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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

References

Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Davies, B. (1978) Integral Transforms and their Applications. Springer-Verlag, New York.CrossRefGoogle Scholar
Doetsch, G. (1955) Handbuch der Laplace Tranformation. Birkauser, Basel.CrossRefGoogle Scholar
Fayolle, G. (1975) Étude du comportement d&un canal radio partagé entre plusieurs utilisateurs. Thèse Docteur Ingénieur, Université Paris VI.Google Scholar
Fayolle, G. and Hofri, M. (1982) On the capacity of a collision channel under stack-based collision resolution algorithms. Dept of Computer Science, Technion.Google Scholar
Knuth, D. E. (1973) The Art of Computer Programming, Vol. 3: Sorting and Searching. Addison-Wesley, Reading, Mass.Google Scholar
Massey, J. L. (1981) Collision resolution algorithms and random access communication. In Multiuser Communications, ed. Longo, G., CISM Courses and Lectures 265, Springer-Verlag, New York, 73–137.Google Scholar
Tsybakov, B. A. and Vvedenskaya, N. D. (1980) Random multiple-access stack algorithm. Probl. Inf. Trans. 16, 230–243.Google Scholar
Vvendskaya, N. D. and Tsybakov, B. S. (1983) Random multiple-access of packets on a channel with errors (in Russian). Probl. Perd. Inf. 19, 52–68.Google Scholar

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