## 5 Performance Evaluation of Multi-Hop Wired Network
### Multi-Stage Case  
Figure 7 shows the results for \(n=1\) to \(10\) servers with through and cross traffic in series. The parameters used were \(\mu_{c}=0.5\) and \(\lambda_{c}=0.5\), and the peak rate \(h=60\) kbps was chosen. The system is unstable up to \(\lambda_{c}=0.43\). The instability indicates that the average rate is the same or closer to the peak rate during this time. After that, with the addition of each server to the network topology, the delay and PDV bounds show an increase in values. The results confirm the scalability of the performance bounds. The same traffic continues through a series of network servers, and thus, the burstiness due to a change in the average rate is accumulated with each new server and subjected to the subsequent stage. Thus,  
Figure 5: Impact of \(\lambda_{c}\) on the PDV bound in a single-stage network (constant \(\mu_{c}\)).  
Figure 6: Impact of \(\mu_{c}\) on the PDV bound in a single-stage network (constant \(\lambda_{c}\)).  
the PDV bound shows a steeper rise with the addition of servers compared to delay. The increment in the end-to-end delay and the PDV bound for each additional concatenated server is greater than the bound that would be obtained for the server in isolation.  
Figure 8 shows the results of the delay and PDV bound curves for \(n=1\) to 10 servers with the peak rate \(h\) and \(\mu_{c}\), kept constant at 60 kbps and 0.7, respectively. With increasing \(\lambda\) from 0 to 1, the delay and PDV bound curves corresponding to the number of servers (1 to 10) follow the decrement pattern in values identical to that of the single stage. With increasing the number of servers, the PDV bound curve shifts to the right, indicating a late starting point on the \(\lambda_{c}\) scale, but higher values after the start compared to the previous curve. This means that, as we continue to increase the number of stages, the PDV and delay bound values reach higher than the previous number of stages, which is true for all the range of \(\lambda_{c}\) values. The operating region of the \(\lambda_{c}\) values for managing PDV reduces as we continue to increase the number of stages.  
Figure 9 shows the results of the delay and PDV bound curves for \(n=1\) to 10 servers with the peak rate \(h\) and \(\lambda_{c}\), kept constant at 60 and 0.7, respectively. The parameter \(\mu_{c}\) was  
Figure 8: Impact of \(n\) and \(\lambda_{c}\) on the PDV bound in a multi-stage network (constant \(\mu_{c}\)).  
Figure 7: End-to-end concatenation of servers in series.  
varied from 0 to 1. The delay and PDV bound curves follow the same increment pattern as for the single stage; also, the increase is steeper for PDV than delay as observed with the single-stage case. As the number of servers increases, the PDV bound curves start at a lower value and show small value increases compared to previous server configurations. This means that, as we continue to increase the number of stages, the PDV and delay bounds reach lower values for the previous number of stages, which is true for all the range of \(\lambda_{c}\) values. The useful region of \(\mu_{c}\) values for managing the PDV bound reduces as we continue to increase the number of stages.