In a practical blockchain system based on the Practical Byzantine Fault Tolerance (PBFT) protocol, the voting nodes can fail at any time due to non-Byzantine errors, such as autonomous shutdowns, device crashes, and communication link failures caused by mobility or obstacles. These errors may cause voting nodes to exit the PBFT-based blockchain system unpredictably, resulting in a variable number of voting nodes available at any given time. To maintain optimal performance and consistency while adapting a PBFT-based blockchain system to this dynamic change, this paper proposes an extension to the PBFT protocol by introducing a repair process for failed nodes. The new PBFT-based blockchain system with repairable voting nodes is then analyzed for performance and reliability analysis by using multi-dimensional Markov processes, queueing theory, and the first passage time method. Additionally, we validate the accuracy of our theoretical findings by conducting numerical examples and simulation experiments. These experiments demonstrate that the introduction of a repair process can improve the performance and reliability of the PBFT-based blockchain system. Furthermore, we illustrate how various system parameters impact the performance measures of the PBFT-based blockchain system with repairable voting nodes. We hope that the methodology and results presented in this paper will establish a common framework for deriving theoretical analysis of existing PBFT-based blockchain systems and inspire future research efforts in this field.

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 ratios and 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 ratios and 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.

How to generate anthropomorphic reaching movement remains a challenging problem in service robots and human motor function repair/reconstruction equipment. However, there is no universally accepted computational model in the literature for reproducing the motion of the human upper limb. In response to the problem, this article presents a computational framework for generating reaching movement endowed with human motion characteristics that imitated the mechanism in the control and realization of human upper limb motions. This article first establishes the experimental paradigm of human upper limb functional movements and proposes the characterization of human upper limb movement characteristics and feature movement clustering methods in the joint space. Then, according to the specific task requirements of the upper limb, combined with the human sensorimotor model, the estimation method of the human upper limb natural postures was established. Next, a continuous task parametric model matching the characteristic motion class is established by using the Gaussian mixture regression method. The anthropomorphic motion generation method with the characteristics of the smooth trajectory and the ability of natural obstacle avoidance is proposed. Finally, the anthropomorphic motion generation method proposed in this article is verified by a human-like robot. The measurement index of the human-likeness degree of the trajectory is given. The experimental results show that for all four tested tasks, the human-likeness degrees were greater than 90.8%, and the trajectories' jerk generated by this method is very similar to the trajectories' jerk of humans, which validates the proposed method.

A simple and efficient method is proposed to visualize a symmetric second order tensor along 3D curves. It is also applied to an oil well model in which, along the well trajectory, the spatial distribution of geostress is visualized to predict the stability of the oil well bore. To give visual representations several forms are used: (1) we visually compare the magnitude of stresses to determine the mud weight; (2) based on a set of ellipse cross-sections along the well trajectory and the ellipticity of the ellipse we predict the shear failure direction and the stability of the oil well bore; (3) from a hinged curved surface, we emphasize the direction of principal stresses; (4) to give useful information we apply other skills such as magnified details, animation and an interactive interface. Our method avoids two defects in the references, that is, the visual cluster caused by discrete point tensor glyphs, and the integration problem of the hyperstreamline.