## 3 Methodology
### Theory for Computing Unavailability of the Investigated Infrastructure  
This section clarifies the method for calculating the unavailability of the interconnected infrastructure system detailed in previous sections. In Section 3.2, the system was depicted using AGs. Here, internal nodes, represented as triangles in Figure 7 (e.g., u1 or u2), symbolize sub-systems within the network. At any given moment, subsystems and components (which are the internal and terminal nodes, respectively) can either be operational _or_ in a failed state, which may be under restoration. A node is deemed operational if the number of its child nodes matches or surpasses the count encapsulated within the triangle; if not, it is categorized as non-functional. For illustration, Node u1 in Figure 7 is operational if either one or two of its directly subordinate nodes are functional. Understanding the unavailability functions of terminal nodes facilitates the calculation of the unavailability functions for internal nodes. Both of these functions are crucial inputs when determining the unavailability function of the entire network, denoted by the SS node \(U(t)\). This function, \(U(t)\), offers insight into the time-dependent likelihood that the network might experience failure at time \(t\) due to a malfunction or an ongoing repair process. Model for Unavailability Exploration of a Terminal Node with Corrective Maintenance (CM)  
To ascertain the unavailability function of the SS node, \(U(t)\), a model and algorithm for quantifying the unavailability of terminal nodes undergoing CM is essential. In the context of CM, it is crucial to acknowledge two complementary random variables: the lifetime \(X\), characterized by either distribution function \(F(t)\) or Probability Density Function (PDF) \(f(t)\), and the time needed for repair or recovery \(Y\), described by either distribution function \(G(t)\) or PDF \(g(t)\). Based on renewal theory and alternating renewal processes, unavailability function \(U(t)\) can be expressed as follows [1]:  
\[U(t)=1-A(t)=F(t)-\int_{0}^{t}h(x)[1-F(t-x)]dx, \tag{6}\] \begin{table}
\begin{tabular}{l r r r} \hline \hline
**Component** & **MTBF (h)** & **MTTRes (h)** & \(\beta\)**(-)** \\ \hline Edge Router ER1 & 16,246.780 & 0.780 & 2 \\ Edge Router ER2 & 16,246.780 & 0.780 & 2 \\ Core Optical Switch COS1 & 5,000,014.000 & 14.000 & 2 \\ Core Optical Switch COS2 & 5,000,014.000 & 14.000 & 2 \\ Aerial Optic fiber AO4 & 500,000.000 & 6.000 & 2 \\ Aerial Optic fiber AO5 & 1,093,750.000 & 6.000 & 2 \\ Aerial Optic fiber AO6 & 500,000.000 & 6.000 & 2 \\ Aerial Optic fiber AO7 & 1,093,750.000 & 6.000 & 2 \\ Buried Optic fiber BO1 & 821,875.000 & 12.000 & 2 \\ Buried Optic fiber BO3 & 1,753,333.333 & 12.000 & 2 \\ Buried Optic fiber BO4 & 1,753,333.333 & 12.000 & 2 \\ Remote Terminal Unit RTU & 100,048.000 & 48.000 & 2 \\ SCADA operation and data center & 175,200.000 & 184.600 & 2 \\ Base Transceiver Station BTS & 100,000.000 & 4.000 & 2 \\ Wide Area Network WAN & 100,000.000 & 4.000 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristic values of the communication and control components. where \(U(t)\) denotes the instantaneous time-dependent unavailability function, \(A(t)\) denotes instantaneous availability function, while \(h(x)\) represents the renewal density of the pertinent alternating renewal process. Calculating unavailability using Equation (6) demands familiarity with \(h(x)\). This can be intricate since it is numerically represented as an infinite sum of probability densities, each calculated as a convolution. Nonetheless, this equation can be substituted with its counterpart, termed the recurrent linear integral equation, as elucidated in [1]:  
\[U(t)=\int_{0}^{t}f(x)[1-G(t-x)]dx+\int_{0}^{t}(f*g)(x)(t-x)dx. \tag{7}\]  
In this context, * signifies convolution. The efficacy of this methodology for determining unavailability in multifaceted and highly reliable systems is validated in [1].