Fork Rate-Based Analysis of the Longest Chain Growth Time Interval of a PoW Blockchain

Nakamoto's consensus protocol, which is well known for its resistance to sybil attacks by using PoW (Proof of Work), enables us to build public blockchains, such as Bitcoin. In this protocol, miners seek to extend the longest chain by solving blockhash-based cryptographic puzzles and the required time is probabilistically determined. Therefore, the distribution of the time interval affects security, performance and applications which utilize the block height information. Some researchers assumed that the time follows an exponential distribution but this assumption requires that the blockchain network is fully synchronized. To overcome this unreal scenario, the bounded delay model, in which there is an upper bound for block propagation delay on the network, was proposed. However, it is difficult to calculate the upper bound without observing delay and bandwidth on real-world network links. To solve this problem, we proposed another method to analyze the distribution of the longest chain growth time interval by using the observed fork rate. We derived a closed-form lower bound for the CDF (Cumulative Distribution Function) of the time to update the global block height. We also obtained the Pearson distance which can be used as the metric to judge whether the network is approximately synchronous or not. Finally, we conducted network simulations for comparing our lower bound with the lower bound that is based on the bounded delay model. In numerical examples, we show how the block size affects these lower bounds.