Binomial distribution with delay in analysis and parametrization of Ouroboros Praos proof of stake blockchain protocol

Sep 10, 2026

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Decentralized consensus protocols require deployment parameters that must balance security and operational performance. This research analyzes Ouroboros Praos, a proof-of-stake protocol used in Cardano and related systems, focusing on how protocol parameters such as the active slot coefficient and the p2p block propagation time affect stochastic block production. The study derives explicit formulas and expectations for the longest-chain length, the longest unintentional honest fork, and block-generation efficiency, and models the process as a deformation of the binomial distribution using denumerable Markov chains and generating functions.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages31

Reading Time1 h 18 min

AccessFree to download

Introduction

Modeling approach and binomial distribution deformation

Security background of PoW and PoS

Ouroboros Praos objectives and stochastic characteristics

Probability in the Engineering and Informational Sciences (2025), 39:4 520–550 doi:10.1017/S0269964825000014 RESEARCH ARTICLE Binomial distribution with delay in analysis and parametrization of Ouroboros Praos proof of stake blockchain protocol Yuri Bespalov1 , Lyudmila Kovalchuk2 , 3 , Hanna Nelasa4 and Roman Oliynykov3 , 5 1Department of Mathematical Methods in Theoretical Physics, Bogolyubov Institute for Theoretical Physics, Kyiv, 03143, Ukraine 2Department of Mathematical and Computer Modeling, Institute for Modelling in Energy Engineering, Kyiv, 03164, Ukraine 3IOG Research, IOG Singapore Pte Ltd, 049908, Singapore 4Department of Information Security and Nanoelectronics, Zaporizhzhia Polytechnic National University, Zaporizhzhia, 69011, Ukraine 5Department of Cybersecurity of Information Systems, Networks and Technologies, V. N. Karazin Kharkiv National University, Kharkiv, 61022, Ukraine Corresponding author: Roman Oliynykov; Email: [roliynykov@gmail.com](roliynykov@gmail.com) Keywords: Binomial distribution with delay; Denumerable Markov chain; Ouroboros Praos; Proof-of-Stake protocol; Rational generating function Abstract Decentralized consensus protocols have a variety of parameters tobe set during their deployment for practical applications in blockchains. The analysis given in most research papers proves the security state of the blockchain, at the same time usually providing a range of acceptable values, thus allowing further tuning of the protocol parameters. In this paper, we investigate Ouroboros Praos, the proof-of-stake consensus protocol deployed in Cardano and other blockchains. In contrast to its predecessor, Praos allows multiple honest slot leaders that lead to fork creation and resolution, consequently decreasing the block rate per time unit. In our analysis of dependence on protocol parameters such as active slot coefficient and p2p network block propagation time, we obtain new theoretical results and explicit formulas for the expectation of the length of the longest chain created during the Praos epoch, the length of the longest unintentional fork created by honest slot leaders, the efficiency of block generation procedure (the ratio of blocks included in the final longest chain vs the total number of created blocks), and other characteristics of the blockchain throughput. We study these parameters as stochastic characteristics of the block generation process. The model is described in terms of the two-parametric family bij of independent Bernoulli random variables which generate deformation of the binomial distribution by a positive integer parameter—the delay (deterministic or random) . An essential part of our paper is a study of this deformation in terms of denumerable Markov chains and generating functions. 1. Introduction The main purpose of this paper is to investigate the stochastic characteristics of the block generation process in the Ouroboros Praos Proof-of-Stake protocol. The probabilistic model that emerged in this study is a deformation of the binomial distribution with an additional positive integer parameter (interpreted as time delay), deterministic or random finitely distributed. This deformation seems interesting in itself and is also systematically studied here. Proof-of-Work (PoW) [17] and Proof-of-Stake (PoS) [19] (first discussion),[12], [11] are the most wide-used approaches for reaching consensus in blockchain technology. Much research has been conducted, mainly regarding their security properties, including resistance to double-spending attacks [21],[18], [7], [13], [10] and splitting attacks [11] . © The Author(s), 2025 . Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence ([http://creativecommons.org/licenses/by/4.0](http://creativecommons.org/licenses/by/4.0)), which permits unrestricted re-use, distribution and reproduction, provided th

FAQs

What is the focus of the study on Ouroboros Praos?

The study focuses on stochastic characteristics of the block generation process and how they depend on protocol parameters in Ouroboros Praos.

Which parameters are analyzed in relation to block generation?

The analysis considers the active slot coefficient and the p2p network block propagation time, treating them as factors affecting the block production process.

How is the probabilistic model for delayed binomial behavior constructed?

The model uses a two-parameter family of distributions generated from independent Bernoulli random variables, where a positive integer parameter represents deterministic or random time delay.

Sep 10, 2026

Upload by Sarah