Summary and Conclusions

A large number of methods have been developed for analysing the reliability of non-repairable phased missions. The earliest methods relied on transforming a phased mission into an equivalent non-phased mission, resulting in computationally expensive analysis. Methods using Boolean phase algebras were then introduced and give better computational efficiency, whilst the most recent, and most computationally efficient, methods use the BDD technique. The Boolean phase algebra based methods offer clear advantages over the earlier methods; the computational efficiency is far higher and most are capable of directly evaluating exclusive phase failure probabilities directly. A wide range of different Boolean phase algebras have been developed. All except the earliest method are able to resolve combinations that include success, in addition to failure, events, allowing them to be used with methods that can find probability of exclusive phase failure in a specific phase. The only algebra able to resolve all resolvable combinations for single failure mode components, was developed by Kohda et al and also has the benefit of being succinct - consisting of just three, easily implemented rules. The early Boolean phase algebra methods are still computationally expensive, since the inclusion-exclusion expansion must be used in each of the probability calculations. The full expansion of all terms is particularly important with the method from Kohda et al when calculating the probability of exclusive phase failure or mission success, since the component success events are unlikely to have the low probabilities that would allow

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the approximate expansions based on the rare event assumption to be used. However, the direct modelling the success of earlier phases in the method by Kohda et al results in lower computational expense than the indirect approach taken in Dazhi and Xiaozhong.

The use of the BDD technique in the later methods reduces the computational expense significantly. Zang et al presented a method utilising phased mission specific BDD conversion and evaluation techniques to offer very fast analysis of phased missions for systems with single failure mode components. Tang and Dugan presented a method and algebra for dealing with multiple failure mode component phased mission components, although there are some problems with the methods presented in their research. The evaluation of the BDD in this method is not as fast as the methods for single failure mode component based systems given by Zang et al, due to the potential distance between a node and the nodes with cached probabilities that are used in its probability calculation. The fast evaluation of phased missions for systems with multiple failure mode components is therefore an area for further research.

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4 New methods for the analysis of non-repairable phased

missions

4.1 Introduction

The literature review presented a wide range of methods developed for the analysis of non-repairable phased missions. The most recent methods use Boolean phase algebra together with the Binary Decision Diagram technique to resolve dependencies in event combinations and provide computationally efficient analysis. The method developed by Tang and Dugan [36] is the first for the analysis of phased missions that includes components with multiple failure modes. Systems with multiple failure mode components are likely to become more common as systems become more sophisticated, more integrated, and are required to carry out an increasing number of tasks. This increased sophistication increases the likelihood that the mode of failure for a component will affect the system reliability in different ways, thus increasing the importance of modelling multiple failure modes. Unfortunately, the method for the analysis of this type of system presented by Tang et al contains a number of errors that can lead to incorrect results, as was shown in section 3.3.2.2 of the literature review.

The ability to analyse the reliability of a system in real time, where its reliability prediction is constantly updated to reflect its current situation, has recently received attention [37] and is likely to be of increasing significance as systems for which automation in decision making is important, such as UAVs [38], become more common. Recently, importance measures for phased missions have been developed by Andrews [2], as well as those developed in chapter 8 of this thesis. Both real time and importance measure analysis often require the re- evaluation of the probability of the same reliability structure but with different component reliabilities and phase durations, and, particularly for real time analysis, require that it is performed in very small time scales. Whilst the method for phased mission systems with single failure mode components by Zang, Sun and Trivedi [31] has similar efficiency to the original BDD methods from Rauzy, the method from Tang et al does not perform so well. This is due to the potential for a substantial increase in the number of node variable comparisons and traversal distance during the probability evaluation procedure as was shown in section 3.3.2.2.

This chapter presents a new method that has been developed to address the problems outlined above. It accurately analyses the reliability of phased mission systems with multiple failure mode components and offers increased efficiency compared to the method from Tang and Duggan – particularly when probability re- evaluations of the same reliability structure are required. A software tool that implements this method has also been developed and is described in this chapter.