# Performability Evaluation of Autonomous Underwater Vehicles Using Phased Fault Tree Analysis
## 3 Performability Evaluation Method Based on Phased Fault Trees
### Performability Evaluation  
The mission performability of an AUV is evaluated by utilizing the reliability and performance of each subsystem based on the phased FT and achievement levels. First, the likelihood for each sub-FT accomplishment level and the performability are determined based on the FT's reliability, as shown in Equation (10).  
\[\begin{split}\text{Pr}(A_{3})&=P(\{\omega|Y_{S}( \omega)\geq A_{3}\})=1-P(T_{sub})\\ \text{Pr}(A_{2})&=P(\{\omega|A_{1}>Y_{S}(\omega) \geq A_{2}\})\\ &=\left(\sum_{p_{i}\in E_{R}}\left((1-p_{i})\cdot\prod_{p_{j}\in E _{R}j\
eq i}p_{j}\right)\right)\cdot\prod_{p_{k}\in E_{C}}(1-p_{k})\\ \text{Pr}(A_{1})&=P(\{\omega|Y_{S}(\omega)\geq A_{1 }\})=1-(\text{Pr}(A_{2})+P(T_{sub}))\end{split} \tag{10}\]  
where \(T_{sub}\) is the likelihood that the sub-FT's top event will occur, \(E_{R}\) is the set of basic event failure probabilities in which the sub-FT contains redundant components, \(E_{C}\) is the set of basic event failure probabilities that the sub-FT undergoes critical failures, and \(n\) is the number of events in the sub-FT.  
By merging the performability of each sub-FT, the overall system performance was determined using the UGF approach. Reliability and performance assessments of multi-state systems (MSSs) have been successfully handled by the UGF technique [68; 69; 70]. In order to assess the probability distribution of the overall performance for systems with varying attributes and performance, the U-function expands upon the ordinary moment generation function. Various composition operators can be introduced and applied to the UGF. When every component of the system is statistically independent of every other component,the definition of the U-function of an independent discrete random variable \(X\) is given by Equation (11).  
\[u(z)=\sum_{k=1}^{K}q_{k}z^{x_{k}} \tag{11}\]  
In this case, the performability of subsystem \(j\) can be defined by the polynomial \(u_{j}(z)\), which means that it can represent all of the potential subsystem states. It is possible to define and introduce a composition operator to combine the performability of two subsystems. In this study, we ascertain the two subsystem's performabilities using algebraic operations. The composition operator has the following form [71] (Equation (12)):  
\[\begin{split} u_{i}(z)\otimes u_{j}(z)&=\sum_{k=1} ^{K_{i}}p_{ik}z^{A_{ik}}\otimes\sum_{h=1}^{K_{j}}p_{jh}z^{A_{jk}}\\ &=\sum_{k=1}^{K_{i}}\sum_{h=1}^{K_{j}}p_{ik}p_{jh}z^{\varphi(A_{ik },A_{jk})}\end{split} \tag{12}\]  
where the terms \(A_{ik}\) and \(p_{ik}\) denote the achievement level and performability (\(Pr(A_{i}k)\)) of sub-FT \(k\), respectively. Depending on the metrics (reliability, performance, and performability) that need to be considered, there are various ways to define \(\varphi(A_{ik},A_{jk})\). The accomplishment level is discussed in this work and is defined as shown in Equation (13).  
\[\varphi(A_{i},A_{j})=\begin{cases}A_{i},&\text{if }A_{i}<A_{j}\\ A_{j},&\text{if }A_{i}\geq A_{j}\\ A_{3},&\text  
A phased mission system, where the mission executed and the system configuration may change over time, can benefit from the application of a phased FTA as a reliability analysis technique [73]. A system that completes several consecutive, non-overlapping missions in succession is known as a phased mission system. In each phase, the system needs to carry out various subtasks in order to complete its purpose. For every mission, a new system configuration might be used. The dependability of the current system can be computed explicitly by creating an FT that is appropriate for the mission and system configuration at each mission.