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In a practical Byzantine fault tolerance (PBFT) blockchain network, thevoting nodes may always leave the network while some new nodes can also enterthe network, thus the number of voting nodes is constantly changing. Such a newPBFT with dynamic nodes is called a dynamic PBFT. Clearly, the dynamic PBFT canmore strongly support the decentralization and distributed structure ofblockchain. However, analyzing dynamic PBFT blockchain systems will become moreinteresting and challenging.  In this paper, we propose a large-scale Markov modeling technique to analyzethe dynamic PBFT voting processes and its dynamic PBFT blockchain system. Tothis end, we set up a large-scale Markov process (and further amulti-dimensional Quasi-Birth-and-Death (QBD) process) and provide performanceanalysis for both the dynamic PBFT voting processes and the dynamic PBFTblockchain system. In particular, we obtain an effective computational methodfor the throughput of the complicated dynamic PBFT blockchain system. Finally,we use numerical examples to check the validity of our theoretical results andindicate how some key system parameters influence the performance measures ofthe dynamic PBFT voting processes and of the dynamic PBFT blockchain system.Therefore, by using the theory of multi-dimensional QBD processes and theRG-factorization technique, we hope that the methodology and results developedin this paper shed light on the study of dynamic PBFT blockchain systems suchthat a series of promising research can be developed potentially.

Dynamic Practical Byzantine Fault Tolerance and Its Blockchain System: A Large-Scale Markov Modeling