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Loss probabilities in a simple circuit-switched network

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

J. A. Morrison
J. A. Morrison*
Affiliation:
AT&T Bell Laboratories, Murray Hill
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Abstract

In this paper a particular loss network consisting of two links with C1 and C2 circuits, respectively, and two fixed routes, is investigated. A call on route 1 uses a circuit from both links, and a call on route 2 uses a circuit from only the second link. Calls requesting routes 1 and 2 arrive as independent Poisson streams. A call requesting route 1 is blocked and lost if there are no free circuits on either link, and a call requesting route 2 is blocked and lost if there is no free circuit on the second link. Otherwise the call is connected and holds a circuit from each link on its route for the holding period of the call.

The case in which the capacities C1, and C2, and the traffic intensities v1, and v2, all become large of O(N) where N » 1, but with their ratios fixed, is considered. The loss probabilities L1 and L2 for calls requesting routes 1 and 2, respectively, are investigated. The asymptotic behavior of L1 and L2 as N→ ∞ is determined with the help of double contour integral representations and saddlepoint approximations. The results differ in various regions of the parameter space (C1, C2, v1, v2). In some of these results the loss probabilities are given in terms of the Erlang loss function, with appropriate arguments, to within an exponentially small relative error. The results provide new information when the loss probabilities are exponentially small in N. This situation is of practical interest, e.g. in cellular systems, and in asynchronous transfer mode networks, where very small loss probabilities are desired.

The accuracy of the Erlang fixed-point approximations to the loss probabilities is also investigated. In particular, it is shown that the fixed-point approximation E2 to L2 is inaccurate in a certain region of the parameter space, since L2 « E2 there. On the other hand, in some regions of the parameter space the fixed-point approximations to both L1 and L2 are accurate to within an exponentially small relative error.

Keywords

ASYMPTOTICSERLANG FIXED-POINT APPROXIMATIONLOSS NETWORKSSADDLEPOINT APPROXIMATIONS

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 2 , June 1994 , pp. 456 - 473
Copyright
Copyright © Applied Probability Trust 1994 

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References

[1] Bleistein, N. (1966) Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, 353–370.Google Scholar
[2] Bleistein, N. and Handelsman, R. A. (1975) Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York.Google Scholar
[3] Fedoryuk, M. V. (1989) Asymptotic methods in analysis. In Analysis I, ed. Gamkrelidze, R. V.. Springer-Verlag, Berlin.Google Scholar
[4] Hunt, P. J. and Kelly, F. P. (1989) On critically loaded networks. Adv. Appl. Prob. 21, 831–841.CrossRefGoogle Scholar
[5] Jagerman, D. L. (1984) Methods in traffic calculations. Bell System Tech. J. 63, 1283–1310.Google Scholar
[6] Kelly, F. P. (1986) Blocking probabilities in large circuit-switched networks. Adv. Appl. Prob. 18, 473–505.Google Scholar
[7] Kelly, F. P. (1991) Loss networks, Ann. Appl. Prob. 1, 319–378.Google Scholar
[8] Kogan, Y. (1992) Another approach to asymptotic expansions for large closed queueing networks. Operat. Res. Letters 11, 317–321.Google Scholar
[9] Magnus, W., Oberhettinger, F. and Soni, R. P. (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, New York.Google Scholar
[10] Whittle, P. (1988) Approximation in large-scale circuit-switched networks. Prob. Eng. Inf Sci. 2, 279–291.Google Scholar
[11] Zachary, S. (1991) On blocking in loss networks. Adv. Appl. Prob. 23, 355–372.Google Scholar