Bivariate Issues in Leader Election Algorithms with Marshall-Olkin Limit Distribution

Cheng Zhang and Hosam Mahmoud ()
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Cheng Zhang: The George Washington University
Hosam Mahmoud: The George Washington University

Methodology and Computing in Applied Probability, 2016, vol. 18, issue 2, 401-418

Abstract: Abstract Most prior work in leader election algorithms deals with univariate statistics. We consider multivariate issues in a broad class of fair leader election algorithms. We investigate the joint distribution of the duration of two competing candidates. Under rather mild conditions on the splitting protocol, we prove the convergence of the joint distribution of the duration of any two contestants to a limit via convergence of distance (to 0) in a metric space on distributions. We then show that the limiting distribution is a Marshall-Olkin bivariate geometric distribution. Under the classic binomial splitting we are able to say a few more precise words about the exact joint distribution and exact covariance, and to explore (via Rice’s integral method) the oscillatory behavior of the diminishing covariance. We discuss several practical examples and present empirical observations on the rate of convergence of the covariance.

Keywords: Randomized algorithm; Stochastic recurrence; Marshall-Olkin distribution; Weak convergence; Rice’s integral method; 60C05; 60E05; 68W40 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s11009-014-9428-1

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