In this paper, we investigate a wear and mixed shock model in which the system can fail due to internal aging or external shocks. The lifetime of the system, due to internal wear, follows continuous phase-type (PH) distributions. The external random shocks arrive at the system according to a PH renewal process. The system will fail when the internal failure occurs or $k_1$ consecutive external shocks, the size of at least $d_1$ or $k_2$ consecutive external shocks the size of at least $d_2$ occur, where $d_1<d_2$, $k_1>k_2$. The failed system can be repaired immediately, and the repair times of the system are governed by continuous PH distributions. The system can be replaced by a new and identical one based on a bivariate replacement policy $(L,N)$. The long-run average profit rate for the system is obtained by employing the closure property of the PH distribution. Finally, a numerical example is also given to determine the optimal replacement policy.

In the study of uncertain autoregressive models, how to estimate the unknown parameters and uncertain disturbance term in the models is always a key problem. In view of this, this paper proposes a statistical inference method based on the principle of least squares to determine the unknown parameters and uncertain disturbance term in an uncertain autoregressive model, and designs a numerical algorithm to calculate the numerical solutions of the corresponding estimators. Then, the uncertain hypothesis test is used to verify the applicability of the estimated uncertain autoregressive model, and point forecast and interval forecast are also made for the time series of future moments. Finally, a case study of the Consumer Price Index for all items in U.S. cities is provided to illustrate the effectiveness of the approach proposed in this paper.

A class of complex systems can be structurally decomposed into several typical multi-state weighted k/n systems. In view of this, this paper proposes a reliability model to evaluate the reliability of such systems. The system reliability evaluation process is divided into two critical stages: (1) Primary stage: Structural reliability analysis (system-level) and (2) Secondary stage: Subsystem reliability verification (component-level). In practical applications, when there is a lack of operational data, expert experience is needed to evaluate the states and availability weights of components, which is a typical application scenario for uncertainty theory. Under the framework of uncertainty theory, two-stage reliability measures and importance measures are defined, and corresponding calculation formulas are derived. To improve computational efficiency, a binary search algorithm is proposed. Finally, an application case of SDN was proposed to demonstrate the effectiveness of the theory and method. This study provides an idea for the reliability modeling of complex systems (multiple pipeline transmissions, power systems, distributed computing, etc.) in the absence of operational data.

This paper proposes a two-stage weighted k-out-of-n (W-k/n) system reliability model based on uncertainty theory to address the problem of traditional W-k/n system theory being difficult to apply to the reliability evaluation of complex systems with two-layer structures. This model effectively solves the evaluation problem in the case of insufficient data by decomposing the reliability assessment of complex systems into two stages: system structure reliability and subsystem reliability. In the model, the system state and weights are described by uncertain variables that follow a certain distribution. Under the framework of uncertainty theory, two-stage reliability measures and importance measures are defined, and corresponding calculation formulas are derived. In order to reduce computational complexity, a corresponding numerical solution method including traverse search algorithm and binary search algorithm is proposed. The effectiveness of the method is also demonstrated through a project management case of electronic consumer goods. This study provides new theoretical ideas and methodological support for reliability modeling of complex systems under data loss conditions.

In the field of statistics, uncertain regression analysis occupies an important position. It can thoroughly analyze data sets contained in complex uncertainties, aiming to quantify and reveal the intricate relationships between variables. It is worth noting that the traditional least squares method only takes into account the reduction in the deviations between predictions and observations, and fails to fully consider the inherent characteristics of the correlation uncertainty distributions under the uncertain regression framework. In light of this, this paper constructs a statistical invariant with symmetric uncertainty distribution based on the observations and the disturbance term. It also proposes the least squares estimation of unknown parameters and disturbance term in the uncertain regression model based on the least squares principle and, combined with the mathematical properties of the normal uncertainty distribution, gives a numerical algorithm for solving specific estimates. Finally, in order to verify the effectiveness of the least squares estimation method proposed in this paper, we also design two numerical examples and an empirical study of forecasting of electrical power output.