# Advanced Tensor Solution to the Problem of Inter-Domain Routing with Normalized Quality of Service
## 3 Decomposition Model of Inter-Domain QoS Routing in Communication Network
During network decomposition, the boundary between domains passed through network routers, as implemented, for example, in OSPF [1,11]:  
\[R^{p}\cap R^{q}\
eq\varnothing\text{ and }W^{p}\cap W^{q}=\varnothing,\ p\
eq q. \tag{1}\]  
In other words, some network routers may belong to several adjacent domains at a time. For each \(p\)th domain, we also define a set of border routers, \(B^{p}\) (\(B^{p}\in R^{p}\)). In turn, the entire set of the \(p\)th domain border routers can be divided into two subsets: \(B^{p,k}_{in}\) is the subset of the border routers through which the packets of the \(k\)th flow income into the \(p\)th domain, and \(B^{p,k}_{out}\) is the subset of the border routers through which the packets of the \(k\)th flow outgoing from the \(p\)th domain. For each communication link, \(W^{p}_{i,j}\), we denote its bandwidth, \(\varphi^{p}_{i,j}\), which is measured in packets per second (pps). As a result of solving the hierarchical inter-domain routing problem for each \(p\)th domain, it is necessary to calculate the routing variables, \(x^{p,k}_{i,j}\), that characterize the fraction of the \(k\)th packet flow transmitted in the link, \(W^{p}_{i,j}\in W^{p}\). Then, for each \(p\)th domain router, the \(k\)th flow conservation conditions must be met to ensure the connectivity of intra-domain sections of the inter-domain routes. If the \(p\)th domain is transit for the \(k\)th packet flow, then such conditions take the following form:  
\[\left\{ \begin{array}{l}\sum\limits_{B^{p,k}_{in}\in B^{p}}\left(\sum\limits _{W^{p}_{i,j}\in W^{p}}x^{p,k}_{i,j}-\sum\limits_{W^{p}_{i,j}\in W^{p}}x^{p,k} _{j,i}\right)=1\\ \sum\limits_{W^{p}_{i,j}\in W^{p}}x^{p,k}_{i,j}-\sum\limits_{W^{p}_{i,j}\in W ^{p}}x^{p,k}_{j,i}=0;\\ \sum\limits_{B^{p,k}_{out}\in B^{p}}\left(\sum\limits_{W^{p}_{i,j}\in W^{p}}x^{ p,k}_{i,j}-\sum\limits_{W^{p}_{i,j}\in W^{p}}x^{p,k}_{j,i}\right)=-1.\end{array} \right. \tag{2}\]  
The conditions within the system of equations (2) must be met separately for each \(k\)th packet flow. The first condition of the system (2) covers all the border routers through which the \(k\)th flow incomes at the \(p\)th domain. The second condition in (2) is introduced for internal \(p\)th domain routers that are transit for the \(k\)th packet flow. The third condition must be satisfied for all border routers through which the \(k\)th flow outgoes from the \(p\)th domain. \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Meaning** \\ \hline \(\kappa_{p}^{\sim}\) & Number of linearly independent interpolar paths in the \(p\)th domain with imaginary network elements \\ \(\theta_{p}^{\sim}\) & Number of internal node pairs \\ \(Q\) & Mixed bivalent tensor \\ \(T\) & Univalent covariance tensor of average packet delays \\ \(\Lambda\) & Univalent contravariant tensor of the average flows intensities \\ \(E_{p}\) & Projection of the metric tensor, \(E\), in the coordinate system of edges \\ \(E_{\pi\epsilon}\) & Projection of the metric tensor, \(E\), in the coordinate system of interpolar paths \\  & and internal node pairs \\ \(\overset{\rightarrow}{\mu}_{p,q}^{k}\) & Lagrange multiplier vector \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._If the \(k\)th packet flow arrives at a network through the \(p\)th domain, and its source is the router \(R_{i}^{p}\), for example, then, for this network, the first condition of the system (2) will be somewhat simplified, as shown below:  
\[\sum_{W_{i,j}^{p}\in W^{p}}x_{i,j}^{p,k}=1.\]  
The rest of the equations in the system (2) will remain unchanged.