Theoretical Analysis on Block Time Distributions in Byzantine Fault-Tolerant Consensus Blockchains

TL;DR

This paper develops a mathematical model to analyze block time fluctuations in Byzantine fault-tolerant blockchain networks, revealing that broadcast times follow a Gumbel distribution and providing an approximation formula for empirical data analysis.

Contribution

It introduces a stochastic model for validator communication and proves that broadcast times follow a Gumbel distribution, offering a new theoretical understanding of block time variability.

Findings

Broadcast times follow a Gumbel distribution.

An approximate formula for block time distribution is derived.

The model fits real-world data effectively.

Abstract

Some blockchain networks employ a distributed consensus algorithm featuring Byzantine fault tolerance. Notably, certain public chains, such as Cosmos and Tezos, which operate on a proof-of-stake mechanism, have adopted this algorithm. While it is commonly assumed that these blockchains maintain a nearly constant block creation time, empirical analysis reveals fluctuations in this interval; this phenomenon has received limited attention. In this paper, we propose a mathematical model to account for the processes of block propagation and validation within Byzantine fault-tolerant consensus blockchains, aiming to theoretically analyze the probability distribution of block time. First, we propose stochastic processes governing the broadcasting communications among validator nodes. Consequently, we theoretically demonstrate that the probability distribution of broadcast time among validator nodes adheres to the Gumbel distribution. This finding indicates that the distribution of block time typically arises from convolving multiple Gumbel distributions. Additionally, we derive an approximate formula for the block time distribution suitable for data analysis purposes. By fitting this approximation to real-world block time data, we demonstrate the consistent estimation of block time distribution parameters.