## 5 Discussion on Unbalanced Lines Performance
This explains why the PWC model has such a simple and effective form since the sum of the processing times collapses in calculations dependent always on \(T_{0}\) and \(r_{b}\). In the case of an unbalanced line, these terms vary as the imbalance of the machines becomes more pronounced. It is of interest to investigate whether these summations are related to each other, i.e., whether fixing a sufficient number of them, the others cannot vary anymore. Without loss of generality, let us consider the case of a production line with five machines and analyze what happens if a defined amount of average processing time (\(\Delta t\)) is taken from one machine to another, without altering the \(T_{0}\) and the \(r_{b}\) of the considered line, as shown in Figure 10. This scenario is equivalent to the two machines swapping places, and the summations of processing time indeed remain constant in both cases. This observation confirms what the commutative property of addition would suggest that when the order of processing time sum is changed (whether they are squares, cubes, or otherwise), the result of their sum remains unaltered. In fact, as the considered production line constitutes a closed queue network (CONWIP), the position of the unbalanced production machine does not affect performance variations. Now, consider the case of distributing the same amount of average processing time \(\Delta t\) not to a single machine but dividing this amount between two other machines. In this scenario, it is possible to state that a fraction of this \(\Delta t\), denoted as \(\alpha\), will be assigned to one machine and the \(1-\alpha\) to another. This situation is illustrated in Figure 11, in which the fraction of average processing time \(\Delta t\) of machine 2 is assigned to machines 3 and 5, determining:  
\[t_{2a}=t_{2a}-\Delta t\]  
Figure 10: Exchange of processing time between two machines. \[\text{Line }2:12;12;11;8;7\text{ [min]}\]0  
\[\text{Line }2:12;12;11;8;7\text{ [min]}\]1  
where \(t_{m_{u}}\) represents the new average processing time of the \(m\)-th machine, and \(t_{m_{o}}\) denotes the previous average processing time. It is of interest to investigate whether there exist combinations of \(\alpha\) and \(\Delta t\) for which the \(\sum(t_{o}^{3})\) remains unchanged for \(\sum(t_{n}^{3})\), in order to analyze the change of these solutions for the other summation terms. Considering the position of the equations above, it is possible to write:  
\[\text{Line }2:12;12;11;8;7\text{ [min]}\]2  
\[\text{Line }2:12;12;11;8;7\text{ [min]}\]3  
Solving for \(\Delta t\) yields:  
\[\text{Line }2:12;12;11;8;7\text{ [min]}\]4  
Equation (7) highlights the relationship between \(\Delta t\) and \(\alpha\), providing a means to explore various scenarios where the redistribution of processing time does not impact the cubic summation of processing times. By analyzing these combinations, further insights can be gained into the behavior of the system under different redistributions and their effects on other performance metrics. We solved the Equation (7) numerically, showing that the only combinations that satisfy the equation are those where the amount of processing time involved in the exchange results in a situation where the machines have moved positions. For example, the actual processing time of machine \(t_{2}\) is moved to machine \(t_{3}\), and the processing time of machine \(t_{3}\) to machine \(t_{5}\). Having demonstrated this phenomenon in the case of two machines and in the case of three machines, we can inductively state that this reasoning is valid for other combinations of time shifting, always considering the case of production lines whose summation of average processing times remains constant, with a fixed \(T_{0}\). Therefore, it is possible to conclude that even a small variation in the processing time distribution leads to a consequent change in the summations and, consequently, in the performance metrics they determine.