## 6 Numerical Results  
In a first test, the effect of layout discretization in the benchmark layout is assessed by setting the control period to \(T=1\) s, and recording the ending times as described in Section 5, for IDRR, IDRRP, DRR-2m and DRR-1m, with the number of AGVs varying from 1 to 10. The results are displayed in Figure 9, while the data are collected in Table 1.  
Figure 8: The industrial layout. W1–W7: pick-up stations; W8–W9: drop-off stations; P1–P10: parking stations. Figure 9: Benchmark layout—Test 1: Ending times (in [s]) for \(T=1\) s, from 1 to 10 AGVs. IDRR (black), IDRRP (dark grey), DRR (grey), DRR-2m (light grey), and DRR-1m (white). As already reported in [16], IDRR and IDRRP clearly outperform DRR. Only for 1 and 2 AGVs are the IDRRP and DRR times close, in the former because IDRRP matches DRR, as there are no conflicts, and in the latter because the number of conflicts between only 2 AGVs in this layout is not null but still very low. Moreover, the difference increases with the number of AGVs in the layout, as the number of conflicts scale up and the DRR waiting times increase. This corroborates the superior performance of IDRR with respect to DRR not only because of the parking policy, but also from the pure traffic management side, as the IDRRP and DRR numbers show. However, it is most interesting that although DRR works with the sparse discretization inherent to IDRR and IDRRP, the use of denser discretizations in DRR-2m and DRR-1m make no significant difference either between them or with DRR. Hence, the fact that denser discretizations entail lower waiting times in the face of conflicts is definitely compensated by the corresponding increase associated with the request of a higher number of permissions to resume. The latter is clearly deduced from the results when there is just one AGV in the layout; the higher ending times observed for denser discretizations can be solely attributed to the additional time required to handle the scaling up of permit requests, because the absence of conflicts excludes the compensating possibility. Therefore, it turns out that (i) DRR works finely with sparse discretizations, and (ii) the use of denser discretizations with IDRR in the general case is not encouraged, as it would complicate the algorithm and there is no expectation of improvement. A second test is conducted to evaluate the effect of the control period, \(T\), on the time completion of the experiment. Simulations were carried out again for the five traffic management algorithms with \(T=1\) s, 0.5 s, 0.2 s, 0.1 s, 0.05 s, 0.02 s, and a number of AGVs ranging from 1 to 4. The results, gathered in Table 2, confirm those of Test 1. Namely, no significant differences are observed between the sparse discretization of DRR and the denser ones in DRR-2m, and DRR-1m, while for the 1 AGV case, the mismatch decreases with \(T\); in fact, when \(T=0.02\) s, times are almost equal, which indicates that for this control period, the scaling up of resuming requests in denser discretizations has no effect on the time completion of the task. It is noticeable that the ending times become better as the period of the control loop decreases. AGVs have to wait for the control loop to be activated at the specific interval when the AGVs have requested for allocation between the interval; thus, for lower control periods, the negative cumulative waiting effect in time completions is minor, hence speeding up the whole system. The improvement with respect to the highest ending time, always given by \(T=1\) s, increases with lower \(T\)'s and tends to stabilize around \(T=0.02\) s, irrespective of the algorithm and number of AGVS. For IDRR and IDRRP, the maximums are 10.91% and 10.95%, respectively, while DRR, DRR-2m and DRR-1m score better for 1 or 2 AGVs, the maximum being 13.5% for DRR-1m and 1 AGV, in all cases for \(T=0.02\) s. The relative errors are displayed in Figure 10.