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March 2014 Perpetuities in fair leader election algorithms
Ravi Kalpathy, Hosam Mahmoud
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Adv. in Appl. Probab. 46(1): 203-216 (March 2014). DOI: 10.1239/aap/1396360110

Abstract

We consider a broad class of fair leader election algorithms, and study the duration of contestants (the number of rounds a randomly selected contestant stays in the competition) and the overall cost of the algorithm. We give sufficient conditions for the duration to have a geometric limit distribution (a perpetuity built from Bernoulli random variables), and for the limiting distribution of the total cost (after suitable normalization) to be a perpetuity. For the duration, the proof is established via convergence (to 0) of the first-order Wasserstein distance from the geometric limit. For the normalized overall cost, the method of proof is also convergence of the first-order Wasserstein distance, augmented with an argument based on a contraction mapping in the first-order Wasserstein metric space to show that the limit approaches a unique fixed-point solution of a perpetuity distributional equation. The use of these two steps is commonly referred to as the contraction method.

Citation

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Ravi Kalpathy. Hosam Mahmoud. "Perpetuities in fair leader election algorithms." Adv. in Appl. Probab. 46 (1) 203 - 216, March 2014. https://doi.org/10.1239/aap/1396360110

Information

Published: March 2014
First available in Project Euclid: 1 April 2014

zbMATH: 1291.60018
MathSciNet: MR3189055
Digital Object Identifier: 10.1239/aap/1396360110

Subjects:
Primary: 60C05
Secondary: 60F05 , 68W40

Keywords: contraction method , fixed point , functional equation , Leader election , metric space , perpetuity , recurrence , weak convergence

Rights: Copyright © 2014 Applied Probability Trust

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Vol.46 • No. 1 • March 2014
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