The practical Byzantine fault tolerant (PBFT) consensus protocol is one of the basic consensus protocols in the development of blockchain technology. At the same time, the PBFT consensus protocol forms a basis for some other important BFT consensus protocols, such as Tendermint, Streamlet, HotStuff, and LibraBFT. In general, the voting nodes may always fail so that they can leave the PBFT-based blockchain system in a random time interval, making the number of timely available voting nodes uncertain. Thus, this uncertainty leads to the analysis of the PBFT-based blockchain systems with repairable voting nodes being more challenging. In this paper, we develop a novel PBFT consensus protocol with repairable voting nodes and study such a new blockchain system using a multi-dimensional Markov process and the first passage time method. Based on this, we provide performance and reliability analysis, including throughput, availability, and reliability, for the new PBFT-based blockchain system with repairable voting nodes. Furthermore, we provide an approximate algorithm for computing the throughput of the new PBFT-based blockchain system. We employ numerical examples to demonstrate the validity of our theoretical results and illustrate how the key system parameters influence performance measures of the PBFT-based blockchain system with repairable voting nodes. We hope the methodology and results developed in this paper will stimulate future research endeavors and open up new research trajectories in this field.

翻译：实用拜占庭容错（PBFT）共识协议是区块链技术发展的基本共识协议之一。同时，PBFT共识协议构成了Tendermint、Streamlet、HotStuff和LibraBFT等其他重要BFT共识协议的基础。通常，投票节点可能发生故障，从而在随机时间间隔内离开基于PBFT的区块链系统，导致实时可用投票节点数量不确定。因此，这种不确定性使得对具有可修复投票节点的PBFT区块链系统分析更具挑战性。本文提出了一种具有可修复投票节点的新型PBFT共识协议，并采用多维马尔可夫过程和首次通过时间方法研究了该新型区块链系统。基于此，我们为具有可修复投票节点的PBFT区块链系统提供了性能与可靠性分析，包括吞吐量、可用性和可靠性。此外，我们提出了一种计算新型PBFT区块链系统吞吐量的近似算法。通过数值算例验证了理论结果的有效性，并说明了关键系统参数如何影响具有可修复投票节点的PBFT区块链系统的性能指标。希望本文提出的方法和成果能够推动该领域的未来研究，并开辟新的研究路径。