  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
a higher complexity of its consensus work in the peer-to-peer network. This
paper describes a simple stochastic performance model of the PBFT consensus
mechanism, which is refined as not only a queueing system with complicated
service times but also a level-independent quasi-birth-and-death (QBD) process.
From the level-independent QBD process, we apply the matrix-geometric solution
to obtain a necessary and sufficient condition under which the PBFT consensus
system is stable, and to be able to numerically compute the stationary
probability vector of the QBD process. Thus we provide four useful performance
measures of the PBFT consensus mechanism, and can numerically calculate the
four performance measures. Finally, we use some numerical examples to verify
the validity of our theoretical results, and show how the four performance
measures are influenced by some key parameters of the PBFT consensus. By means
of the theory of multi-dimensional Markov processes, we are optimistic that the
methodology and results given in this paper are applicable in a wide range
research of PBFT consensus mechanism and even other types of consensus
mechanisms.
