# Advanced Tensor Solution to the Problem of Inter-Domain Routing with Normalized Quality of Service
## 4 Conditions for Ensuring End-to-End Quality of Service in Multidomain Network
In the first case, the normalizing of the QoS level between the network domains by the average delay was carried out as follows:  
\[\tau_{req}^{1,1}=30\;\text{ms},\;\tau_{req}^{2,1}=50\;\text{ms}. \tag{23}\]  
Then Figure 5 shows the initial problem solution of inter-domain QoS routing, i.e., before starting the coordination procedure (22). This figure shows the following data in the link gaps (from top to bottom): packet flow intensity, bandwidth, and the average packet delay in this communication link (ms). A characteristic feature of the obtained solution (Figure 5) is that the conditions for providing normalized QoS (8) are met. The average end-to-end delay in the first domain from the source router, \(R_{1}^{1}\), to the domain boundary was 25.6 ms, while in the second domain, from the domain boundary to the receiving router, \(R_{4}^{2}\), it was equal to 34.88 ms (Table 3). However, there was no interconnection of inter-domain routes, because the conditions of inter-domain interaction (5) in the case of Lagrangians (21) minimization were not  
Figure 5: Initial solution to the inter-domain QoS routing problem following the normalized requirements (23). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Domain**} & \multirow{2}{*}{\(\lambda_{req}^{1}\)**\(\mathbf{pps}\)**} & \multirow{2}{*}{1\(P\)1} & \multicolumn{5}{c}{**Characteristics of the Calculated Paths**} & \multirow{2}{*}{\(\tau_{MP}^{p,1}\)**, ms**} \\ \cline{5-6}  & & & **Rout** & \(\lambda_{p}\)**, pps** & \(\tau_{pr}\)**, ms** & \\ \hline \multirow{4}{*}{1} & & & & \(R_{1}^{1}\to R_{1}^{1}\to R_{1}^{1}\) & 45.5 & 13 & \\  & & 3 & \(R_{1}^{1}\to R_{1}^{2}\to R_{1}^{1}\) & 130.5 & 12.7 & 25.6 \\ \cline{1-1}  & & & \(R_{1}^{1}\to R_{1}^{1}\) & 174 & 38.5 & \\ \cline{1-1}  & & & \(R_{1}^{2}\to R_{1}^{2}\) & 129.43 & 48.6 & \\ \cline{1-1}  & & & \(R_{2}^{2}\to R_{2}^{2}\to R_{2}^{2}\) & 76.69 & 17.1 & 34.88 \\ \cline{1-1}  & & & \(R_{2}^{2}\to R_{3}^{2}\to R_{4}^{2}\) & 143.88 & 32 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the calculated routes in Figure 5.  
met, since the route for each domain is calculated independently on the appropriate SDN domain controllers. To ensure the connectivity of inter-domain routes within the presented method, a coordination procedure (22) was used that is proposed for implementation on the SDN controller. Thus, Figure 6 shows the routing order in a multidomain network after the first iteration of the coordination procedure (22), when conditions (5) were not met again. However, the average end-to-end packet delays in the domains met the norms (23) and amounted to 26.16 and 32.64 ms, respectively. Inter-domain route connectivity was ensured only after the third iteration of the coordination procedure (22). The coordinated solution of the inter-domain QoS routing problem is presented in Figure 7. The specified normalized values of average packet delays (23) were provided in each domain. In the first domain, the maximum end-to-end delay was 26.84 ms, and in the second, it was 44.24 ms (Table 4). Figure 6: Solution to the inter-domain QoS routing problem after the first iteration of the coordination procedure following the normalized requirements (23). Figure 7: Final (coordinated) solution of inter-domain QoS routing problem (after the third iteration of the coordination procedure) following the normalized requirements (23).