## 5 Performance Evaluation of Multi-Hop Wired Network
### Single-Stage Case  
Figure 5 shows the results for a single server with a peak rate of cross traffic during the on state \(h\) and the off to on transition probability \(\mu_{c}\) being maintained constant at 60 kbps and 0.7, respectively. As the on to off transition probability \(\lambda_{c}\) is varied from 0 to 1, the system shows instability up to \(\lambda_{c}=0.43\), and after that, the delay and PDV (variance) bound curves show a decline in values. The decline is steeper in the case of PDV compared to delay. As \(\lambda_{c}\) increases, the cross traffic process tends to stay more in the off state and continuously reduces the traffic in the network. The average rate of cross traffic reduces with an increase in \(\lambda_{c}\), raising the peak-to-average rate with it. This means that less bursty cross traffic is experienced by through traffic flows. This network behavior causes the delay and PDV bound curves to experience a decline. The PDV expression in (A14) indicates that the part of the PDV bound depends on the changes in the average rate of cross traffic, and hence, its decline is steeper. The practical range of \(\lambda_{c}\) for the PDV bound control is greater than 0.43 for this configuration. The values below 0.43 indicate unstable conditions when the average rate is the same or closer to the peak rate.  
Similarly, Figure 6 shows the results for a single server if the off to on transition probability \(\mu_{c}\) is varied from 0 to 1. The peak rate of cross traffic during the on state \(h\) and the on to off transition probability \(\lambda_{c}\) are kept constant at 60 kbps and 0.7, respectively. With the increase in \(\mu_{c}\), the traffic in the network continuously increases, resulting in delay and PDV bound curves experiencing a rise; however, this rise is sharper for PDV compared to its delay counterpart. As \(\mu_{c}\) increases, the cross traffic process tends to stay more in the on state and continuously increases the traffic in a network. The average rate of cross traffic increases with the rise in \(\mu_{c}\). Thus, the peak-to-average rate increases with it. This means that more bursty cross traffic is experienced by through traffic flows. This network behavior causes the delay and PDV bound curves to experience a rise. The PDV bound  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Description** \\ \hline \(\sigma(\theta)\) & 20 kb & Through traffic burst \\ \hline \(\rho(\theta)\) & 40 kbps & Through traffic rate \\ \hline \(\sigma_{c}(\theta)\) & 0 & Cross traffic burst \\ \hline \(C\) & 100 kbps & Channel capacity \\ \hline \(\epsilon\) & \(10^{-4}\) & Violation probability \\ \hline \(\mu\) & 0–1 & Off-on transition probability \\ \hline \(\lambda\) & 0–1 & On–off transition probability \\ \hline \(n\) & 1 (single-stage) & Number of stages \\  & 10 (multi-stage) & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters used in the analysis.  
Figure 4: Two state on–off Markov model.  
expression in (A14) indicates that the part of PDV depends up the changes in the average rate of cross traffic, whereas the delay bound comprises the latency due to the burst size of the through flows and the latency of the server induced by cross traffic. Thus, lowering the average rate makes PDV rise steeper. All the range of the \(\mu_{c}\) values can be used to control the PDV bounds for this configuration.