  Blockchain has many benefits including decentralization, availability,
persistency, consistency, anonymity, auditability and accountability, and it
also covers a wide spectrum of applications ranging from cryptocurrency,
financial services, reputation system, Internet of Things, sharing economy to
public and social services. Not only may blockchain be regarded as a by-product
of Bitcoin cryptocurrency systems, but also it is a type of distributed ledger
technology through using a trustworthy, decentralized log of totally ordered
transactions. By summarizing the literature of blockchain, it is found that
more papers focus on engineering implementation and realization, while little
work has been done on basic theory, for example, mathematical models (Markov
processes, queueing theory and game models), performance analysis and
optimization of blockchain systems. In this paper, we develop queueing theory
of blockchain systems and provide system performance evaluation. To do this, we
design a Markovian batch-service queueing system with two different service
stages, while the two stages are suitable to well express the mining process in
the miners pool and the building of a new blockchain. By using the
matrix-geometric solution, we obtain a system stable condition and express
three key performance measures: (a) The number of transactions in the queue,
(b) the number of transactions in a block, and (c) the transaction-confirmation
time. Finally, We use numerical examples to verify computability of our
theoretical results. Although our queueing model is simple under exponential or
Poisson assumptions, our analytic method will open a series of potentially
promising research in queueing theory of blockchain systems.
