Perpetuities in Fair Leader Election Algorithms

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.

A Binomial Splitting Process in Connection with Corner Parking Problems

“…They would also arise in a leader election algorithm that advances a truncated binomial number of contestants at each stage. Namely, X n is the number of rounds till the election comes to an end; see [15,16] for a broad framework for these types of problems.…”

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

“…Many interesting parameters have been studied in fair leader election, ranging from the duration of the algorithm till termination (see Fill et al (1996) and Janson and Szpankowski (1997)), the total cost till termination (Kalpathy and Mahmoud 2014;Prodinger 1993), to the number of rounds a contestant survives (Kalpathy and Mahmoud 2014;Kalpathy et al 2011), and the number of survivors after a prespecified number of rounds (Kalpathy et al 2013). All these investigations are concerned with univariate statistics in leader election.…”

“…Only very recently did research branch out of the specific case of binomial splitting to more general splitting protocols. For instance, the reference (Kalpathy and Ward 2014) considers truncated geometric distributions as splitting protocols, while Janson et al (2008), Kalpathy and Mahmoud (2014), Kalpathy et al (2013) consider broader theories that cover large classes of probability distributions.…”

“…As we mention in the preceding paragraphs, Kalpathy and Mahmoud (2014) looks at the distribution of the duration of a particular individual contestant, a statistic that is important from the vantage point of that individual contestant. In this note we look at the various layers at which contestants can fall and examine the covariance structure between the various durations.…”

A Binomial Splitting Process in Connection with Corner Parking Problems

“…It also arose in a leader election algorithm, in which a truncated binomial number of contestants were advanced at each stage during a contest. Namely, X n is the number of rounds taken until the election comes to an end; see [13] and [14] for a broad framework for these types of problems.…”