## 3 Dynamic Reliability Framework
### Validation of the Maintenance Box  
To demonstrate the functioning of the Maintenance Box, a validation use case has been solved and compared with an equivalent Stochastic Activity Network (SAN) model, solved with the Mobius(r) Software [46]. The Stochastic Activity Network is a high-level formalism that generalizes Stochastic Petri Nets (SPN). Whereas the latter are constituted by places and transitions (timed or instantaneous), SAN offers other elements and mechanisms for the construction of complex stochastic models. The basic components of Stochastic Activity Networks are places, activities (or transitions), Input Gates (IG), and Output Gates (OG). Places and activities (or transitions) can be used with the same logic of a Stochastic Petri Nets; but, in SAN the place marking can be either discrete (like in SPN) and continuous by the means of Extended Place (EP). Timed activities can support different types of distribution functions that define the firing time of an activity. Among them, we can recall that both deterministic timed and stochastic activities are allowed. The parameters of the distributions can be marking dependent. Finally, Input and Output Gates allow enrichment of the logic that controls the activation and the reactivation of a transition and can be also used to modify the marking of the network. Table 6 illustrates the main elements of the SAN formalism as implemented in the Mobius(r) tool. It is important to point out that the Mobius(r) tool allows users to code the logic of the Input and Output Gates with the programming language C++; moreover, it supports the definition of new types of entity (structs and vector of structs) with properties that can be referred inside the Input and Output Gates. Figure 4: Maintenance Box coupled with the “generic hybrid basic event” block. The use case implemented for the validation of the Maintenance Box is constituted by a simple AND Gate that takes as input a Hybrid Basic Event (HBE1) and a traditional Basic Event, BE2 has been tested. Table 7 illustrates the failure/repair rates parameters adopted for the system components, considering a mission time T\({}_{\text{m}}=8760\) h.  
The proposed benchmark of test is characterized by a hybrid Basic Event, HBE1, that is affected by aging. Two scenarios have been tested: the corrective (2C) maintenance and a preventive maintenance (2P). In fact, as a result of the maintenance activities, this component can be replaced as good as new (aging is reset to 0). Moreover, for the maintenance policy 2P, it is assumed that it occurs in the system every 2000 h.  
As said, in order to account for the aging process of HBE1, a Weibull probability density function with shape parameter \(\beta=1.2\) (\(\beta>1\) models the increasing of failure rate with time) can be used:  
\[\text{f(t)}=\text{t}/\gamma\cdot(\text{t}/\gamma)^{\beta-1}\;\text{e}^{-(\frac {1}{\gamma})^{\beta}} \tag{6}\]  
If in the previous equation \(\beta=1\), the probability density function takes the form of the exponential distribution, typical of random failures, with failure rate \(\lambda=1/\gamma\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**ID** & **Failure Distribution** & **Failure Parameters** & **Repair Distribution** & **Repair Rate** \\ \hline HBE1(A) & Weibull & \(\beta=1.2\); \(\gamma=10,000\) & Exponential & \(1/48\) (h\({}^{-1}\)) \\ BE2 (B) & Exponential & \(5\times 10^{-4}\) (h\({}^{-1}\)) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters of the system components. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Description** & \\ \hline \hline  
\hline
**TP** & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Main elements of the Stochastic Activity Network (SAN) formalism.