## 2 Theoretical Background and Problem Statement
The most interesting case for this paper discussion is the practical worst case (PWC), which assumes that the line is balanced over the long term (i.e., the average processing times for all stations are the same), each station is formed by a single machine, and, most importantly, processing time variability is exponential. This choice is motivated by the fact that closed form analytical equations can be easily obtained only at the steady state when stochastic processes are \"memoryless\". Thus, while the PWC model is mathematically exact and solvable in closed form, it is only valid if the assumptions of exponential processing times and balanced line hold. Unfortunately, given the multitude of products and manufacturing operating condition, these assumptions can only serve as a reference, as the authors intended, that is, to provide a baseline against which the performance of an actual production system can be compared to. Indeed, the same authors developed an iterative model [25], based on an approximation \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Cycle Time (CT)** & **Throughput (TH)** \\ \hline Best Case & \(CT_{min}=\begin{cases}T_{0},&\text{if }w\leq W_{0}\\ \frac{w}{r_{b}},&\text{otherwise}\end{cases}\) & \(TH_{max}=\begin{cases}\frac{w}{T_{0}},&\text{if }w\leq W_{0}\\ r_{b},&\text{otherwise}\end{cases}\) \\ Worst Case & \(CT_{max}=w\cdot T_{0}\) & \(TH_{min}=\frac{1}{T_{0}}\) \\ Practical Worst Case & \(CT_{PWC}=T_{0}+\frac{w-1}{r_{b}}\) & \(TH_{PWC}=\frac{w}{W_{0}+w-1}\cdot r_{b}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The performances models for single-routing and single-product CONWIP line [33]. Figure 2: The CONWIP flow-shop production system under consideration. of the mean value analysis, to obtain TH and CT estimations as a function of WIP in the more general case of unbalanced line and general variability of processing times. However, as already shown in [35], their estimation suffers from an approximation error that increases in magnitude as one moves away from the case of exponential variability and the balanced line hypotheses, because the model assumptions are no longer adequately verified. The model proposed in Hopp and Spearman [33] can be summarized in the steps (1)-(4), with the following notation:  
\(CT_{j}(w)=\) cycle time at station \(j\) with WIP level \(w\);  
\(CT(w)=\sum_{j=1}^{n}CT_{j}(w)=\) cycle time with WIP level \(w\);  
\(TH(w)=\) throughput with WIP level \(w\);  
\(WIP_{j}(w)=\) average WIP level at station \(j\) with WIP level \(w\). Letting \(WIP_{j}(0)=0\) and \(TH(0)=0\), cycle time, throughput, and station-by-station WIP levels can be calculated iteratively as a function of the number of jobs:  
\[CT_{j}(w)=\frac{t_{\varepsilon}^{2}(j)}{2}[c_{\varepsilon}^{2}(j)-1]TH(w-1)+[ WIP_{j}(w-1)+1]t_{\varepsilon}(j) \tag{1}\]  
\[CT(w)=\sum_{j=1}^{n}CT_{j}(w) \tag{2}\]  
\[TH(w)=\frac{w}{CT(w)} \tag{3}\]  
\[WIP_{j}(w)=TH(w)\cdot CT_{j}(w) \tag{4}\]  
In light of the research questions posed, the scientific methodology for this study is structured as follows:  
1. Develop a simulation tool capable of generating data for the analyzed CONWIP flow-shop production system, ensuring the ability to capture the effect of generalized variability in processing times on the performances indicators (throughput and cycle time). 2. Conduct a preliminary study to develop and validate a normalization approach for the performance of flow-shop balanced lines, thereby enabling one-time training on a larger dataset and facilitating the use of machine learning algorithms. 3. Propose an artificial-intelligence-based model able to estimate line throughput at the steady-state, taking into account characteristic parameters and the WIP under processing. 4. Demonstrate the development process of a deep learning algorithm tailored for the balanced CONWIP production line scenario, showcasing its applicability and accuracy. 5.