## 3 DCMS Model  
### Establishment of the DCMS Model  
The real intention of proposing DCMS in the circulation system is to disrupt the inherent order of the rollers without collision, to allow rollers that are far apart to enter the processing area for lapping at the same time, and to achieve size comparison. This facilitates a full-size comparison between the entire batch of rollers, thereby achieving rapid convergence of the batch diameter variation. However, in the actual application of DCMS, it is necessary to consider the impact of roller distribution probability (\(k\)) and conveyor belt speed (\(v_{1},v_{2}\)) on the convergence speed of batch diameter variation. The existing DDSGL physical system can be improved to carry out the experimental study of DCMS. However, a single experiment can only simply determine the relationship between the controlled mixing parameters and the convergence speed of the roller size accuracy. It cannot explore how DCMS promotes full comparison between rollers and consequently improves roller size consistency. Therefore, this study aims to establish a high  
Figure 4: Dual-channel mixing scheduling strategy diagram, (**a**) Single-line operation mode before improvement, (**b**) Dual-channel operation mode after improvement, (**c**) The flowchart of the study on DCMS. fidelity DCMS model based on the DDSGL system, select the optimal DCMS parameters through the evaluation system established by the calculation results of the DCMS model, and verify the accuracy of the optimal results in the actual processing process through lapping experiments, ultimately achieving the purpose of the DCMS model guiding the DDSGL system to achieve high-precision and efficient production. The operation mode of the roller after adding DCMS is shown in Figure 5. The yellow area represents the area where DCMS is embedded. The leap probability \(k\) and the leap space \(S_{n}\) are two important input parameters of the DCMS model. The operating rule is to select a space node (space B\({}_{3}\)) in the external circulation space. When the roller passes through this space node, there are two possible paths in the next processing step, which are the possibility of leaping \(S_{n}\) spaces with a probability of \(k\) and the possibility of moving forward into the next space with a probability of 1-\(k\). In physical space (Figure 4), the DCMS parameters controlled by the system are the dwell time of the swing channel outlet at the entrances of the two conveyor belts and the speeds of the two conveyor belts. The relationship between these parameters and the leap probability \(k\) and the number of leap spaces \(S_{n}\) in the model is:  
\[k=\frac{t_{1}}{t_{1}+t_{2}} \tag{1}\]  
\[n=(\frac{L}{v_{1}}+\frac{L}{v_{2}})Qk \tag{2}\]  
In the formula, \(t_{1}\) represents the dwell time of the swing channel outlet at conveyor belt 1, \(t_{2}\) represents the dwell time of the swing channel outlet at conveyor belt 2, L represents the length of conveyor belts 1 and 2, and \(Q\) represents the roller flow rate of the swing channel. \(v_{1}\) and \(v_{2}\) represent the running speeds of conveyor belts 1 and 2. Figure 5: DDSGL roller operation mode after adding DCMS. Due to the inclusion of DCMS, some interfaces between mechanisms will experience local random swaps of rollers. This phenomenon changes the order of rollers within a small range, and these areas are defined as small-scale random mixing. To represent this phenomenon in the model, we can adopt the mapping rules in the green area in Figure 5. One external space node is selected in the circulation system of the model (such as the external space B\({}_{202}\) in Figure 5). After the roller passes through this external space, it will randomly leap forward a certain number of spaces with a random probability.