Yan-Xia Chang

Quan-Lin Li, Yan-Xia Chang, Qing Wang

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.

Quan-Lin Li, Yan-Xia Chang, Chi Zhang

It is interesting but difficult and challenging to study Ethereum with multiple mining pools. One of the main difficulties comes from not only how to represent such a general tree with multiple block branches (or sub-chains) related to the multiple mining pools, but also how to analyze a multi-dimensional stochastic system due to the mining competition among the multiple mining pools. In this paper, we first set up a mathematical representation for the tree with multiple block branches. Then we provide a block classification of Ethereum: Regular blocks (in the main chain), orphan blocks, uncle blocks, stale blocks, and nephew blocks, and give some key probabilities of generating the different types of blocks by applying the law of large numbers. Based on this, we further discuss the growth rate of blockchain, and the reward allocation among the multiple mining pools through applying the renewal reward theorem. Finally, we use some simulation experiments to verify our theoretical results, and show that the approximate computation approaches developed, such as the key probabilities, the long-term growth rate of blockchain, and the long-term reward allocation (rate) among the multiple mining pools, can have a faster convergence. Therefore, we provide a powerful tool for observing and understanding the influence of the selfish mining attacks on the performance of Ethereum with multiple mining pools. We believe that the methodology and results developed in this paper will shed light on the study of Ethereum with multiple mining pools, such that a series of promising research can be inspired potentially.

Fan-Qi Ma, Quan-Lin Li, Yi-Han Liu et al.

The practical Byzantine fault tolerant (PBFT) consensus mechanism is one of the most basic consensus algorithms (or protocols) in blockchain technologies, thus its performance evaluation is an interesting and challenging topic due to a higher complexity of its consensus work in the peer-to-peer network. This paper describes a simple stochastic performance model of the PBFT consensus mechanism, which is refined as not only a queueing system with complicated service times but also a level-independent quasi-birth-and-death (QBD) process. From the level-independent QBD process, we apply the matrix-geometric solution to obtain a necessary and sufficient condition under which the PBFT consensus system is stable, and to be able to numerically compute the stationary probability vector of the QBD process. Thus we provide four useful performance measures of the PBFT consensus mechanism, and can numerically calculate the four performance measures. Finally, we use some numerical examples to verify the validity of our theoretical results, and show how the four performance measures are influenced by some key parameters of the PBFT consensus. By means of the theory of multi-dimensional Markov processes, we are optimistic that the methodology and results given in this paper are applicable in a wide range research of PBFT consensus mechanism and even other types of consensus mechanisms.

Quan-Lin Li, Yan-Xia Chang, Xiaole Wu et al.

In this paper, we provide a new theoretical framework of pyramid Markov processes to solve some open and fundamental problems of blockchain selfish mining under a rigorous mathematical setting. We first describe a more general model of blockchain selfish mining with both a two-block leading competitive criterion and a new economic incentive mechanism. Then we establish a pyramid Markov process and show that it is irreducible and positive recurrent, and its stationary probability vector is matrix-geometric with an explicitly representable rate matrix. Also, we use the stationary probability vector to study the influence of many orphan blocks on the waste of computing resource. Next, we set up a pyramid Markov reward process to investigate the long-run average profits of the honest and dishonest mining pools, respectively. As a by-product, we build three approximative Markov processes and provide some new interesting interpretation on the Markov chain and the revenue analysis reported in the seminal work by Eyal and Sirer (2014). Note that the pyramid Markov (reward) processes can open up a new avenue in the study of blockchain selfish mining. Thus we hope that the methodology and results developed in this paper shed light on the blockchain selfish mining such that a series of promising research can be developed potentially.

Quan-Lin Li, Jing-Yu Ma, Yan-Xia Chang et al.

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.