# Advanced Tensor Solution to the Problem of Inter-Domain Routing with Normalized Quality of Service
## 7 Discussion  
The demands that motivated the creation and modification of SDN inter-domain routing solutions can be formulated as follows:  
* Accounting for multidomain network structure;
* Ensuring the connectivity of inter-domain routes and the order of balancing across inter-domain routes;
* Guarantying the QoS level, both within domains and end-to-end Quality of Service;
* Support for multipath routing with load balancing to ensure more efficient use of the network resource;
* Implementation of hierarchical routing with coordination of decisions obtained in separate domains;
* Providing a high convergence rate of the coordination procedure to reduce the time required to solve the problem of hierarchical multidomain routing and decrease the amount of overhead traffic associated with coordination. The decomposition model of inter-domain routing (1)-(18) was further developed to ensure the Quality of Service in the network, consisting of many series-connected domains. The model was based on the conditions for the implementation of multipath routing (4), conditions for the flow conservation (2), preventing overload of links and the network in general (3), and conditions for inter-domain interaction (5) guaranteeing the routes connectivity. The flow-based model of inter-domain routing was supplemented by the conditions of providing end-to-end and normalized Quality of Service (6)-(9). The novelty of the proposed routing model is the modification of flow-conservation conditions (10), which allowed us to formulate conditions for ensuring the end-to-end \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Calculation Results on the First Domain Controller**} \\ \hline Destination \(\mid\) domain & Next Hop \(\mid\) domain & Fraction of flow (balancing order) \\ \hline For \(R_{1}\mid 1\) & & & \\ \(R_{4}\mid 2\) & \(R_{2}\mid 1\) & & 0.48 \\ For \(R_{2}\mid 1\) & & & \\ \(R_{4}\mid 2\) & \(R_{4}\mid 1\) & & 0.52 \\ \(R_{4}\mid 2\) & & & \\ \(R_{4}\mid 2\) & & & \\ \hline \hline \end{tabular}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{\(\mid\)} & & \\ \(R_{4}\mid 2\) & & & 0.28 \\ For \(R_{2}\mid 2\) & & & \\ \(R_{4}\mid 2\) & & & 0.72 \\ For \(R_{3}\mid 2\) & & & \\ \(R_{4}\mid 2\) & & & 1 \\ \(R_{4}\mid 2\) & & & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Forwarding order for inter-domain QoS routing shown in Figure 10. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Domain**} & \multirow{2}{*}{\(\lambda_{req}^{1}\)**\(\mathbf{p}\)**ps**} & \multirow{2}{*}{\(\mid\)**\(\mid\)**\(\mid\)**} & \multicolumn{4}{c}{**Characteristics of the Calculated Paths**} & \\ \cline{4-7}  & & & **Route** & \(\lambda_{p}\)**\(\mathbf{p}\)**ps** & \(\tau_{p}\)**, \(\mathbf{ms}\)** & \(\tau_{MP}^{p,1}\)**, \(\mathbf{ms}\)** \\ \hline \multirow{4}{*}{1} & & & & \(R_{1}^{1}\to R_{2}^{1}\to R_{4}^{1}\) & 146 & 17.1 & \multirow{4}{*}{37.81} \\  & & 3 & \(R_{1}^{1}\to R_{2}^{1}\to R_{4}^{1}\) & 21.4 & 10.5 & & \\ \cline{1-1}  & & & \(R_{1}^{1}\to R_{2}^{1}\) & 182.6 & 57.5 & & \\ \cline{1-1}  & & & \(R_{1}^{1}\to R_{4}^{1}\) & 105.2 & 22.3 & & \\ \cline{1-1}  & & 3 & \(R_{1}^{2}\to R_{3}^{2}\to R_{4}^{2}\) & 40.8 & 22 & & \\ \cline{1-1}  & & & \(R_{2}^{2}\to R_{3}^{2}\to R_{4}^{2}\) & 204 & 21.7 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Characteristics of the calculated routes in Figure 10.  
Quality of Service in multidomain networks in terms of bandwidth and average end-to-end packet delay ((17) and (18)). It was possible to formulate in analytical form the conditions of QoS assurance ((17) and (18)) based on tensor network modeling in geometric space, which was created by coordinate paths of edges (links), interpolar paths, and internal node pairs. A method of hierarchical coordination routing in multidomain Software-Defined Networks (18)-(22) with the provision of normalized Quality of Service is a further development of the solutions presented in References [22; 23; 24].