Reliability evaluation method and system for the ventilation door cylinder based on Bayes Monte Carlo simulation | Scientific Reports

Feb 19, 2025

Scientific Reports volume 15, Article number: 5871 (2025) Cite this article

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The automatic ventilation door in mining operations is a crucial component for ensuring production safety and maintaining ventilation system stability. However, the primary power element of this equipment—the cylinder—often lacks effective monitoring, which can compromise operational reliability. To address this gap, this study proposes a Weibull life prediction method, integrating Bayesian inference and Monte Carlo simulation, aiming at anticipating changes in cylinder reliability. This proactive approach supports timely maintenance to prevent. Given the unknown shape and scale parameters of the Weibull distribution, Bayesian methodology is applied, alongside accelerated life testing principles, to analyze the life characteristics of cylinder. By deriving the posterior distribution function of Weibull parameters, Monte Carlo simulation is employed to estimate these parameters across various operational conditions. This method reveals how life characteristics relate to environmental factors such as temperature. Following the constant-failure-mechanism assumption used in accelerated life testing, the characteristic parameters of cylinder characteristic parameters under standard operating conditions are predicted. Results show that this method is effective for life prediction using truncated small-sample data, overcoming the limitations of conventional approaches. Its applicability is proven in the life assessment of automatic ventilation doors, offering a robust tool for reliability. A reliability evaluation system for mine emergency control equipment is developed. This system provides real-time assessments and visualizations of equipment reliability, enhancing maintenance and management practices essential for mining operations.

As a fundamental component within mine ventilation systems, the ventilation door is integral for isolating, directing, and enabling air circulation. The reliability of this critical control equipment is paramount for safeguarding the ventilation system of mine against potential disaster conditions. Although some coal mining operations continue to employ traditional, manual methods for monitoring and recording ventilation door performance data. These practices lead to substantial data processing burdens, reduced operational efficiency, disorganized data management, and compromised accuracy in information storage. Hence, developing a mathematical model for assessing the reliability of ventilation doors is essential, providing a scientific foundation for ensuring their safe and dependable operation, especially under emergency conditions.

System reliability assessment focuses on determining whether a system can consistently perform its intended functions under defined conditions. Over recent decades, advancements in mining technology have broadened the scope of system reliability engineering, leading to extensive research into evaluation methodologies tailored for mining applications. Since the 1990s, this field has attracted growing interest from researchers aiming to improve the reliability of mining systems and equipment, reflecting the critical role of reliability engineering in modern mining operations.

By using a failure selection mechanism in equipment life tests, Yang presented a quantitative method for evaluating reliability that addresses the challenges of limited failure samples and low calculation accuracy1. Cheng integrated the fuzzy analytic hierarchy process and multi-objective decision theory to assess the reliability of mine ventilation systems2. Guo used the truncated Bayes method and constant stress acceleration method with fixed frequency truncated to study the reliability of substation3. Jiang proposed a remaining useful life prediction model and tested the accuracy of the model under error4. Guo established the system operation reliability evaluation model, which combined reliability evaluation and statistical fault data, and integrated Dynamic Bayesian Network (DBN) and XGBoost in an evaluation framework5.Liu established a model based on Markov chain that can quickly evaluate the reliability of the ventilation system. Liu established a rapid evaluation model of ventilation system reliability based on Markov chain, which can be combined with the current operating state of the system to realize the rapid evaluation of ventilation system operation reliability6.Chen used a hybrid Arrhenius-inverse power-law model to fit the test data to predict cylinder reliability under normal operating conditions7.

Analysis of current mine ventilation door reliability evaluation methods reveals several critical challenges: (1) Existing research primarily focuses on the overall reliability of the mine ventilation system, with limited emphasis on evaluating the reliability of ventilation doors as standalone components. (2) Traditional reliability models entail high computational demands, posing challenges to real-time and rapid ventilation door reliability assessments. (3) The mine environment is inherently complex and variable, with factors such as the uncertain operational states of ventilation door cylinders, making it difficult for conventional models to accurately capture reliability fluctuations under such conditions.

The estimation of parameters using the Weibull distribution function has been extensively researched8,9,10,11. Based on field data conforming to the 3-P Weibull distribution model, Han analyzed the reliability of wind turbine components by maximum likelihood estimation (MLE)12. Saeed used an artificial intelligence optimization method based on the Chebyshev metric to improve the convergence and accuracy of Weibull distribution estimation13. Song and Lu proposed a mechanical reliability evaluation method based on the Weibull distribution, specifically designed for small failure data samples, and substantiated its efficacy through practical illustrations14. Park and Ham established a probabilistic fatigue life model for Ni-based alloys based on Weibull distribution, and quantified the improved reliability according to the failure probability level15. Kohout proposed a Weibull distribution function suitable for experimental curves with convex shapes16. Hua presents a novel approach to fatigue life prediction by integrating the Weibull distribution with the nominal stress method and accounting for the impact of stress gradient on fatigue life17. Li and Guan used the Weibull distribution method to analyze the test data of concrete specimens under different loading modes18. Yang focused on the high reliability life of the three-parameter Weibull distribution for lifting lug fatigue data and proposed a new method to generate reliability life estimates based on generalized reference point inference19. Jia analyzed the reliability of Weibull distribution under homogeneous, heavily censored data and proposed an improved method based on Bayesian inference and least squares method20.

The Weibull distribution is widely recognized as a natural choice for modeling failure characteristics across various products and materials. Its shape parameters provide considerable flexibility, enabling effective adaptation to diverse failure datasets. Furthermore, the analytical form of the Weibull distribution supports straightforward mathematical handling, enhances visualization, and facilitates efficient numerical computation.

Within the domain of reliability analysis, an extensive corpus of failure data finds modeling and analysis through the utilization of the Weibull distribution21,22,23,24,25,26,27,28. The Weibull distribution is a widely used method for modeling life distribution in reliability analysis, especially for equipment with significant wear periods. The parameter estimation of Weibull distribution includes several commonly used methods: graph estimation, least squares estimation, moment estimation, and maximum likelihood estimation. The first three methods usually provide rough estimates with limited accuracy. The maximum likelihood estimation method solves the transcendental equation by an iterative method and has high accuracy when dealing with large samples. However, achieving convergence in some cases is challenging and exhibits significant bias in the face of censored small samples. As the main structure of the ventilation door, the cylinder has a high service life, and it is difficult to collect large sample failure data in a short time. Therefore, the reliability analysis of ventilation door cylinder faces the small sample problem.

The analysis of small-scale data typically entails augmenting the sample size and employing tailored analysis methodologies. Both domestic and international research endeavors have explored the reliability of small-sample data, employing notable methods such as the Bootstrap method based on computer simulation29,30, the Monte Carlo method31,32, the historical data fusion method for estimating fatigue life distribution33 and the extended algorithm based on the Bayesian principle34, etc. While widely used, the Bootstrap method can lead to an excessive dependency on subsamples, affecting the accuracy of the estimated parameter means. The accuracy of Monte Carlo simulation results heavily relies on the underlying mathematical model. The historical data fusion method, which integrates weighted least squares to estimate fatigue life distribution, combines historical and current small-sample data. However, it requires significant experimental data under different conditions, making it unsuitable for analyzing cylinder failure data. In contrast, the Bayesian approach, as a specialized method for processing small-sample data, effectively combines various sources of prior information to derive comprehensive posterior information. It enables more accurate probability estimates without the need for a large subsample. However, it is worth noting that different forms of prior distributions can yield distinct statistical analysis results.

To address the outlined challenges, this study introduces a novel Bayesian approach tailored to parameter estimation of the Weibull distribution, particularly suited for truncated small-sample datasets. Additionally, Monte Carlo simulations are employed to enhance parameter estimation accuracy in these truncated sample scenarios. By comparing the probability density function and life curve of the Weibull distribution with the original data, the proposed method demonstrates reliable predictability for ventilation door cylinder lifespan while accounting for multiple influencing factors.

The primary innovations of this research are threefold. First, leveraging a Bayesian-Monte Carlo simulation-based reliability evaluation model, the sample data size is effectively expanded, addressing the limitations in accuracy commonly observed with small sample sizes of cylinders. Second, the primary power components of mine automatic ventilation door cylinder are studied. Experimental data across varying temperatures and dust concentrations are analyzed, leading to a mathematical model that characterizes parameter variations with temperature and dust concentration. Finally, a comprehensive reliability evaluation system for mine emergency control equipment is established, enabling real-time, rapid assessments of ventilation door reliability.

To accurately assess equipment reliability, it is essential to characterize the failure mechanisms based on the distribution laws obtained from test data analysis. Typically, failure data is subjected to various mathematical fitting techniques, enabling the estimation of unknown distribution parameters.

By incorporating an accelerated life statistical model, experimental data provide a robust theoretical basis for evaluating equipment reliability. Accelerated life tests allow for the determination of the lifetime characteristics of equipment. Utilizing the Weibull distribution model, the precision of reliability calculations is significantly improved.

Several variants of the Weibull distribution have found practical applications in addressing various problems. Assuming that the failure time of the equipment adheres to the Weibull distribution, the corresponding probability density function can be represented by Eq. (1):

The cumulative failure probability can be written as:

And the reliability equation is:

In Eqs. (1), (2),and (3), β is the shape parameter, η is the scale parameter.

The formulation of a prior distribution is pivotal within the Bayesian framework. However, due to the limited test duration for cylinder fault data, small sample sizes, and issues with truncated timing data, the Jeffreys prior distribution is applied to enhance the accuracy of the reliability evaluation results.

Bayesian analysis begins by establishing an initial understanding of parameters using prior information. This prior distribution is then updated with sample data to derive the posterior distribution of the parameters. The steps for constructing a reliability assessment based on Bayesian theory are as follows:

Obtain the a priori information \(\:{I}_{a}\) relevant to the reliability assessment and transform it into the a priori distribution π(β, η | Ia);

Construct a likelihood function L(β, η | x) based on the observed data x and the chosen statistical model;

Substitute the distribution density function into the corresponding Bayesian prior distribution function to obtain the posterior distribution g(β, η | Ia) with unknown parameters;

The reliability of the product is assessed based on the statistical model chosen and the posterior distribution obtained g(β, η | x).

In Bayesian theory, determining an appropriate prior distribution is essential. When prior information is scarce or difficult to obtain, an uninformative prior can be employed. Typically, a uniform distribution is used to represent the prior when data collection is limited or challenging.

Conversely, when prior information is available, an informative prior can be applied. This is achieved by gathering failure time data from similar models or utilizing historical data from comparable systems. The model is characterized by leveraging this information to better inform the prior distribution.

In cases where both the shape and scale parameters are unknown, the lack of a suitable conjugate prior for the Weibull distribution necessitates the use of an uninformative prior. According to Jeffreys’ law, the uninformative prior distributions for parameters β and η are defined by Eq. (4):

The likelihood function is the product of the probability density function of each failure time. According to Eqs. (1) and (4), the likelihood function of sample x is as follows:

The range of values [β1, β2] for the parameter β can be given empirically. According to Bayes’ theorem, the joint posterior distribution density function of the parameters β and η is as follows:

According to Eq. (6), β and η follows the gamma distribution.

Building on this foundation, the posterior density function for the parameters can be derived by integrating parameters β and η. Applying the principle of minimum square loss, the Bayesian estimates for parameters β and η are obtained as follows:

Since Eqs. (7) and (8) are difficult to be solved analytically directly, the random walk Metropolis-Hastings algorithm35 in Markov chain-Monte Carlo (MCMC) numerical simulation method is adopted to solve them.

The suggested density function of the candidate point β* of parameter β is the uniform distribution on the interval [β1, β2], and the suggested density function of the candidate point η* is as follows:

In Eq. (9), j is the current number of simulations; ηj-1 is the JTH simulation value of η. σ is constant. The receiving probabilities of candidate points β* and η* are respectively as follows36:

In Eqs. (10) and (11), βj-1 is the JTH simulation value of m, and the initial value β0 when j = 0. And η0 can be set according to the actual situation.

The Markov chain Monte Carlo method (MCMC) is also called the Markov chain Monte Carlo sampler, which combines the Markov chain with the Monte Carlo method. The Markov chain refers to a series of random samples x(x1, x2,…, xn), P(xn+1| x1, x2,…, xn). The value \(\:{x}_{n+1}\) is related only to its current value, not its previous value. In the Markov chain, the conversion can take place between adjacent samples (xi, xi+1), and its conversion probability P(xi→xi+1) is determined by the conversion probability distribution function. In homogeneous Markov chains, the generated sample distribution will gradually converge and become stable. In the context of Bayesian inference, the stationary distribution is the posterior distribution of the parameters to be estimated. However, the initial sample of the Markov chain does not come from the stationary distribution, so it cannot represent the stationary distribution, and this process is called the burn-in stage, which should be removed from the Markov chain.

The distribution is the posterior distribution of parameters to be estimated. However, the initial sample of the Markov chain does not come from the stationary distribution, so it cannot represent the stationary distribution. This process is called the burn-in stage, which should be removed from the Markov chain.

Suppose the number of samples Nb is generated during the maturity period, and the total number of samples N is generated by the Markov chain. After Nb times of calculation, the resulting simulation sequence can be considered as samples from a posteriori distribution, while the samples before Nb times belong to the old refining stage of the algorithm and cannot be considered as samples from a posteriori distribution. Therefore, MCMC creates a Markov chain whose invariant distribution is the posterior probability distribution.

According to the cylinder failure data, the values of parameters β and η are calculated using the maximum likelihood estimation method. The results are substituted into η conditional posterior density function and β conditional posterior density function.

The primary objective of this test was to collect time-truncated failure data for the cylinder under study. This data was subsequently used to calculate the parameters of the Weibull distribution. The complex structure and substantial size of automatic control ventilation doors in mines pose significant challenges for testing the system as a whole. Upon analyzing the overall structure, it becomes evident that the primary power component is the cylinder, which functions as the closure mechanism for the doors.

As the cylinder is critical to the system’s operation, damage to this component can result in complete device failure. Therefore, this study focuses on the cylinder valve as the test object.

The automatic ventilation door control system primarily consists of key components such as a Programmable Logic Controller (PLC)-based integrated monitoring system, a substation linkage control system, and an underground linkage control ventilation door. Central to the operation of system is the power cylinder, which plays a critical role in controlling the ventilation door.

In this experiment, the SC 32 × 75 cylinder model was utilized. The cylinder operates optimally at a maximum input pressure of 1 MPa and is constructed from heat-resistant sealing materials that ensure reliable performance at temperatures up to 150 °C.

Cylinder failures can stem from a variety of factors, with the most common failure modes being excessive leakage due to worn seals and increased starting friction. Notably, excessive cylinder leakage is often observed following episodes of elevated starting friction. To effectively monitor cylinder performance degradation, the minimum operating pressure (MOP) is selected as a suitable index, given its ease of measurement and its ability to quantify starting friction. Cylinder life, typically measured by the number of piston reciprocations within the cylinder barrel, is a key metric for evaluating the longevity and durability of the cylinder. The accelerated life testing device for the cylinder is shown in Fig. 1.

Cylinder accelerated life test device.

The cylinder was placed in an enclosed test chamber equipped with four round apertures positioned on its top, bottom, and rear surfaces. The round holes at the top and bottom were used for cylinder connection pipes for pressure detection, while the rear aperture allowed for the introduction of dust to simulate high dust concentration environments. The cylinder and test chamber were further enclosed within a temperature-controlled chamber, with overall temperature adjustments managed via a temperature control unit. A computer connected to sensors monitored the cylinder’s pressure values, recording both the minimum operating pressure and the number of operational cycles when the cylinder functioned normally.

The reliability testing of the power cylinder requires the collection of its minimum starting pressure. The inlet pressure for the cylinder is set at 1 MPa, with precise control maintained through a pressure-reducing valve. A pressure sensor is used to record the minimum pressure at which the cylinder operates without failure. The detailed experimental procedure is as follows:

In a controlled dry environment, temperature tests are conducted at 50 °C, 100 °C, 200 °C, and 300 °C. For each condition, 1 MPa of air pressure is applied through the inlet and then sealed. The cylinder is firmly mounted onto a hydraulic fatigue testing machine using a rigid fixture. The hydraulic device is connected to the screw at the top of the cylinder piston, which has a stroke length of 60 mm and operates at a loading frequency of 180 cycles per minute (cpm).

Under standard temperature and humidity conditions, dust concentrations of 10, 50, 100, and 200 mg/m³ are tested. Dust particles, with a diameter of 50 μm, are accurately measured and introduced into the 1 m³ test chamber via a blower. Afterward, 1 MPa of air pressure is applied to the cylinder’s inlet and sealed. The cylinder is again secured to the hydraulic fatigue testing machine, which is connected to the screw at the top of the piston, with a stroke length of 60 mm and a loading frequency of 180 cpm.

Cylinder failure data was collected at the appropriate working conditions according to the cylinder accelerated life test setup. The data for tests is presented in Table 1.

The Metropolis-Hastings algorithm was set with 20,000 sampling iterations and a burn-in period of 3,000 iterations (Nburn-in=3000). The least squares method and the maximum likelihood estimation method based on Newton iteration (Newton-MLE) were used to estimate the two parameters, respectively. The posterior density distribution and the highest density region are shown in Fig. 2, while the comparative results are presented in Fig. 3.

Each simulation yields a set of estimates for parameter β and η, resulting in 20,000 sets of parameter estimates after 20,000 simulations. A statistical analysis of these results provides the probability density distributions for the estimates of β and η. In Fig. 2, the darker regions indicate the areas with the highest frequency of parameter estimates, and the mean of all samples is selected as the final estimate for parameters β and η.

Figure 3 shows that the Weibull reliability evaluation method based on Bayesian inference falls between the least squares method and Newton-MLE when fault data is used. Under consistent fault data conditions, the parameter η estimates obtained using this method are more accurate than those from the least squares method and more reasonable than those from Newton-MLE. For the shape parameter β, the estimated value is greater than 1, indicating that the failure rate increases over time. The failure pattern identified by this method aligns with the results from the least squares and Newton-MLE methods, but the parameter estimates for β and η are more precise.

The posterior density distribution diagram and the highest density region diagram. (a) Temperature 50 ℃; (b) Temperature 100 ℃; (c) Temperature 200 ℃; (d) Temperature 300 ℃; (e) Dust concentration 10mg/m3; (f) Dust concentration 50mg/m3; (g) Dust concentration 100mg/m3; (h) Dust concentration 200mg/m3.

Comparing results of \(\:\beta\:\) and \(\:\eta\:\) under different experimental conditions. (a) Temperature 50 ℃; (b) Temperature 100 ℃; (c) Temperature 200 ℃; (d) Temperature 300 ℃; (e) Dust concentration 10mg/m3; (f) Dust concentration 50mg/m3; (g) Dust concentration 100mg/m3; (h) Dust concentration 200mg/m3.

The Weibull distribution parameters obtained under different conditions using the Monte Carlo simulation, least squares method, and Newton-MLE were substituted into Eq. (3). Figure 4 illustrates the reliability function curves for the cylinder using the three methods at a temperature of 30 °C and a dust concentration of 200 mg/m3.

Cylinder reliability curve.

Figure 4 shows that the reliability curve derived from the least squares method begins to decline at approximately 1.8 × 106 cycles, reaching zero at around 4.3 × 106 cycles. This curve exhibits the fastest rate of decline. In contrast, the reliability curve from the Newton-MLE method starts to decrease around 1.4 × 106 cycles and reaches zero near 4.7 × 106 cycles, demonstrating the slowest rate of decline. The reliability curve obtained from the Bayesian-Monte Carlo simulation-based model falls between these two, with a similar trend. This consistency in the variation of the reliability curves confirms the accuracy of the Bayesian-Monte Carlo simulation model for reliability evaluation.

According to the estimated values of parameters β and η under different conditions obtained by Monte Carlo simulation, the mathematical model of parameters β and η changing with temperature and dust concentration ratio was obtained. Figure 5 shows the fitting results of the parameters β and η.

Substituting the fitting formula of parameters β and η in Fig. 5 into Eq. (15), the mathematical model of cylinder reliability under different temperatures and dust concentrations can be obtained:

Where, t is operation times of equipment, cycles; C is the dust concentration, mg/m3; T is the temperature, ℃.

The posterior mean of Weibull distribution parameters of cylinder under normal working conditions is obtained through fitting analysis. The results of parameter estimation are shown in Table 2.

The fitting results of feature parameters. (a) Parameter β; (b) Parameter η.

According to the obtained parameters β and η probability distribution, the mean cycle between failure(MCBF) can be:

The function curves of each reliability index are shown in Fig. 6.

The MCBF obtained by maximum likelihood estimation, least square method and original experimental data were compared, with the results presented in Table 3.

Table 3 shows that the MCBF values obtained from the other two methods are lower than those from the reliability assessment method based on Bayesian-Monte Carlo simulation. This is attributed to the fact that the cylinder experimental data are small-sample censored data. The proposed method mitigates the variability introduced by small, truncated datasets, thereby enhancing the accuracy of the reliability evaluation.

Results of cylinder parameters at 30 ℃ under different times. (a) Cumulative failure probability, (b) Failure probability density, (c) Degree of reliability.

A mathematical model for evaluating the reliability of emergency control equipment was developed, and a reliability evaluation system was constructed using the GUI editing capabilities of Matlab.

The system comprises a data monitoring interface for mine working faces and a reliability evaluation interface for emergency control equipment. It collects data from fast-sealing and combined wind control sensors, transmitting this information to the three-dimensional dynamic visualization and reliability evaluation systems for real-time analysis. The process framework of the emergency control equipment reliability evaluation system is illustrated in Fig. 7.

Process framework of reliability evaluation system for emergency control equipment.

The reliability evaluation system for emergency control equipment utilizes the TCP/IP protocol for data integration. After detecting underground environmental conditions, mine sensors transmit the collected data to an SQL (Structured Query Language) Server database via the mine’s industrial Ethernet ring network. Upon receiving data packets from Ethernet, the database parses them and periodically sends the data to the host end of the reliability evaluation system. The host processes the data and displays the final reliability results.

The system was developed using the GUI visual interface of Matlab, enabling real-time visualization of reliability evaluation results for airflow emergency control equipment.

The main interface consists of an emergency control equipment data monitoring panel, a reliability evaluation panel, and an equipment fault prediction panel. The data monitoring panel displays preset equipment timings and environmental data. The evaluation panel assesses the reliability of the equipment’s current state and displays the results. Mine monitoring sensors upload data to the SQL Server database in real time, allowing the working face data monitoring interface and database server to issue operation commands to the emergency control equipment and visually display the data. Simultaneously, sensor data are transmitted to the reliability evaluation system via the working face data monitoring interface, enabling the system to evaluate the reliability of the emergency control equipment. The evaluation results are then transmitted back to the monitoring interface. Figure 8(a) shows the working face data monitoring interface and the reliability evaluation interface.

Reliability Evaluation Interface. (a) Main Interface; (b) Parameter interface.

As shown in Fig. 8(b), the reliability of the equipment decreases as the preset time increases. With rising temperature and dust concentration, the slope of the reliability curve steepens, indicating that these factors significantly affect the working life and reliability of the equipment. This trend is consistent with the behavior observed in the test data, confirming that the system effectively evaluates the operational state of the emergency control equipment.

In response to the limited experimental data and difficulty in real-time reliability prediction of mine automatic air door cylinders, this study uses Weibull distribution and a combination of Bayesian method and Monte Carlo simulation to predict equipment reliability under small sample conditions. A reliability evaluation system for mine emergency control equipment is constructed to achieve real-time evaluation and visualization of mine automatic air door reliability. The following conclusions are drawn:

In the absence of a conjugate prior distribution, it is assumed that the parameter to be estimated follows a Gamma prior distribution, and the posterior probability density of the parameter is derived. By leveraging existing data and expert knowledge, the Metropolis-Hastings sampling method in MCMC is applied to obtain point estimates of the parameters, enabling the simulation of complex density functions.

The functional relationship between Weibull distribution parameters and operating conditions is derived. Based on the failure mechanisms observed in accelerated life tests, the proposed method is extended to equipment reliability evaluation, ensuring the accuracy of life prediction. The analysis confirms that the model’s output is both accurate and reliable, expanding the applicability of Bayesian inference while maintaining high precision.

The model is adaptable, allowing for the estimation of parameters and reliability functions for different Weibull-type products by adjusting the prior distribution. Once the prior distribution of the parameters is determined using prior information and expert experience, parameter estimates can be refined, facilitating reliability evaluation and management of equipment with small sample data.

A mathematical model relating temperature and dust concentration to Weibull distribution parameters is established based on the accelerated life model. The modified Weibull parameters are calculated, and the detected environmental parameters are uploaded to the SQL Server database via the industrial Ethernet ring network of mine, enabling real-time, accurate reliability calculations. A reliability evaluation system for emergency control equipment has been developed using Matlab, allowing real-time monitoring of mine environmental indicators and equipment through direct integration with the mine’s monitoring data.

The datasets used and analysed during the current study available from the corresponding author on reasonable request.

Yang, X., Fang, Z. & Tao, L. Gert network model for life evaluation of equipment reliability test. J. Nanjing Univ. Aeronaut. Astronaut. 48 (05), 689–695. https://doi.org/10.16356/j.1005-2615.2016.05.012 (2016).

Article MATH Google Scholar

Cheng, G. et al. Comprehensive evaluation of mine ventilation system reliability based on fahp-modm. J. Saf. Sci. Technol. 14 (02), 99–105. https://doi.org/10.11731/j.issn.1673-193x.2018.02.016 (2018).

Article MATH Google Scholar

Guo, J. & Zhang, Y. Study of reliability of substation for coal mine supervision system. Energy Procedia 17, 513–519. https://doi.org/10.1016/j.egypro.2012.02.129 (2012).

Article MATH Google Scholar

Jiang, D. et al. A mechanical system reliability degradation analysis and remaining life estimation method—with the example of an aircraft hatch lock mechanism. Reliab. Eng. Sys Safe 230, 108922. https://doi.org/10.1016/j.ress.2022.108922 (2023).

Article MATH Google Scholar

Guo, Y., Wang, H. & Guo, Y. System operational reliability evaluation based on dynamic bayesian network and XGBoost. Relia Eng. Sys Safe 225, 108622. https://doi.org/10.1016/j.ress.2022.108622 (2022).

Article MATH Google Scholar

Liu, L., Liu, J. & Zhou, Q. Mine ventilation system reliability evaluation based on a Markov chain. Sci. Rep. 12 (1), 17115. https://doi.org/10.1038/s41598-022-22098-z (2022).

Article ADS CAS PubMed PubMed Central MATH Google Scholar

Chen, J. et al. : Accelerated Life Test for Reliability Evaluation of Pneumatic Cylinders. IEEE Access 6, 75062–75075. https://doi.org/10.1109/ACCESS.2018.2882767 (2018).

Charongrattanasakul, P. & Pongpullponsak, A. Optimizing the cost of integratedmodel for fuzzy failure weibull distribution using genetic algorithm. Eng. J. 21 (02), 253–268. https://doi.org/10.4186/ej.2017.21.2.253 (2017).

Article MATH Google Scholar

Taereem, K. & Shin, H. J. A study on empirical distribution function with unknown shape parameter and extreme value weight for three parameter weibull distribution. J. Kor Water Resour. Assoc. 46 (06), 643–653. https://doi.org/10.3741/JKWRA.2013.46.6.643 (2013).

Article MATH Google Scholar

Aydogdu, H., Senog, L. B. & Parameter, M. K. Parameter estimation in geometric process with weibull distribution. Appl. Math. Comput. 217 (06), 2657–2665. https://doi.org/10.1016/j.amc.2010.08.003 (2010).

Article MathSciNet MATH Google Scholar

Zheng, R. Parameter estimation of three parameter weibull distribution and its applicat-ion in reliability analysis. Shock Vib. 34 (05), 78–81. https://doi.org/10.13465/j.cnki.jvs.2015.05.014 (2015).

Article MATH Google Scholar

Han, F. et al. Reliability analysis of wind turbine subassemblies based on the 3-p weibull model via an ergodic artificial bee colony algorithm. Probabilistic Eng. Mech. 73. https://doi.org/10.1016/j.probengmech.2023.103476 (2023).

Saeed, M. A., Ahmed, Z., Yang, J. & Zhang, W. An optimal approach of wind power assessment using chebyshev metric for determining the weibull distribution parameters. Sustain. Energy Techn. 37. https://doi.org/10.31801/cfsuasmas.455276 (2020).

Song, M., Lu, W. & Fang, X. A research on mechanical reliability assessment based on small sample failure data. Ind. Eng. J. 20 (05), 87–93. https://doi.org/10.3969/j.issn.1007-7375.e17-4099 (2017).

Article MATH Google Scholar

Park, J. P. et al. Statistical analysis of sn type environmental fatigue data of Ni-base alloy welds using weibull distribution. Nucl. Eng. Technol. 55 (05), 1924–1934. https://doi.org/10.1016/j.net.2022.08.005 (2022).

Article CAS MATH Google Scholar

Kohout, J. Three-parameter weibull distribution with upper limit applicable in reliability studies and materials testing. Microelectron. Reliab. 137. https://doi.org/10.1016/j.microrel.2022.114769 (2022).

Hua, F. et al. Research on multiaxial fatigue life of notched specimens based on weibull distribution and bayes estimation. Int. J. Fatigue 166, 107271. https://doi.org/10.1016/j.ijfatigue (2022).

Li, L. et al. A weibull distribution-based method for the analysis of concrete fracture. Eng. Fract. Mech. 256. https://doi.org/10.1016/j.engfracmech.2021.107964 (2021).

Yang, X. et al. Inference on the high-reliability lifetime estimation based on the three-parameter Weibull distribution. Probabilistic Eng. Mech., 77,103665. https://doi.org/10.1016/j.probengmech.2024.103665 (2024).

Jia, X. Reliability analysis for Weibull distribution with homogeneous heavily censored data based on bayesian and least-squares methods. Appl. Math. Model. 83, 169–188. https://doi.org/10.1016/j.apm.2020.02.013 (2020).

Article MathSciNet MATH Google Scholar

Li, J., Huang, M. & Zhao, Y. Accuracy analysis of maximum likelihood estimation of weibull distribution. J. Beijing Univ. Aeronaut. Astronaut. 32 (08), 930–932. https://doi.org/10.13700/j.bh.1001-5965.2006.08.013 (2006).

Article CAS MATH Google Scholar

Zhang, H. Application of three parameter weibull distribution in mechanical reliability analysis. Mech. Manage. Dev. 24 (03), 59–60. https://doi.org/10.16525/j.cnki.cn14-1134/th.2009.03.019 (2009).

Article CAS MATH Google Scholar

Tiryakiolu, M. & Hudak, D. On estimating weibull modulus by the linear regression method. J. Mater. Sci. 42 (24), 0–73079. https://doi.org/10.1007/s10853-007-2060-5 (2009).

Article CAS MATH Google Scholar

Razali, A. M. & Al-Wakeel, A. A. Mixture weibull distributions for fitting failure times data. Appl. Appl. Math. Comput. 219 (24), 11358–11364. https://doi.org/10.1016/j.amc.2013.05.062 (2013).

Article MathSciNet MATH Google Scholar

Ali, A. Progressive stress accelerated life test for inverse weibull failure model: a parametric inference. J. King Saud Univ. Sci. 34 (04). https://doi.org/10.1016/j.jksus.2022.101994 (2022).

He, J. et al. Using two and three-parameter weibull statistical model for predicting the loading rate effect on low-temperature fracture toughness of asphalt concrete with the endb specimen. Theor. Appl. Fract. Mech. 121. https://doi.org/10.1016/j.tafmec.2022.103471 (2022).

Surucu, B. & Sazak, H. S. Monitoring reliability for a three-parameter weibull distribution. Reliab. Eng. Syst. Safe 94 (02), 503–508. https://doi.org/10.1016/j.ress.2008.06.001 (2009).

Article MATH Google Scholar

Zhang, Q., Hua, C. & Xu, G. A mixture weibull proportional hazard model for mechanical system failure prediction utilising lifetime and monitoring data. Mech. Syst. Signal. Pr. 43 (1–2), 103–112. https://doi.org/10.1016/j.ymssp.2013 (2014).

Article MATH Google Scholar

Duan, X. & Wang, Z. Applicability of bootstrap method in small sample case. J. Ballistic. 15 (03), 103–112. https://doi.org/10.3969/j.issn.1004-499X.2003.03.001 (2003).

Article MATH Google Scholar

Hu, W. & Qian, Q. Small data reliability analysis in concrete three-point bending tests: a Weibull mixture model approach based on Weibull fracture theory. Eng. Fract. Mech. 110344 https://doi.org/10.1016/j.engfracmech.2024.110344 (2024).

Gong, H. et al. A Monte Carlo and PSO based virtual sample generation method for enhancing the energy prediction and energy optimization on small data problem: an empirical study of petrochemical industries. Appl. Energy. 197, 405–415. https://doi.org/10.1016/j.apenergy.2017.04.007 (2017).

Article ADS MATH Google Scholar

Betz, W., Papaioannou, I. & Straub, D. Bayesian post-processing of monte carlo simulation in reliability analysis. Reliab. Eng. Syst. Safe 227. https://doi.org/10.1016/j.ress.2022.108731 (2022).

Ye, G., Kong, X. & Sun, W. Study of power system transient stability assessment with monte carlo method. Proc. CSU-EPSAc 24(05), 71–76. https://doi.org/10.3969/j.issn.1003-8930.2012.05.013 (2012).

Zhang, Y., Dong, Y. & Feng, R. Bayes-informed mixture distribution for the evd estimation and dynamic reliability analysis. Mech. syst. Signal. Pr. 197. https://doi.org/10.1016/j.ymssp.2023.110352 (2023).

Li, Y. & Yi, Z. An improved mcmc detection algorithm for massive mimo systems. Appl. Electron. Tech. 42 (05), 101–104. https://doi.org/10.16157/j.issn.0258-7998.2016.05.028 (2016).

Article MATH Google Scholar

Wang, G. Bayesian regression models for ecological count data in pymc3. Ecol. Inf. 63. https://doi.org/10.1016/j.ecoinf.2021.101301 (2021).

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The authors gratefully acknowledge the financial support provided by the National Key Research and Development Program of China (No. 2018YFC0808100) and the National Natural Science Foundation of China (No. 52174182).

College of Mining Engineering, North China University of Science and Technology, Tangshan, 063210, Hebei, China

Peiyang Su

College of Emergency Management and Safety Engineering, North China University of Science and Technology, Tangshan, 063210, Hebei, China

Liwen Guo & Jiayong Zhang

College of Safety Science and Engineering, Xi’an University of Science and Technology, Xi’an, 710054, Shanxi, China

Li Ma

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Peiyang Su: Validation, Formal analysis, Writing-original draft, Writing-review & editing, Data curation, Visualization. Liwen Guo: Conceptualization, Methodology, Writing-review & editing, Supervision, Funding acquisition. Jiayong Zhang: Investigation, Resources, Data curation, Writing-review & editing. Li Ma: Writing-review & editing, Data curation.

Correspondence to Liwen Guo or Jiayong Zhang.

The authors declare no competing interests.

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Su, P., Guo, L., Zhang, J. et al. Reliability evaluation method and system for the ventilation door cylinder based on Bayes Monte Carlo simulation. Sci Rep 15, 5871 (2025). https://doi.org/10.1038/s41598-025-90311-w

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Received: 31 July 2024

Accepted: 12 February 2025

Published: 18 February 2025

DOI: https://doi.org/10.1038/s41598-025-90311-w

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