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

Akihiro Fujihara

TL;DR

The paper addresses fluctuations in block time for Byzantine fault-tolerant PoS blockchains by building a stochastic model of block propagation and verification. It shows that broadcast time among validator nodes naturally follows a Gumbel distribution for large $N$, and that the overall block time arises from the convolution of three independent Gumbel times, with a saddle-point-based approximation enabling practical data fitting. Fitting to Cosmos data ($N=175$) yields a consistent per-hop transfer time $\Delta t$ around $0.0011$ s and an estimated transfer rate of approximately $9.1\times 10^2$ transfers per second, validating the theoretical framework. The results offer a theoretically grounded method to analyze and predict block-time variability, informing design choices and parameter estimation without solely relying on empirical measurements, and suggesting that the aggregated Gumbel components trend toward normality as their number increases.

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.

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

TL;DR

The paper addresses fluctuations in block time for Byzantine fault-tolerant PoS blockchains by building a stochastic model of block propagation and verification. It shows that broadcast time among validator nodes naturally follows a Gumbel distribution for large , and that the overall block time arises from the convolution of three independent Gumbel times, with a saddle-point-based approximation enabling practical data fitting. Fitting to Cosmos data () yields a consistent per-hop transfer time around s and an estimated transfer rate of approximately transfers per second, validating the theoretical framework. The results offer a theoretically grounded method to analyze and predict block-time variability, informing design choices and parameter estimation without solely relying on empirical measurements, and suggesting that the aggregated Gumbel components trend toward normality as their number increases.

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.
Paper Structure (12 sections, 31 equations, 4 figures)

This paper contains 12 sections, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Theoretical analysis
  4. Components of block time
  5. Stochastic processes on broadcast communications of candidate block between validator nodes
  6. Distribution of block verification time
  7. Block time distribution
  8. Approximate representation of block time distribution
  9. Discussion
  10. Fitting the result of theoretical analysis with Cosmos block time data
  11. Skewness of the convolution of many independent Gumbel distributions
  12. Conclusion

Figures (4)

  • Figure 1: Histograms of block time in Byzantine fault--tolerant consensus blockchains and Ethereum 2.0. Those for Cosmos and Ethereum 2.0 are depicted at the top and bottom of the figure, respectively. The solid red line at the top of the figure is provided by non-linear fitting using a theoretical result in Eq. (\ref{['eq:approximate_result3']}) to be explained later.
  • Figure 2: Three phases of repeating broadcasts of block information among validator nodes in Byzantine fault--tolerant consensus blockchains
  • Figure 3: State transition diagram in the process on broadcast communications of candidate block between validator nodes
  • Figure 4: Results of numerical analyses to evaluate the theoretically derived formula $f^{(k)}$ for $k=1,2,3$. The histogram at the top illustrates a pattern of randomly sampled data according to the Gumbel distribution and the fitted curve of the approximate representation, $f^{(1)}$. One at the middle illustrates a pattern of randomly sampled data according to the sum of two independent Gumbel distributions and the fitted curve of $f^{(2)}$, and one at the bottom illustrates a pattern of randomly sampled data according to the sum of three independent Gumbel distributions and the fitted curve of $f^{(3)}$.