مدل سازی قابلیت اطمینان بهبود یافته با استفاده از شبکه های بیزی و گسسته دینامیکی

This paper shows how recent Bayesian network (BN) algorithms can be used to model time to failure distributions and perform reliability analysis of complex systems in a simple unified way. The algorithms work for so-called hybrid BNs, which are BNs that can contain a mixture of both discrete and continuous variables. Our BN approach extends fault trees by defining the time-to-failure of the fault tree constructs as deterministic functions of the corresponding input components’ time-to-failure. This helps solve any configuration of static and dynamic gates with general time-to-failure distributions. Unlike other approaches (which tend to be restricted to using exponential failure distributions) our approach can use any parametric or empirical distribution for the time-to-failure of the system components. We demonstrate that the approach produces results equivalent to the state of the practice and art for small examples; more importantly our approach produces solutions hitherto unobtainable for more complex examples, involving non-standard assumptions.. The approach offers a powerful framework for analysts and decision makers to successfully perform robust reliability assessment. Sensitivity, uncertainty, diagnosis analysis, common cause failures and warranty analysis can also be easily performed within this framework.

Most published reliability analysis methods are based on parametric and non-parametric statistical models of time-to-failure data and their associated metrics [1]. The underlying assumption of these methods is that a coherent, statistical model of system failure time can be developed that will prove stable enough to accurately predict a system’s behaviour over its lifetime. However, given the increasing complexity of the component dependencies and failure behaviours of today’s real-time safety and mission critical systems, the statistical models may not be feasible to build or computationally tractable. This has led to an increasing interest in more flexible modeling frameworks for reliability analysis. The most notable such frameworks are combinatorial models such as fault trees (FTs), Markov chain based approaches such as dynamic fault trees (DFTs), which are described in Section 2.1 and Bayesian networks (BNs), which are described in 2.2 and 2.3. While the DFT approach [2] and [3] is very flexible, in practice it has severe limitations, such as the problem of state space explosion and the inability to handle non-standard statistical distributions. The Bayesian network (BN) framework has provided a compact representation of the event-dependent failure behaviours, characteristic of fault-tolerant systems, avoiding the state space explosion problem of the Markov chain based approaches, [4] and [5]. However, for real world applications, the BN models necessary for reliability assessment are invariantly ‘hybrid’ models [6], [7] and [8], meaning that they contain both discrete variables (e.g., the possible system states) and continuous variables (e.g., operating and environmental covariates influencing the reliability of a system). Due to limitations in inference algorithms, previous attempts to apply reliability assessment to such models have been unreasonably constrained, such as for example having to assume Gaussian or exponential distributions for all continuous variables that are not adequately handled . This paper presents (in Section 3) a simple event-based hybrid BN modeling method for reliability assessment that scales up to large, complex dynamic systems, and overcomes the limitations of both dynamic fault trees and previous BN approaches. The basic idea of our BN approach is to define the time-to-failure of the fault tree constructs as deterministic functions of the corresponding input components’ time-to-failure. The new approach incorporates a recent powerful approximate inference algorithm for hybrid BNs, based on a process of dynamic discretization of the domain of all continuous variables in the BN [6]. By efficiently integrating the iterative approximation scheme within existing robust propagation algorithms on BN architectures, such as junction tree (JT) [9], robust inference on complex hybrid models (without any constraints on the distributions of continuous variables) can be performed. The power and flexibility of the approach is demonstrated (in 3 and 4) by comparing the results with traditional state space approaches like DFTs, used in a number of popular reliability tools. The accuracy of the algorithm is tested on a range of classical dynamic fault trees constructs, allowing the system components to adopt any time to failure distribution occurring in practical applications. In each case the results are compared with the analytical solution of the Markov chain representation or the approximated solutions generated by numerical integration schemes, as appropriate. The results are very close to the analytical solutions and are achieved with much less effort. In several cases the approach provides predictions of situations that simply cannot be modelled by alternative approaches. This shows that the approach can not only replicate the results provided by existing algorithms but also extends the state of art by providing solutions to problems that the past generation of algorithms cannot solve. All the example models shown in this paper are built and executed using the commercial general-purpose Bayesian network software tool AgenaRisk [10], in which the dynamic discretization algorithm is now implemented.

This paper has presented a new, effective and flexible event-based BN framework for system reliability modeling. By combining the modeling capabilities of BNs with the dynamic discretization inference algorithm it offers a unified technique for reliability analysis of large, safety critical dynamic systems. The BN framework is mathematically sound and at the same time simple enough to allow interaction with domain experts and decision makers. By modeling the failure distributions of the DFT constructs using deterministic functions (weighted uniform density functions) and using the dynamic discretization algorithm, the BN framework is able to solve any configuration of static and dynamic gates with general time-to-failure distributions. The approach overcomes most of the problems inherent in state-space based reliability models, like DFTs. In particular, it avoids the state-space explosion problem of the Markov chain based approaches, and the limitation on the modeling of dynamic gates with general failure distributions. Any configuration of static and dynamic gates and general failures distributions can be modelled using our BN framework. Within the framework, approximated solutions for both Boolean and dynamic constructs are obtained simultaneously. This paper has demonstrated the approach using a range of examples and achieves results that are in the simple case almost as good as analytical results. Moreover, in many complex cases the approach is able to obtain results that cannot be computed analytically, and the final example presents a result that cannot be computed using any competing approach. Currently it can only partially solve the voting k/n gate since it leads to the generation of prohibitively large cliques in the junction trees, currently produced by the algorithms embedded in most BN software packages. However the reliability of a voting k/n gate can be easily computed from input component reliabilities even if the time-to-failure distribution cannot.