Stochastic performance modeling for practical byzantine fault tolerance consensus in the blockchain

Abstract

The practical Byzantine fault tolerant (PBFT) consensus mechanism is one of the most basic consensus algorithms (or protocols) in blockchain technologies. Thus its performance evaluation is an interesting and challenging topic due to the higher complexity of its consensus work in a peer-to-peer network. This study describes a simple stochastic performance model of the PBFT consensus mechanism. This model is refined not only as a queuing system with complicated service times but also as a level-independent quasi-birth-and-death (QBD) process. With regard to the level-independent QBD process, we apply the matrix-geometric solution to obtain the necessary and sufficient condition under which the PBFT consensus system is stable and then numerically compute the stationary probability vector of the QBD process. Thus, we provide four useful performance measures for the PBFT consensus mechanism, and we can numerically calculate these performance measures. Finally, we use numerical examples to verify the validity of our theoretical results and demonstrate how the four performance measures are influenced by certain key parameters of the PBFT consensus. Considering theory of multi-dimensional Markov processes, we are optimistic that the methodology and results presented in this study are applicable to a wide range of PBFT consensus mechanism and even other types of consensus mechanisms.