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

Decentralized consensus protocols have a variety of parameters to be 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 ξij 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.