## 7 Performance Evaluation
### Numerical Results and Discussion
#### 7.2.2 Implementing Overlapping Constraints Based on Critical Time Deadlines  
Based on the apparent fluctuation in TT latency bounds under window overlapping situations, we can add a new constraint for the overall TSN scheduling limitations, which was formulated in [42]. As the critical time traffic requires different WCD deadlines, each TT frame should be forwarded to a priority queue, where it should experience an end-to-end latency no higher than its deadline. We can differentiate between TT priority queues according to different overlapping ratios between corresponding windows. Moreover, the overlapped queue's priority is a significant issue to determine which maximum allowable overlapping ratio guarantees the targeted latency deadline. Based on the results presented in Figure 16, each WCD bound can be met if the related TT overlapping does not exceed a  
Figure 15: WCD vs. higher-priority overlapping with different sizes of the fourth-priority frame (\(f_{4}^{max}\)): (a) opening-edge overlapping and (b) closing-edge overlapping. specific limit. These OR limits are differentiated in Figure 16 concerning the priority of the overlapped queues and the overlapping position. It can be noticed that lower-priority overlapping causes a small effect on WCD boundaries, with a trivial difference between the opening, closing, and two-sided overlapping cases, as shown in Figure 16. In particular, slightly after the WCD bound in the entire isolation case (282.7 \(\upmu\)s), the latency deadline can be met under any lower-priority overlapping. At most, the deadline of 287.7 \(\upmu\)s can be met under any lower-priority overlapping. However, large ORs are not acceptable in GCL design as it is considered higher overlapping on the overlapped queue. In contrast, higher-priority overlapping has the highest impact on the latency, and stricter overlapping constraints should be guaranteed. Opening-edge overlapping has a larger influence than closing-edge overlapping. For example, to meet the 400 \(\upmu\)s latency deadline, the highest allowable OR at the opening and closing edges is 38.62% and 54.62%, respectively. Under two-sided higher overlapping, the performance is almost the same as under closing-edge higher overlapping, and the latency deadline of 400 \(\upmu\)s is met until 54.31% overlapping. For mixed overlapping cases, the OR limits are between pure lower and pure higher overlapping. For instance, the latency deadline of 400 \(\upmu\)s is guaranteed until 76.84% OR with higher priority at the opening edge and lower priority at the closing edge, and until 85.2% with the opposite. As noted, the lowest WCD bound that can be achieved is 282.7 \(\upmu\)s. This limit is based on specific framing and timing settings. Adjusting these values increases/decreases the WCD bounds accordingly, as examined in the previous results. Thus, it is worth noting that Figure 16 can be considered a reference chart for window overlapping vs. latency deadlines based on the above networking and data settings. For each specific required setting, we can easily implement related overlapping vs. the latency reference chart. Accordingly, we can implement, in the general case, a relaxed window overlapping constraint that will guarantee meeting latency requirements for TT traffic and can be used to improve the overall solution space, as follows:  
Constraint definition: By considering \(Q_{k}^{h}\) as a targeted TT queue, where \(k\in\left\{1,2,\ \dots,\ N_{q}\right\}\) and \(h\) is a node in the selected path (\(h\in\left\{1,2,\ \dots,\ N\right\}\)), and the lower-priority queue as \(k^{+}\), it is assumed that the largest size of \(Q_{k}^{h}\) and \(Q_{k^{+}}^{h}\) frames in \(h\) is given as \(f_{k}^{h,max}\) and \(f_{k^{+}}^{h,max}\). Figure 16: Maximum allowable overlapping ratio (OR) vs. guaranteed WCD deadlines under one-sided and two-sided overlapping conditions by lower-, higher-, and mixed-priority queues.