## 4 Case Study and Hybrid Modelling  
The system under analysis is characterized by a production line where items are enqueued and processed one by one in a service station. It is assumed that both the interarrival time and the service time of the system items are random. Moreover, the service station can be failure prone. It is assumed that smart sensors can monitor the working temperature of operation which, in this model, will affect the dynamic failure rate of the service station. Figure 4 shows a schema of the system, where it is possible to identify within the bold contour the main components of the model, namely the enqueueing conveyor and the service station. Each item (or job) is characterized by its own time for being processed (time of service), whereas they are enqueued one by one by the picker robot (interarrival time).  
The main assumptions for this generic production line are thereby listed:  
* Interarrival Time follows a rectified Normal Distribution (see Table 1 parameters);
* Service Time follows a rectified Normal Distribution (see Table 1 parameters);
* Mean Time to Failure follows a Weibull Distribution (see Table 2 parameters);
* Mean Time to Repair follows a Uniform Distribution (see Table 2 parameters);
* The machine tool is subjected to aging that depends on the operating temperature, following the Equations (14) and (15).  
Table 1 shows the main parameters of the queueing model. It was assumed that the interarrival time and the service time follow two rectified Normal distributions to avoid negative time values and overcome Hypotheses 1 and 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Symbol** & **Value** & **Measuring Unit** \\ \hline Expected service time & ts\({}_{\text{mean}}\) & 5 & h \\ Standard deviation service time & \(\sigma_{\text{ts}}\) & 1 & h \\ Expected interarrival time & ti\({}_{\text{mean}}\) & 10 & h \\ Standard deviation between jobs & \(\sigma_{\text{ti}}\) & 2 & h \\ \hline \hline \end{tabular}
\end{table}
Table 1: Normal distribution parameters characterizing the automated factory.  
Figure 4: Generic production line of an automated factory.  
The failure of the service station is modelled using a Weibull distribution characterized by a shape factor, \(\beta=3\), and an expected machine tool life \(\alpha\)(T) of 100 hours which represents the mean time to failure of the service machine if it operates in the same fixed operational conditions in which it was tested by the Original Equipment Manufacturer (OEM). Therefore, the failure rate can be written as follows:  
\[h(t,\text{T})=\frac{\beta}{\alpha\text{(T)}}\bigg{(}\frac{\text{t}}{\alpha \text{(T)}}\bigg{)}^{\beta-1} \tag{13}\]  
The time to repair for the machine tool follows a uniform distribution time between a maximum repair and a minimum repair time (\(\text{TTR}_{\text{min}}\), \(\text{TTR}_{\text{max}}\)). Table 2 resumes the main data of the failure and repair model of the machine tool.  
In order to model a daily variation of the temperature during the season of the year, the equation used to model the working temperature is the following:  
\[\text{T}(t)=\text{Tm}+\text{A}_{\text{T}}\sin\bigg{(}\frac{2\pi}{\text{T}}t \bigg{)} \tag{14}\]  
where Tm = 20 \({}^{\circ}\)C is the mean operation temperature, T = 24 h is the oscillation period of the temperature, \(\text{A}_{\text{T}}=5\)\({}^{\circ}\)C is the amplitude of the oscillation, and t is the independent time variable.  
As it was said, the machine tool is subjected to aging that depends also on the operating temperature. Under this assumption, the expected machine tool life takes the following form:  
\[\text{a}(\text{T})=\frac{\text{a}_{0}}{\text{T}_{0}} \tag{15}\]  
where \(\text{a}_{0}\) is the expected lifetime at the nominal working temperature \(\text{T}_{0}\).