn. Chelsea, New York.Google Scholar
12. Gondran, M. and Minoux, M. (1984) Graphs and Algorithms. Wiley, Chichester.Google Scholar
13. Haberman, S. J. (1974) The Analysis of Frequency Data. University of Chicago Press, Chicago.Google Scholar
14. Hall, P. (1983) On the roles of the Bessel and Poisson distributions in chemical kinetics. J. Appl. Prob. 20, 585–599.Google Scholar
15. Harvey, C. and Hills, C. R. (1979) Determining grades of service in a network. 9th Internat. Teletraffic Congress.Google Scholar
16. Holtzman, J. M. (1971) Analysis of dependence effects in telephone trunking networks. Bell System Tech. J. 50, 2647–2662.Google Scholar
17. Iazeolla, G., Courtois, P. J. and Hordijk, A., eds. (1984) Mathematical Computer Performance and Reliability. North-Holland, Amsterdam.Google Scholar
18. Katz, S. S. (1967) Statistical performance analysis of a switched communications network. 5th Internat. Teletraffic Congress, 566–575.Google Scholar
19. Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
20. Kelly, F. P. and Ziedins, I. (1985) Loss probabilities in circuit-switched star networks. Unpublished.Google Scholar
21. Krupp, R. S. (1982) Stabilization of alternate routing networks. IEEE International Communications Conference, Philadelphia.Google Scholar
22. Lin, P. M., Leon, B. J. and Stewart, C. R. (1978) Analysis of circuit-switched networks employing originating-office control with spill-forward. IEEE Trans. Comm. 26, 754–765.Google Scholar
23. Mckenna, J. and Mitra, D. (1982) Integral representations and asymptotic expansions for closed Markovian queueing networks: normal usage. Bell System Tech. J. 61, 661–683.CrossRefGoogle Scholar
24. Marbukh, V. V. (1981) Asymptotic investigation of a complete communications network with a large number of points and bypass routes. Probl. Pered. Inform. 7, 89–95.Google Scholar
25. Mitrani, I. (1984) Fixed-point approximations for distributed systems. In Iazeolla, et al. [17], 245–257.Google Scholar
26. Nakagome, Y. and Mori, H. (1973) Flexible routing in the global communication network. 7th Internat. Teletraffic Congress.Google Scholar
27. Pittel, B. (1979) Closed exponential networks of queues with saturation: the Jackson-type stationary distribution and its asymptotic analysis. Math. Operat. Res. 4, 357–378.Google Scholar
28. Poston, T. and Stewart, I. (1978) Catastrophe Theory and its Applications. Pitman, London.Google Scholar
29. Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
30. Whittle, P. (1968) Equilibrium distributions for an open migration process. J. Appl. Prob. 5, 567–571.Google Scholar
31. Whittle, P. (1971) Optimization under Constraints. Wiley, Chichester.Google Scholar