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The practical Byzantine fault tolerant (PBFT) consensus mechanism is one ofthe most basic consensus algorithms (or protocols) in blockchain technologies,thus its performance evaluation is an interesting and challenging topic due toa higher complexity of its consensus work in the peer-to-peer network. Thispaper describes a simple stochastic performance model of the PBFT consensusmechanism, which is refined as not only a queueing system with complicatedservice times but also a level-independent quasi-birth-and-death (QBD) process.From the level-independent QBD process, we apply the matrix-geometric solutionto obtain a necessary and sufficient condition under which the PBFT consensussystem is stable, and to be able to numerically compute the stationaryprobability vector of the QBD process. Thus we provide four useful performancemeasures of the PBFT consensus mechanism, and can numerically calculate thefour performance measures. Finally, we use some numerical examples to verifythe validity of our theoretical results, and show how the four performancemeasures are influenced by some key parameters of the PBFT consensus. By meansof the theory of multi-dimensional Markov processes, we are optimistic that themethodology and results given in this paper are applicable in a wide rangeresearch of PBFT consensus mechanism and even other types of consensusmechanisms.

Stochastic Performance Modeling for Practical Byzantine Fault Tolerance Consensus in Blockchain