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On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel

Published online by Cambridge University Press:  01 July 2016

Guy Fayolle ,
Philippe Flajolet  and
Micha Hofri
Guy Fayolle*
Affiliation:
INRIA
Philippe Flajolet*
Affiliation:
INRIA
Micha Hofri*
Affiliation:
Technion-Israel Institute of Technology
*
∗INRIA, Domaine de Voluceau, BP 103, 78153 Le Chesnay Cedex, France.
∗INRIA, Domaine de Voluceau, BP 103, 78153 Le Chesnay Cedex, France.
∗∗Computer Science Department, Technion–Israel Institute of Technology, Haifa 32000, Israel.
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Abstract

We analyse a stack protocol of the Capetanakis–Tsybakov–Mikhailov type for resolving collisions in a random multiple-access channel. We obtain a functional equation for the generating function of the expected collision resolution interval (CRI) durations, which is non-local with a non-commutative iteration semigroup. Using Mellin transform techniques and geometric properties of the iteration semigroup we show that for arrival rates smaller than a fixed threshold, the mean CRI duration for n initial colliders is asymptotically proportional to n. Ergodicity conditions are also demonstrated.

Keywords

ASYMPTOTIC ANALYSISFUNCTIONAL EQUATIONMELLIN TRANSFORMPROTOCOLRANDOM ACCESSTREE ALGORITHM
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 441 - 472
Copyright
Copyright © Applied Probability Trust 1986 

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References

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Reference added in proof

Fayolle, G., Flajolet, P., Hofri, M. and Jacquet, P. (1985) Analysis of a stack algorithm for random multiple-access communication. IEEE Trans. Inf Theory 31, 244–254.CrossRefGoogle Scholar