## 2 Problem Statement
### Availability Metrics of SCOS  
In the era of Industry 4.0 and intelligent manufacturing, integrated systems of industrial software such as ERP, MES, and WMS are required to operate collaboratively on a 24/7 basis without interruption. The continuity and efficiency of these systems are directly linked to the economic performance of enterprises. Table 2 presents the theoretical maximum allowable downtime over a 90-day and one-year period under different levels of availability. As a key functional metric for evaluating the sustainable operational capability of services, availability is defined as the ratio of the time a system remains in an operational state to the total operational period [21]. The modeling process for composite availability in multi-source heterogeneous industrial software integration is outlined as follows. 1. Definition of the service component set: In the context of heterogeneous industrial software integration, assume there are \(n\) industrial software components originating from different vendors and built on different technology stacks. The expression is given in Equation (3). \begin{table}
\begin{tabular}{c c c c} \hline
**QoS** & **Function** & **Meaning** & **Attribute** \\
**Indicator** & & & \\ \hline \(T\) & \(Q_{T}=\sum_{l=1}^{n}q_{T}(ST^{j})/n\) & \begin{tabular}{c} Includes deployment and debugging time for \\ industrial software \\ \end{tabular} & Negative \\ \(C\) & \(Q_{C}=\sum_{l=1}^{n}q_{C}(ST^{j})/n\) & \
\begin{tabular}{c} Costs of procurement, maintenance, hardware, \\ implementation, and licensing of industrial software \\ \end{tabular} & Negative \\ \(Re\) & \(Q_{Re}=\sqrt{\prod\limits_{j=1}^{n}q_{Re}(ST^{j})}\) &
\begin{tabular}{c} Reputation of candidate industrial software resources \\ \end{tabular} & Positive \\ \(Q\) & \(Q_{Q}=\sum_{l=1}^{n}q_{Q}(ST^{j})/n\) & Product quality & Positive \\ \hline \end{tabular}
\end{table} 
Table 1: The sequential type’s aggregate QoS attributes. \begin{table}
\begin{tabular}{c c c} \hline
**Availability** & **Downtime/90 Days** & **Downtime/1 Year** \\ \hline
99\% & 21 h and 36 min & 3 days 15.6 h \\
99.9\% & 2 h and 10 min & 8 h and 45 min \\
99.99\% & 12 min and 58 s & 52 min and 34 s \\
99.999\% & 1 min and 18 s & 5 min and 15 s \\
99.9999\% & 8 s & 32 s \\ \hline \end{tabular}
\end{table} 
Table 2: The maximum allowable downtime of the system theory. \[S=\{s_{1},s_{2},\ldots,s_{n}\} \tag{3}\]
2. Computation of individual component availability: for each component \(s_{i}\), its availability \(A_{i}\) is calculated using Equation (4). \[A_{i}=\frac{MTBF_{i}}{MTBF_{i}+MTTR_{i}}\] (4) where \(MTBF\) represents the average time the system operates without failure, reflecting its inherent reliability, and \(MTTR\) denotes the average time required to diagnose, repair, and restore the system to operational status. For example, if the MTBF of an industrial software component is 200 h and the \(MTTR\) is 20 h, its availability \(A_{i}\) can be calculated as follows according to Equation (4): \[A_{i}=\frac{MTBF_{i}}{MTBF_{i}+MTTR_{i}}=\frac{200}{200+20}=0.91\] This value indicates that the component remains in a functional service state approximately 91% of the time during long-term operation. 3. Construction of the dependency matrix: Based on the modeling of functional and logical dependencies, a dependency matrix \(D=[d_{ij}]\) is constructed to represent the strength of the dependency of component \(s_{i}\) on component \(s_{j}\). The criteria for determining functional relationships are presented in Table 3.
4. Incorporation of integration uncertainty factors: Due to the unpredictability associated with the initial collaboration between services, additional integration risks often arise. Therefore, it is necessary to introduce an \"integration uncertainty factor\" in the modeling of composite availability to penalize the potential failure probability caused by first-time integration. A confidence factor \(\theta_{i}\) is introduced to represent the reliability level of component \(s_{i}\) when integrated with other components.