Traditional importance measures seldom consider how the number of failed components influences the network reliability. This paper proposes two importance measures under the circumstance that the failure sequence of the components follows a counting process. The first importance measure aims to assess the contribution of the individual component (edge) to the network failure. The second evaluates the contribution of the individual component to the network functionality. Both importance measures are time-dependent functions, and their values are jointly determined by the network structure and the distribution of the number of failed components at a particular time. We prove that the proposed importance measures are able to generate consistent rankings based on edge's impact on the network reliability behavior. When networks possess special structure or the number of failed edges follows the special distribution, the rankings are coincident with the results generated from some traditional importance measures. When component's failure sequence follows a saturated nonhomogeneous Poisson process, the proposed importance measures are equivalent to the structural importance measure as time approaches zero or infinite. Finally, numerical examples are provided to demonstrate the application and performance of the proposed measures.

Studies related to oil and gas wells have attracted worldwide interest due to the increasing energy shortfall and the requirement of sustainable development and environmental protection. However, the state of oil and gas wells in terms of research characteristics, technological megatrends, article-produced patterns, leading study items, hot topics, and frontiers is unclear. This work is aimed at filling the research gaps by performing a comprehensive bibliometric analysis of 6197 articles related to oil and gas wells published between 1900 and 2021. VOSviewer and CiteSpace software were used as the main data analysis and visualization tools. The analysis shows that the annual variation of article numbers, interdisciplinary numbers, and cumulative citations followed exponential growth. Oil and gas well research has promoted the expansion of research fields such as engineering, energy and fuels, geology, environmental sciences and ecology, materials science, and chemistry. The top 10 influential studies mainly focused on shale gas extraction and its impact on the environment. More studies were produced by larger author teams and inter-institution collaborations. Elkatatny and Guo have greatly contributed to the application of artificial intelligence in oil and gas wells. The two most contributing institutions were the Southwest Petr Univ and China Univ Petr from China. The People’s Republic of China, the US, and Canada were the countries with the most contributions to the development of oil and gas wells. The authoritative journal in engineering technology was J Petrol Sci Eng, in environment technology was Environ Sci Technol, in geology was Aapg Bull, and in materials was Cement Concrete Res. The keyword co-occurrence network cluster analysis indicated that oil well cement, new energy development, machine learning, hydraulic fracturing, and natural gas and oil wells are the predominant research topics. The research frontiers were oil extraction and its harmful components (1992–2016), oil and gas wells (1997–2016), porous media (2007–2016), and hydrogen and shale gas (2012–2021). This paper comprehensively and quantitatively analyzes all aspects of oil and gas well research for the first time and presents valuable information about active and authoritative research entities, cooperation patterns, technology trends, hotspots, and frontiers. Therefore, it can help governments, policymakers, related companies, and the scientific community understand the global progress in oil and gas well research and provide a reference for technology development and application.

Bayesian importance measures (BIMs) are useful tools for quantifying the contribution of an edge to the up or down state of the network. This article investigates BIMs for the K-terminal networks under the assumption that the failures of edges occur according to a branching process in which the total number of the failed edges follows a saturated Lagrangian Poisson distribution (SLPD). First, we derive two types of BIM equations when the total number of the failed edges follows a certain probability distribution. Both BIMs are represented in terms of network spectra that depend only on the network structure. It is also found that both BIMs are equivalent as they lead to the identical ranking order of network edges. Next, when the total number of the failed edges has an SLPD, several unique properties of BIMs rankings are explicitly derived. We further prove that under certain conditions, the rankings based on the BIMs belong to the structural ranking, namely, the spectra-based rankings solely depend on the network structure. Finally, the numerical analyses of a transportation network show that the BIMs can effectively measure the edge importance for the medium and relatively large networks.

Facilitated by advanced digital technologies, reliability managers can monitor system working cycles during the whole life cycle. Such a technological realization can help reliability managers ensure system reliability in real time by monitoring working cycles. In this paper, by incorporating a limited random working cycle, rebate and charge into warranty theory, a random free repair warranty with rebate and charge (RFRW-RC) is devised to ensure system reliability during the warranty stage. Under RFRW-RC, the rebate removes manufacturers’ responsibility for continuing to ensure system reliability, while the charge is a support where manufacturers continue to ensure system reliability. The warranty cost of RFRW-RC is derived, and a random discrete free repair warranty (RDFRW) is presented by simplifying RFRW-RC. By mixing random age replacement last (RARL) and classic age replacement (CAR), a random hybrid age replacement (RHAR) is designed in order to ensure system reliability during the post-warranty stage. In such an RHAR, RARL is applied to extend the replacement time during the post-warranty stage in order to maximize the remaining life of the system through warranty, and CAR is used to lower the maintenance cost of the system through warranty. The cost rate of RHAR is modeled, and the cost rate of RDFRW is offered as well by discussing parameter values. The decision variable is optimized by minimizing the cost rate model. The properties of the presented models are explored from numerical perspectives.

A monitoring system (MS) has been used to monitor products’ job cycles. It is indicated that by incorporating the job cycle into the product’s life cycle, warrantors can devise novel warranty models and consumers can define and model random maintenances sustaining the reliability of the product through warranty. In this study, by incorporating limited job cycles and a refund into the traditional free repair warranty, a two-dimensional free repair warranty with a refund (2DFRW-R) is devised for guaranteeing the product reliability to consumers. Under the condition that 2DFRW-R is planned to guarantee product reliability, a bivariate random periodic replacement (BRPR) (i.e., a random periodic replacement where the accomplishment of the Nth job cycle and the replacement time T are designed as replacement limits) is modeled to sustain the post-warranty reliability from the point of view of the consumer. From the point of view of the warrantor, the warranty cost related to 2DFRW-R is derived, and the characteristics of 2DFRW-R are explored. From the point of view of consumers, the expected cost rate related to BRPR is constructed, and the existence and uniqueness of the optimal BRPR are summarized as well. By discussing parameters, several special cases are derived. The characteristics of the proposed models are analyzed in numerical examples.