Bayesian Reliability, Availability, and Maintainability Analysis for Hardware Systems Described Through Continuous Time Markov Chains

Bayesian approaches have been demonstrated as effective methods for reliability analysis of complex systems with small-amount data, which integrate prior information and sample data using Bayes’ theorem. However, there is an assumption that precise prior probability distributions are available for unknown parameters, yet these prior distributions are sometimes unavailable in practical engineering. A possible way to avoiding this assumption is to generalize Bayesian reliability analysis approach by using imprecise probability theory. In this paper, we adopt a set of imprecise Dirichlet distributions as priors to quantify uncertainty of unknown parameters and extend traditional Bayesian reliability analysis approach by introducing an imprecise Dirichlet model (IDM). When the prior information is rare, the result of imprecise Bayesian analysis method is too rough to support engineering decision-making, so we proposed an optimization model to reduce the imprecision of the new method. Spindles are crucial for machine tools and reliability data related to spindles of new-developed machine tools are often rare. We can then use the imprecise Bayesian reliability analysis method to assess its reliability. In this paper, we mainly investigate the reliability assessment of a motorized spindle to illustrate the effectiveness of the proposed method.

Bayesian reliability analysis (BRA) technique has been actively used in reliability assessment for engineered systems. However, there are two key controversies surrounding the BRA: the reasonableness of the prior and the consistency among all data sets. These issues have been debated in Bayesian analysis for many years. As we observed, they have not been resolved satisfactorily. These controversies have seriously hindered the applications of BRA as a useful reliability analysis tool to support engineering design. In this paper, a Bayesian reliability analysis methodology with a prior and data validation and adjustment scheme (PDVAS) is developed to address these issues. As the part of the PDVAS development, a consistency measure is first defined that judges the level of consistency among all data sets including the prior. The consistency measure is then used to adjust either the prior or the data or both to the extent that the prior and the data are statistically consistent. This prior and data validation and adjustment scheme is developed for Binomial sampling with Beta prior, called Beta-Binomial Bayesian model. The properties of the scheme are presented and discussed that provides some insights of PDVAS. Various forms of the adjustment formulas are shown, and a selection framework of a specific formula, based on engineering design and analysis knowledge, is established. Several illustrative examples are presented, which show the reasonableness, effectiveness, and usefulness of PDVAS. General discussion of the scheme is offered to enhance the Bayesian reliability analysis in engineering design for reliability assessment.

A two-component parallel or series system is studied where the failure time of each component follows the Burr distribution. The testing environment of the system is assumed different from the operating one, causing the parameters (i=1, 2) to increase or decrease by a positive, random amount η. Two priors are considered for the behaviour of η, a gamma and a uniform. The Bayesian reliability is derived in closed form. Three hypothetical examples show how this approach could be used by reliability engineers.

The contribution first describes the implementation of Continuous-Time Markov Chains (CTMC) in the language Modelica; this enables the simulation by powerful solvers making typical Functional Safety applications numerically more efficient and reliable than with classically used Discrete-Time Markov Chains (DTMC). Secondly, the CTMC are extended to Multi-Phase Markov Chains (CTMC MP ) enabling the inclusion of maintenance (restoration) actions as well as online diagnostics, which is finally elaborated to the online prediction of an individual system's safety quantities.

Prior information and elicitation are the prerequisite in Bayesian reliability inference. Multiple sources for priors such as probability fitting based on historical data and expert judgment are often available when estimating the reliability of complex systems. This paper investigates the integration of multiple priors in Bayesian reliability analysis. Specifically, methods for multiple priors' integration based on Bayesian Melding are investigated. The performance of the studied methods with different prior information integration algorithms such as the arithmetic and geometric averaging is investigated. The impacts of the prior misspecification and the pooling parameter selection for prior integration algorithms are also studied. In numerical examples, simulation methods are applied for posterior reliability inference under the proposed prior integration methods and the performance of the two methods are compared.

Optimizing gas turbine maintenance can reduce total operating costs but raises the risk of unit failures. A large pipeline fleet requires constant up time, but experiences variation in unit-to-unit reliability and availability. There is an ongoing desire to optimize maintenance budget to maximize fleet availability. This can be done using detailed maintenance records; however, analyzing these in-depth for the entire fleet can be cumbersome. In this study, a Bayesian reliability and availability analysis is used to predict failure risks for both the fleet and individual units using only high level summary information including MTBF, MTTR, and asset service factor. Bayesian statistics enable transformation of these summary metrics into predictive distributions for reliability and availability accounting for seasonal and annual variations. The study considers a fleet of gas turbines and compares individual units to the fleet to identify outliers. This provides the first step in prioritizing maintenance decisions.

Failure Models. Qualitative System Analysis. Systems of Independent Components. Component Importance. Markov Models. Counting Processes. Dependent Failures. Life Data Analysis. Accelerated Life Testing. Bayesian Reliability Analysis. Reliability Data Sources. Appendices. References. Indexes.

The state space approach is a common method for conducting reliability assessment of protective systems. However, as more factors affecting the reliability of protective relay are considered, the system Markov model is becoming more and more complex and its solving efficiency is decreasing. To avoid this, a method to simplify the Markov model of protective system is proposed in this paper. Based on the continuous-time Markov process (CTMP), the theory underlying the state space approach, the method illustrates that the duration of the protective system in the switching states can be neglected. In other words, the protection switching state can be ignored when developing a Markov model of protective system. This method is applied to two existing models, one for a protection/component system and the other for redundant protective systems. The study results show that the method significantly reduces the complexity of the Markov model and improves the solving efficiency without reducing the capability of Markov model to describe the reliability of protective systems.

The predominant technique for computing the transient distribution of a Continuous Time Markov Chain (CTMC) exploits uniformization, which is known to be stable and efficient for non-stiff to mildly-stiff CTMCs. On stiff CTMCs however, uniformization suffers from severe performance degradation. In this paper, we report on our observations and analysis of an alternative technique using Krylov subspaces. We implemented a Krylov-based extension to MRMC (Markov Reward Model Checker) and conducted extensive experiments on five case studies from different application domains. The results reveal that the Krylov-based technique is an order of magnitude faster on stiff CTMCs.

Integrating dynamic capability theory and social exchange theory, this study reveals the impact of organizational resilience (OR) on firm performance (FP) from both short-term and long-term perspectives. The mediating roles of R&D innovation (RD) and employee turnover (ET), as well as the moderating role of artificial intelligence (AI), are also explored. Using a sample of Chinese A-share listed companies in the manufacturing industry from 2009 to 2022, this study demonstrates that OR contributes to both short-term performance (SP) and long-term performance (LP). RD serves as a negative mediator between OR and SP, while it plays a positive mediating role between OR and LP. ET, in turn, acts as a positive mediator between OR and SP/LP. Our results also suggest that AI positively moderates the relationship between OR and SP/LP. Additionally, the heterogeneity results indicate that the promotion effect of OR on SP is more significant in larger enterprises and low-competition industries. In contrast, the impact of OR on LP is more significant in smaller enterprises and high-competition industries. This study comprehensively describes the effects of OR on FP and explains why there are contradictory findings on the relationship between the two. It also provides practical guidance for manufacturing companies to promote SP and LP through OR in the age of AI.

This work develops asymptotic expansions for solutions of systems of backward equations of time-inhomogeneous Markov chains in continuous time. Owing to the rapid progress in technology and the increasing complexity in modeling, the underlying Markov chains often have large state spaces, which make the computational tasks infeasible. To reduce the complexity, two-time-scale formulations are used. By introducing a small parameter ɛ > 0 and using suitable decomposition and aggregation procedures, it is formulated as a singular perturbation problem. Both Markov chains having recurrent states only and Markov chains including also transient states are treated. Under certain weak irreducibility and smoothness conditions of the generators, the desired asymptotic expansions are constructed. Then error bounds are obtained.

Ovarian cancer is among the leading causes of gynecologic cancer-related death. Past ovarian cancer screening trials using combination of cancer antigen 125 testing and transvaginal ultrasound failed to yield statistically significant mortality reduction. Estimates of ovarian cancer sojourn time-that is, the period from when the cancer is first screen detectable until clinical detection-may inform future screening programs. We modeled ovarian cancer progression as a continuous time Markov chain and estimated screening modality-specific sojourn time and sensitivity using a Bayesian approach. Model inputs were derived from the screening arms (multimodal and ultrasound) of the UK Collaborative Trial of Ovarian Cancer Screening and the Prostate, Lung, Colorectal and Ovarian cancer screening trials. We assessed the quality of our estimates by using the posterior predictive P value. We derived histology-specific sojourn times by adjusting the overall sojourn time based on the corresponding histology-specific survival from the Surveillance, Epidemiology, and End Results Program. The overall ovarian cancer sojourn time was 2.1 years (posterior predictive P value = .469) in the Prostate, Lung, Colorectal and Ovarian studies, with 65.7% screening sensitivity. The sojourn time was 2.0 years (posterior predictive P value = .532) in the United Kingdom Collaborative Trial of Ovarian Cancer Screening's multimodal screening arm and 2.4 years (posterior predictive P value = .640) in the ultrasound screening arm, with sensitivities of 93.2% and 64.5%, respectively. Stage-specific screening sensitivities in the Prostate, Lung, Colorectal and Ovarian studies were 39.1% and 82.9% for early-stage and advanced-stage disease, respectively. The histology-specific sojourn times ranged from 0.8 to 1.8 years for type II ovarian cancer and 2.9 to 6.6 years for type I ovarian cancer. Annual screening is not effective for all ovarian cancer subtypes. Screening sensitivity for early-stage ovarian cancers is not sufficient for substantial mortality reduction.

Jump statistics, sojourn times, fluctuation dynamics and ergodic behavior for Markov processes in continuous time with a finite number of states

This chapter concentrates primarily on the general aspects of occupancy times when studying the reliability of systems that are described by finite-state continuous-time Markov or semi-Markov processes. Occupancy times for Markov models for system reliability are discussed together with a brief comment on reward processes. Similar problems, focused on joint distributions and cumulative distributions, are considered for semi-Markov models. Similar distribution problems for time interval omission situations are given briefly in the chapter. Some numerical examples are presented to illustrate the results obtained. The chapter also presents some general results for certain probability distributions of occupancy times in terms of Laplace-Stieltjes transforms. The advantages are that the results are elegant and moments can usually be easily obtained by differentiation. The chapter points out the similarity of some Markov and semi-Markov models for reliability and ion-channel theory. Controlled Vocabulary Terms laplace transforms

Pricing American Drawdown Options under Markov Models