## 5 Research Methodology and Empirical Findings
### Reliability Analysis  
This study will start from the perspective of system reliability before proceeding to the quality of the service system. Reliability analysis can be used to measure the characteristics of items in a stated time interval and conditions [34]. Finding corresponding relationships between variables and defining the probability function assist in system reliability analysis. The system maintenance period is defined as the interval between malfunctions and the unavailability duration [35]. The longer a system operates without failure and with higher availability, the higher its reliability can be interpreted. In reliability engineering, the meantime to failure (MTTF), which is commonly defined for calculating system availability, is the factor for measuring the reliability function [35]. The function of the system reliability is also derived from the probability of nonfailure within the time interval (0, t] [34]. \[\mathrm{R(t)}=Prob(non-Failure\;in\;the\;interval) \tag{1}\]  
For analyzing the reliability, scholars commonly assumed \(\mathrm{R(0)}=1\) defining the system mean time-to-failure (MTTF), as follows [35]:  
\[\mathrm{MTTF_{sys}}=\int_{0}^{\infty}\mathrm{R_{sys}(t)dt} \tag{2}\]  
Except for defining the complex system, the MTT function can also measure the reliability of a single item configuration. For the boundaries, the system lifecycle is tacitly assumed to be infinite for the derivation. After establishing the system lifecycle boundary, we can approximate the probability polynomial to obtain the function of failure ratio \(\gamma(t)\). \[\gamma(t)=-\frac{\mathrm{dR}(t)/\mathrm{dt}}{\mathrm{R}(t)} \tag{3}\]  
Setting \(\mathrm{R}(0)=1\), the reliability function is given by  
\[\mathrm{R}(t)=\mathrm{e}^{-\int_{0}^{t}\gamma(x)\mathrm{dx}}. \tag{4}\]  
In this study, Firm A has several systems combinations for complex manufacturing requirements. Considering that the original system design's characteristics will directly affect its reliability, the proportions \(\mathrm{p}\) and \(\mathrm{q}\) in the equation are used to identify the functional and architectural differences, respectively. To improve the accuracy assessment in reliability analysis, the definition of proportion can be designed regarding third parties' consideration. The adjustment coefficient factors can effectively assist in the differentiation of similar systems. The reliability of the selected system can be formulated as follows:  
\[\mathrm{R}_{\mathrm{sys}}(t)=\mathrm{p}(1-\mathrm{q})\mathrm{R}_{1}(t)+(1- \mathrm{p})\mathrm{q}\mathrm{R}_{2}(t)=\mathrm{p}(1-\mathrm{q})\mathrm{e}^{- \gamma_{1}t}+(1-\mathrm{p})\mathrm{q}\mathrm{e}^{-\gamma_{2}t}, \tag{5}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) represent the failure rate of the system. By integrating systems with similar design structures, companies can simplify simulation models. After eliminating duplicate variables, the model can deal with complex system service structures. Deriving from Equation (3), the failure rate with the time-dependent feature is considered. \[\gamma(t)=\frac{\gamma_{1}\mathrm{p}(1-\mathrm{q})\mathrm{e}^{-\gamma_{1}t}+(1 -\mathrm{p})\mathrm{q}\gamma_{2}\mathrm{e}^{-\gamma_{2}t}}{\mathrm{p}(1- \mathrm{q})\mathrm{e}^{-\gamma_{1}t}+(1-\mathrm{p})\mathrm{q}\mathrm{e}^{- \gamma_{2}t}}, \tag{6}\]  
In the process of observing the failure rate, the value will monotonically decrease from \(t=0\) to \(t=\infty\). We can calculate the defective rate of each time interval by decreasing the value from the average rate to the minimum value of 0. In the failure of different events, the root cause must be analyzed by professionals in related fields. The difference in the degree of decline between the values must still be explored using raw data. While data censorship is usually unavoidable in the extraction process, companies can use different methodologies to overcome the problem [41].