  With rapid development of blockchain technology as well as integration of
various application areas, performance evaluation, performance optimization,
and dynamic decision in blockchain systems are playing an increasingly
important role in developing new blockchain technology. This paper provides a
recent systematic overview of this class of research, and especially,
developing mathematical modeling and basic theory of blockchain systems.
Important examples include (a) performance evaluation: Markov processes,
queuing theory, Markov reward processes, random walks, fluid and diffusion
approximations, and martingale theory; (b) performance optimization: Linear
programming, nonlinear programming, integer programming, and multi-objective
programming; (c) optimal control and dynamic decision: Markov decision
processes, and stochastic optimal control; and (d) artificial intelligence:
Machine learning, deep reinforcement learning, and federated learning. So far,
a little research has focused on these research lines. We believe that the
basic theory with mathematical methods, algorithms and simulations of
blockchain systems discussed in this paper will strongly support future
development and continuous innovation of blockchain technology.
