  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 times
of any transaction and block. Note that an original aim of this paper is to
generalize the two-stage batch-service queueing model studied in Li et al.
\cite{Li:2018} both ``from exponential to phase-type" service times and ``from
Poisson to MAP" transaction arrivals. In general, 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 random
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) By using the
matrix-geometric solution, we first obtain a sufficient stable condition of the
blockchain system. Then we provide simple expressions for the average number of
transactions in the queueing waiting room, and the average number of
transactions in the block. (2) However, comparing with Li et al.
\cite{Li: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 lockchain technologies.
