## 3 DCMS Model
### DCMS Model Simulation Calculation Results and Discussion
#### 3.4.3 Roller Cycle Time Range Statistics  
As can be inferred from Figures 9 and 10d,e, with a further increase in the leap probability \(k\) or leap space \(S_{n}\), a more intense shuffling of the roller sequence order results in a higher peak in the distribution, achieving better size comparison results among rollers. This raises the question of whether a higher intensity of shuffling the roller sequence order is always better and what the impact of a larger leap probability \(k\) or leap space \(S_{n}\) is on the entire lapping process.  
The lapping experiment consists of a total of 90 cycles. This is the mean number of cycle processing times for the entire batch of rollers, but the processing cycle times for individual rollers will vary, resulting in a range value between the rollers with the most and least cycle times in the entire batch. For rollers in the batch with fewer processing cycle times, the amount of material removed is relatively less, resulting in a larger overall size compared to rollers with more processing cycle times. In the DDSGL lapping experiment, the average diameter removal of the roller in each lapping cycle is about 0.08 \(\upmu\)m. Therefore, an increase of 1 in the range will result in an estimated batch diameter variation error of 0-0.08 \(\upmu\)m. Thus, the size of the range of roller cycle processing times will affect the precision level of the batch diameter variation.  
Calculating the range of roller cycle times for each group of experiments in Table 2, the range of cycle times for the entire batch of 500 rollers under different controlled mixing parameter conditions is shown in Figure 11. It should be noted that the entire processing area has 150 processing spaces. A roller is considered to have completed one full cycle when it has undergone 150 processing steps in the processing area. If the number of steps completed is less than 150, two decimal places are retained, so the cycle time range can appear in decimal form.  
When the leap probability \(k\) or leap space \(S_{n}\) is 0, the system has not implemented the controlled mixing strategy, and the rollers rely solely on local random mixing to change their sequence order. At this point, the minimum cycle times are 89, the maximum cycle times are 91, and the batch range of cycle times for all the rollers is 2. When the leap probability \(k\) = 0.1 or leap space \(S_{n}\) = 10, the intensity of disrupting the order of rollers is still small, and the batch range of cycle times for the entire batch of rollers remains at 2.  
When \(0.1<k<0.3\) or \(10<S_{n}<30\), the intensity of shuffling the roller sequence order gradually increases, and the batch range of cycle times for all the rollers begins to slowly rise from 2 to 2.71 times. This may increase the batch diameter variation error by up to 0-0.057 \(\upmu\)m. As the leap probability \(k\) or leap space \(S_{n}\) continues to increase, the rate of increase in the batch range of cycle times for all the rollers becomes larger, which will cause a greater error effect on the batch diameter variation.  
In summary, a larger leap probability \(k\) and leap space \(S_{n}\) will increase the intensity of mixing the roller sequence order, making the total comparison time distribution of roller groups more concentrated and the size comparisons among the rollers more thorough and uniform. However, a larger leap probability \(k\) and leap space \(S_{n}\) will also cause a rapid increase in the cycle time range, which brings about a greater impact on the batch diameter variation error. Therefore, the leap probability \(k\) and leap space \(S_{n}\) need to be considered in combination with the cycle time range, and it is not the case that higher is always better. Under the current conditions, when the leap probability \(k=0.3\) and the leap space \(S_{n}=30\), the total comparison time distribution is normal, and the cycle time range has only increased by 0.71 times compared to the unmixed group, which is a better parameter combination.