Markov processes in blockchain systems

In this paper, we develop a more general framework of block-structured Markov processes in the queueing study of blockchain systems, which can provide analysis both for the stationary performance measures and for the sojourn time of any transaction or block. In addition, an original aim of this paper is to generalize the two-stage batch-service queueing model studied in Li et al. (Blockchain queue theory. In: International conference on computational social networks. Springer: New York; 2018. p. 25–40) both “from exponential to phase-type” service times and “from Poisson to MAP” transaction arrivals. Note that the MAP transaction arrivals and the two stages of PH service times make our blockchain queue more suitable to various practical conditions of blockchain systems with crucial factors, for example, the mining processes, the block generations, the blockchain building and so forth. For such a more general blockchain queueing model, we focus on two basic research aspects: (1) using the matrix-geometric solution, we first obtain a sufficient stable condition of the blockchain system. Then, we provide simple expressions for the average stationary number of transactions in the queueing waiting room and the average stationary number of transactions in the block. (2) However, on comparing with Li et al. (2018), analysis of the transaction–confirmation time becomes very difficult and challenging due to the complicated blockchain structure. To overcome the difficulties, we develop a computational technique of the first passage times by means of both the PH distributions of infinite sizes and the RG factorizations. Finally, we hope that the methodology and results given in this paper will open a new avenue to queueing analysis of more general blockchain systems in practice and can motivate a series of promising future research on development of blockchain technologies.