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The Time to Consensus in a Blockchain: Insights into Bitcoin's "6 Blocks Rule''

Partha S. Dey, Aditya S. Gopalan, Vijay G. Subramanian

TL;DR

The paper addresses quantifying the time to consensus in Nakamoto-style blockchains under nontrivial network delays and a worst-case adversary. It develops a stochastic growth/queueing framework and derives a finite-consensus threshold, along with an exact Laplace transform and tail decay for the stylized Bitcoin model, complemented by simulations. For the general model, it introduces a cycle-based decomposition using quantities $(X_{(n)},Y_{(n)})$ and proves exponential tail bounds via a dominant pole and the parameters $z_*$ and $j_0$, offering a principled way to assess consensus timing. The results illuminate how delays and adversarial blocks shape confirmation times and provide quantitative benchmarks applicable to real-world settings like grocery supply chains that require timely consensus.

Abstract

We investigate the time to consensus in Nakamoto blockchains. Specifically, we consider two competing growth processes, labeled \emph{honest} and \emph{adversarial}, and determine the time after which the honest process permananetly exceeds the adversarial process. This is done via queueing techniques. The predominant difficulty is that the honest growth process is subject to \emph{random delays}. In a stylized Bitcoin model, we compute the Laplace transform for the time to consensus and verify it via simulation.

The Time to Consensus in a Blockchain: Insights into Bitcoin's "6 Blocks Rule''

TL;DR

The paper addresses quantifying the time to consensus in Nakamoto-style blockchains under nontrivial network delays and a worst-case adversary. It develops a stochastic growth/queueing framework and derives a finite-consensus threshold, along with an exact Laplace transform and tail decay for the stylized Bitcoin model, complemented by simulations. For the general model, it introduces a cycle-based decomposition using quantities and proves exponential tail bounds via a dominant pole and the parameters and , offering a principled way to assess consensus timing. The results illuminate how delays and adversarial blocks shape confirmation times and provide quantitative benchmarks applicable to real-world settings like grocery supply chains that require timely consensus.

Abstract

We investigate the time to consensus in Nakamoto blockchains. Specifically, we consider two competing growth processes, labeled \emph{honest} and \emph{adversarial}, and determine the time after which the honest process permananetly exceeds the adversarial process. This is done via queueing techniques. The predominant difficulty is that the honest growth process is subject to \emph{random delays}. In a stylized Bitcoin model, we compute the Laplace transform for the time to consensus and verify it via simulation.

Paper Structure

This paper contains 25 sections, 13 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Blockchain Protocol
  3. Nakamoto Blockchains with Adversaries
  4. Contributions of This Work
  5. Results for Bitcoin
  6. General Results
  7. Related Literature
  8. Stylized Bitcoin Model
  9. Model
  10. Bitcoin's Time to Consensus
  11. Mean and Tail Decay for Bitcoin's Time to Consensus
  12. Numerical Evaluation
  13. General Model and Main Results
  14. Stochastic Growth Process Model
  15. Queueing Structure
  16. ...and 10 more sections

Key Result

Proposition 2.1

For a stable $M/M/1$ queue with arrival rate $\lambda$ and service rate $\mu > \lambda$, the Laplace transform of the busy period is given by Moreover, the Laplace transform of the cycle length and the residual busy period are, respectively, given by

Figures (4)

  • Figure 1: Event leading to the time to consensus. Here, the orange sections are stable busy periods, and the blue sections are unstable ones. The time to consensus occurs at the beginning of the first infintie unstable busy period.
  • Figure 2: Simulations in the stylized Bitcoin model with 25,000 samples.
  • Figure 3: Empirical distribution of time to consensus with $p\in\{.72,.84,.89\}.$ The blue line is the empirical distribution, and the orange line is the theoretical rate predicted by Corollary \ref{['cor:tail-prob-bitcoin']}
  • Figure 4: Queueing Cycle Behavior of our Model

Theorems & Definitions (24)

  • Remark 1
  • Remark 2
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Lemma 3.1
  • ...and 14 more