## 5 Performance Evaluation of Multi-Hop Wired Network
### Discussion  
The stable operating regions of analysis can be obtained by checking the performance of PDV against various burstiness levels of cross traffic. We used the parameters from Table 1 for this analysis. The peak-to-average ratio of cross traffic indicates the burstiness level of cross traffic. Figure 10 shows the impact of the peak-to-average rate ratio of cross traffic on the PDV and delay bounds when \(\mu_{c}\) is kept constant at 0.7. The valid PDV bound values start when the peak-to-average rate reaches 1.6. Beyond 1.6, the PDV bounds decrease exponentially with the peak-to-average rates. The valid \(\lambda_{c}\) values start from 0.43. Thus, the unstable region for this configuration is when the peak-to-average rate is less than 1.6, which corresponds to the \(\lambda_{c}\) values being less than 0.43. Similarly, Figure 11 shows the impact of the peak-to-average rate ratio of cross traffic on the PDV and delay bounds when \(\lambda_{c}\) is kept constant at 0.7. The valid PDV bound values start when the peak-to-average rate reaches 1.7. Beyond 1.7, the PDV bounds decrease exponentially with the peak-to-average rates. All the \(\lambda_{c}\) values can lead to valid PDV values. Thus, the unstable region for this configuration is when the peak-to-average rate is less than 1.7. Figure 9: Impact of \(n\) and \(\mu_{c}\) on the PDV bound in a multi-stage network (constant \(\lambda_{c}\)). The PDV performance was evaluated in different network conditions to cover the corner cases. The different network scenarios were created by varying the burstiness of cross traffic. The approach of varying cross traffic dynamics using different peak-to-average rates of cross traffic was discussed in [22]. This, in turn, can be achieved by changing the transition probabilities \(\lambda_{c}\) and \(\mu_{c}\) from the minimum to the maximum limits. To further strengthen the confidence in the analysis, we simulated the through traffic in the presence of cross traffic in the time domain using the parameters listed in Table 2. The cross traffic was generated with the average rate of 35 kbps using the Markovian process. For a seven-stage network, the PDV output is shown in Figure 12. The average PDV bound value of 1.6 ms seems practical for the given configuration. Figure 11: Impact of the peak-to-average rate ratio of cross traffic on the PDV bound in a single-stage network (constant \(\lambda_{c}\)). Figure 10: Impact of the peak-to-average rate ratio of cross traffic on the PDV bound in a single-stage network (constant \(\mu_{c}\)). Overall, the analysis revealed the behavior of PDV bounds under different network conditions influenced by cross traffic parameters. For a single stage, the upper bound of the PDV changes from 276 ms to 7 ms when \(\lambda_{c}\) increases from 0.52 to 0.81, keeping \(\mu_{c}\) constant at 0.7. Similarly, the upper bound of PDV for a single stage varies from 4 ms to 64 ms when \(\mu_{c}\) increases from 0.29 to 0.91, keeping \(\lambda_{c}\) constant at 0.7. For a seven-stage configuration, the upper bound of PDV changes from 274 ms to 40 ms when \(\lambda_{c}\) increases from 0.59 to 0.83, keeping \(\mu_{c}\) constant at 0.7. Similarly, the upper bound of PDV for a seven-stage configuration varies from 15 ms to 97 ms when \(\mu_{c}\) increases from 0.15 to 0.7, keeping \(\lambda_{c}\) constant at 0.7. Thus, the PDV observations from single- and multi-stage configurations proved the initial hypothesis. When changed in a planned manner, the traffic parameters can control the range in which the PDV can vary. This is an important observation as there are many applications where keeping the PDV under control or within a prescribed range can result in maintaining the QoS for the traffic of interest. A background traffic shaper can be conceived of based on the dependence of the PDV on the traffic parameters. The analysis showed promising results in maintaining the QoS in the presence of unpredictable background traffic using such a shaper. The known limitations of this approach are as follows.