Reliability analysis of phased-mission systems using Bayesian networks

This paper presents a novel and simple approach to reliability analysis of phased-mission systems. The technique presented in this paper transforms a phased-mission system into several non-phased-mission systems, and thus allows each phase to be modeled separately. The mission succeeds as long as every phase successively executes its requirements, and failed component(s) of any phase is(are) repaired so that the system is in a successful state before as well as after it begins the next phase. Alternative mission success scenarios are also readily definable within this context. The benefit of this simplified approach is that it provides a practical implementation for the reliability analysis of phased-mission systems. The different phases of the system are computed independently, and the duration and the structure of the phases are left to the designer or manager. If the requirements of the mission change during execution of the mission (duration of a phase or the sequence of phases), the impact of the change in mission objectives is easily found without recomputing the applicable or operative Markov model. This is due to the fact that all of the reliability analysis data are presented over the entire mission duration consisting for all phases, and the initial and final probabilities can easily be found by examining the output reliability data instead of recomputing the model

In a phased-mission system (PMS), an uncovered component fault may lead to a mission failure regardless of the status of other components, and the reliability can be analyzed with traditional imperfect fault coverage model (IFCM). The IFCM, however, only considers the coverage of faulty components. Recently, an irrelevancy coverage model (ICM) is proposed to cover both faulty components and irrelevant components, but the analysis is limited to normal non-phased mission systems. This paper first demonstrates that, the coverage of irrelevant components is also important in PMSs, as an initially relevant component could also become irrelevant later due to the failures of other components, and an uncovered fault of irrelevant component may threaten the whole mission as well. A method to analyze the reliability of PMS in ICM is proposed using sum of disjoint products (SDP) technique. Experimental results demonstrate not only the effectiveness of the proposed reliability analysis method, but also that the ICM can achieve higher reliability than the IFCM for PMSs in general.

ABSTRACTApplication of the Markov approach in the reliability analysis of phased-mission systems (PMS) may encounter the problem of state space explosion. In this paper, a relationship is derived to compute the non-zero elements of the transition rate matrix (TRM) of a Markov model on the basis of the states binary description. Moreover, a TRM compressed storage scheme (TRMCS) is proposed, and an algorithm for matrix vector multiplication under TRMCS is presented. A method to solve the state probability vector using the Krylov subspace is proposed. The performance and the effectiveness of the proposed methods are demonstrated through a typical example.

Reliability analysis of phased-mission systems (PMS) must consider the statistical dependences of element states across different phases as well as changes in system configuration, success criteria, and component behavior. This paper proposes a recursive method for the exact reliability evaluation of PMS consisting of nonidentical independent nonrepairable multistate elements. The method is based on conditional probabilities and the branch-and-bound principle. It is invariant to changes in system structure, demand, and the elements' state transition rates among phases. The main advantage of this method is that it does not require the composition of decision diagrams and can be fully automated. Both analytical and numerical examples are presented to illustrate the application and advantages of the proposed method. The computational performance of the proposed algorithm is illustrated through comprehensive experimentation on the CPU running time of the algorithm.

The petri net (PN) simulation method is a common-used methodology for mission reliability analysis of phased mission system (PMS). However, traditional PN sometime become unviable for PMS modeling. To solve such problems, an extension of PN named Extended Object-Oriented Petri Net (EOOPN) has been introduced, which proves to be efficient for PMS modeling. In this paper, a software named EOOPN Tool is exploited to trace the functions of PMS mission reliability modeling and simulation. The software bases on EOOPN model. It is developed in WINDOWS-7 Operating System using C# language. A user-friendly interface is provided for EOOPN modeling and result representing. After modeling, a validity checking is permitted then a simulation could be performed. To be specific, several logical transitions are added to fulfill the need of PMS mission logic modeling. Endue transitions with color to express a firing rate or a distribution of the time delay. Furthermore, an EOOPN sub-model base is provided for OOPN modeling and a XML document that complying with the PNML standard is used for input/output file.

This paper discusses some of the difficulties in the paper: A BDD-based algorithm for reliability analysis of phased-mission systems by X. Zang, H. Sun, and K. S. Trivedi.

Many practical systems are phased-mission systems (PMSs), where the mission consists of multiple, consecutive, and non-overlapping phases of operation. An accurate reliability analysis of a PMS must consider statistical dependence of component states across phases, as well as dynamics in system configurations, success criteria, and component behavior. This paper proposes a new method based on multiple-valued decision diagrams (MDDs) for the reliability analysis of a non-repairable binary-state PMS. Due to its multi-valued logic nature, the MDD model has recently been applied to the reliability analysis of multistate systems. In this work, we present a novel way to adapt MDDs for the reliability analysis of systems with multiple phases. Examples show how the MDD models are generated and evaluated to obtain the mission reliability measures. Performance of the MDD-based method is compared with an existing binary decision diagram (BDD)-based method for PMS analysis. Empirical results show that the MDD-based method can offer lower computational complexity as well as a simpler model construction and improved evaluation algorithms over those used in the BDD-based method.

With the extensive application of phased mission system (PMS), the study of phased mission characteristics becomes more and more necessary and urgent. Considering the multi-mode failure of equipment, this paper proposes a reliability modeling and prediction method based on the micro-replacement fault tree, which is used to solve the reliability analysis problem of phased mission system. We take a typical phased mission system as an example to verify the feasibility of the method. This method provides a simpler and more efficient way to evaluate the reliability of complex systems.

This article has been retracted by the publisher.

Many systems are multi-phase mission systems. During different phases, the system reliability configuration, reliability requirements and operation environment can be different. For example, some components may change from active status to standby or the system reliability configuration may change from serial to parallel. In addition to configuration changes, the system may be exposed to different stress conditions; therefore, the time to failure distribution of some components will change due to the change in stress. In general, systems that operate under different phases are considered as dynamic systems. A way to estimate the reliability and availability of such systems is needed. In this paper, a systematic procedure of applying accelerated life tests and simulation to analyze the reliability and availability of such dynamic systems is first proposed. Methods for solving both non-repairable and repairable systems are provided. For non-repairable systems, an analytical solution based on cumulative damage theory is discussed. Therefore, the exponential assumption, which is used in many existing methods, is not required in the proposed method. In addition to the analytical solution, a cumulative- damage-based simulation solution is also provided. For repairable systems, based on different scenarios in real applications, the repairable phased-mission system is classified into three categories. Because of the complexity of the problem, only simulation results are given for repairable systems. The proposed systematic procedure of applying accelerated life tests in phased-mission system analysis provides a general guideline for dealing with real-world applications. The cumulative-damage-based analytical and simulation method provide a practical and useful approach for solving phased-mission system problems. The classification of repairable phased-mission systems gives a clear overview of the phased-mission systems and makes the analysis of complex real applications easier.

This paper presents a fuzzy graph evaluation and review technique (GERT) analytic model for multi-phased mission systems that provides a new approach to calculate mission performance of phased-mission system (PMS). The model extends traditional analytic models of PMS using logic diagram method (such as Fault Trees and Binary Decision Diagram) and state transition method (such as Markova Model). By using fuzzy GERT based on grey number, we provide a stochastic network to build the model of phased-mission which avoids the difficulty to give exact logic relation in reliability analysis method based on logic diagram and the state space explosion and the mapping problems that are encountered by the Markov chain approach, and at the same time, we use gray number to describe the phased-mission transition probability which is more easier to determine than traditional models. In this model, as the key performance indicators of PMS, mission cycle and success probability are solved. Finally, a computation study of typical carrier operational mission is given to illustrate how to apply the approach to calculate mission reliability of PMS. Furthermore, sensitivities of mission cycle and success probability in the model are analyzed.

Bayesian networks (BNs) are being studied in recent years for system diagnosis, reliability analysis, and design of complex engineered systems. In several practical applications, BNs need to be learned from available data before being used for design or other purposes. Current BN learning algorithms are mainly developed for networks with only discrete variables. Engineering design problems often consist of both discrete and continuous variables. This paper develops a framework to handle continuous variables in BN learning by integrating learning algorithms of discrete BNs with Gaussian mixture models (GMMs). We first make the topology learning more robust by optimizing the number of Gaussian components in the univariate GMMs currently available in the literature. Based on the BN topology learning, a new multivariate Gaussian mixture (MGM) strategy is developed to improve the accuracy of conditional probability learning in the BN. A method is proposed to address this difficulty of MGM modeling with data of mixed discrete and continuous variables by mapping the data for discrete variables into data for a standard normal variable. The proposed framework is capable of learning BNs without discretizing the continuous variables or making assumptions about their conditional probability densities (CPDs). The applications of the learned BN to uncertainty quantification and model calibration are also investigated. The results of a mathematical example and an engineering application example demonstrate the effectiveness of the proposed framework.

There are many methods applied including Bayesian network and D-S evidence theory to cope with uncertainty involving aleatory uncertainty and epistemic uncertainty in reliability analysis of complex systems. This paper introduces theories of these two methods briefly, and then conversion rules that convert fault tree into Bayesian network under uncertainty are put forward, including AND node, OR node, XOR node, NOT node and Two-out-of-three vote node. Comparing to probability importance, structural importance and criticality importance, epistemic importance is given to measure the influence of root event to top event. At last, a type of engine is taken for example. Bayesian network model is established by referring to the fault tree of the engine, and D-S evidence theory is used to determine the belief functions and plausibility functions of uncertain nodes by data fusion. Weak nodes in reliability design and distribution are pointed out after reliability assessment, importance analysis, and backward reasoning. And corresponding measures can be taken to improve the reliability of the whole system.

Multi-state components, common cause failures (CCFs) and data uncertainty are the general problems for reliability analysis of complex engineering systems. In this paper, a method incorporating fuzzy probability and Bayesian network (BN) into multi-state systems (MSSs) with CCFs is proposed. In particular, basic theories of multi-state BN and fuzzy probability are developed. Moreover, a model integrating CCFs with BN has also been illustrated. In order to incorporate fuzzy probability into MSSs reliability evaluation considering common parent node generated by CCFs, fuzzy probability has to be translated into accurate probability through defuzzification and normalization methods which are both elaborated. In addition, quantitative analysis based on BN is carried out. In this paper, feed system of boring spindle in computer numerical control machine is analyzed as an example to validate the feasibility of the proposed method. It can improve the ability of BN on reliability evaluation of complex system with uncertainty issues.

A combined physics of failure and Bayesian network reliability analysis method for complex electronic systems