Quan‐Lin Li

Quan-Lin Li

Big networks express various large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to set up a broad class of nonlinear continuous-time block-structured Markov processes, which can be applied to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of interacting big networks with symmetric interactions, each of which is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the Lyapunov functions and the relative entropy are used to analyze stability or metastability of the big network, and several interesting open problems are proposed with detailed interpretation. We believe that the results given in this paper can be useful and effective in the study of big networks.

Jing-Yu Ma, Li Xia, Quan-Lin Li

In this paper, we propose a novel dynamic decision method by applying the sensitivity-based optimization theory to find the optimal energy-efficient policy of a data center with two groups of heterogeneous servers. Servers in Group 1 always work at high energy consumption, while servers in Group 2 may either work at high energy consumption or sleep at low energy consumption. An energy-efficient control policy determines the switch between work and sleep states of servers in Group 2 in a dynamic way. Since servers in Group 1 are always working with high priority to jobs, a transfer rule is proposed to migrate the jobs in Group 2 to idle servers in Group 1. To find the optimal energy-efficient policy, we set up a policy-based Poisson equation, and provide explicit expressions for its unique solution of performance potentials by means of the RG-factorization. Based on this, we characterize monotonicity and optimality of the long-run average profit with respect to the policies under different service prices. We prove that the bang-bang control is always optimal for this optimization problem, i.e., we should either keep all servers sleep or turn on the servers such that the number of working servers equals that of waiting jobs in Group 2. As an easy adoption of policy forms, we further study the threshold-type policy and obtain a necessary condition of the optimal threshold policy. We hope the methodology and results derived in this paper can shed light to the study of more general energy-efficient data centers.

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.

Jing-Yu Ma, Quan-Lin Li

In this paper, we provide a novel dynamic decision method of blockchain selfish mining by applying the sensitivity-based optimization theory. Our aim is to find the optimal dynamic blockchain-pegged policy of the dishonest mining pool. To study the selfish mining attacks, two mining pools is designed by means of different competitive criterions, where the honest mining pool follows a two-block leading competitive criterion, while the dishonest mining pool follows a modification of two-block leading competitive criterion through using a blockchain-pegged policy. To find the optimal blockchain-pegged policy, we set up a policy-based continuous-time Markov process and analyze some key factors. Based on this, we discuss monotonicity and optimality of the long-run average profit with respect to the blockchain-pegged reward and prove the structure of the optimal blockchain-pegged policy. We hope the methodology and results derived in this paper can shed light on the dynamic decision research on the selfish mining attacks of blockchain selfish mining.

Heng-Li Liu, Quan-Lin Li, Chi Zhang

In this paper, we discuss an interesting but challenging bilateral stochastically matching problem: A more general matched queue with matching batch pair (m, n) and two types (i.e., types A and B) of impatient customers, where the arrivals of A- and B-customers are both Poisson processes, m A-customers and n B-customers are matched as a group which leaves the system immediately, and the customers' impatient behavior is to guarantee the stability of the system. We show that this matched queue can be expressed as a novel bidirectional level-dependent quasi-birth-and-death (QBD) process. Based on this, we provide a detailed analysis for this matched queue, including the system stability, the average stationary queue lengthes, and the average sojourn time of any A-customer or B-customer. We believe that the methodology and results developed in this paper can be applicable to dealing with more general matched queueing systems, which are widely encountered in various practical areas, for example, sharing economy, ridesharing platform, bilateral market, organ transplantation, taxi services, assembly systems, and so on.

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.

Li Xia, Zhe George Zhang, Quan-Lin Li et al.

In this paper, we study a dynamic on/off server scheduling problem in a queueing system with multi-class servers, where servers are heterogeneous and can be classified into $K$ groups. Servers in the same group are homogeneous. A scheduling policy determines the number of working servers (servers that are turned on) in each group at every state $n$ (number of customers in the system). Our goal is to find the optimal scheduling policy to minimize the long-run average cost, which consists of an increasing convex holding cost and a linear operating cost. We use the sensitivity-based optimization theory to characterize the optimal policy. A necessary and sufficient condition of the optimal policy is derived. We also prove that the optimal policy has monotone structures and a quasi bang-bang control is optimal. We find that the optimal policy is indexed by the value of $c - μG(n)$, where $c$ is the operating cost rate, $μ$ is the service rate for a server, and $G(n)$ is a computable quantity called perturbation realization factor. Specifically, the group with smaller negative $c - μG(n)$ is more preferred to be turned on, while the group with positive $c - μG(n)$ should be turned off. However, the preference ranking of each group is affected by $G(n)$ and the preference order may change with the state $n$, the arrival rate, and the cost function. Under a reasonable condition of scale economies, we further prove that the optimal policy obeys a so-called $c$/$μ$-rule. That is, the servers with smaller $c$/$μ$ should be turned on with higher priority and the preference order of groups remains unchanged. This rule can be viewed as a sister version of the famous $cμ$-rule for polling queues. With the monotone property of $G(n)$, we further prove that the optimal policy has a multi-threshold structure when the $c$/$μ$-rule is applied.