# 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
The results presented in References [22; 23; 24] allowed analytical expressions to calculate the values of end-to-end QoS indicators, which were evaluated and analyzed between a pair of individual routers. For the case considered in this work, in the structure of an arbitrary network domain, multiple routers may simultaneously belong to each set \(B^{p,k}_{in}\) and \(B^{p,k}_{out}\). This aspect is a unique characteristic of transit domains. The following methodology is proposed to account for the features of the multidomain network architecture. 1. During the \(k\)th flow routing for each \(p\)th domain, the pair of routers \(R^{p}_{in}\) and \(R^{p}_{out}\) is determined, between which \(\tau^{p,k}\) is calculated. In the domain that served as the source of the \(k\)th packet flow, the \(R^{p}_{in}\) was the router through which the \(k\)th flow income to the network. For the domain that served as the \(k\)th packet flow receiver, the \(R^{p}_{out}\) was the router through which the \(k\)th flow outgo from the network. 2. Additional imaginary routers \(R^{p}_{in}\) and \(R^{p}_{out}\), which become adjacent for routers from \(B^{p,k}_{in}\) and \(B^{p,k}_{out}\) subsets, respectively, through imaginary communication links, are introduced to the network structure at the boundaries of the \(p\)th domain that interacts with other ones through two or more routers. If the boundary between the \(p\)th and \(q\)th domains passes through several routers, as shown, for example, in Figure 2, then the imaginary routers \(R^{p}_{in}\) and \(R^{p}_{out}\) coincide (Figure 3). The introduction of imaginary routers is conditioned by the fact that value \(\tau^{p,k}\) can now be evaluated and analyzed between a pair of routers \(R^{p}_{in}\) and \(R^{p}_{out}\) based on the approach proposed in Reference [22]. To prevent the impact of packet delays, \(\tau^{p,k}\), in imaginary communication links connecting real border routers to imaginary ones, their bandwidths must go to \(+\infty\) during the calculations. 3. The continuous numbering of the communication links in the \(p\)th domain is adopted to geometrize the network structure further. For this purpose, the set of network links belonging to the \(p\)th domain is denoted as \(V^{p}=\left\{v^{p}_{z},\ z=\overline{1,m^{\sim}_{p}}\right\}\), where \(n^{\sim}_{p}\) and \(m^{\sim}_{p}\) are the number of communication links and routers in the \(p\)th domain, respectively, taking into account the introduced imaginary network elements. Thus, for the network structure example, as shown in Figure 3, \(n^{\sim}_{1}=n^{\sim}_{2}=6\) and \(m^{\sim}_{1}=m^{\sim}_{2}=5\). Such an increase in the number of communication links and routers leads to an increase in the number of variables, \(x_{i,j}^{p,k}\), with the condition (2) being replaced by the following flow conservation conditions on the \(p\)th domain routers:  
\[\left\{ \begin{array}{ll}\sum\limits_{w_{j,i}^{p,k}\in W^{p}}x_{i,j}^{p,k}-\sum \limits_{w_{j,i}^{p,k}\in W^{p}}x_{j,i}^{p,k}=1,\text{ if }R_{i}^{p}=R_{in}^{p};\\ \sum\limits_{w_{j,i}^{p,k}\in W^{p}}x_{i,j}^{p,k}-\sum\limits_{w_{j,i}^{p,k} \in W^{p}}x_{j,i}^{p,k}=0,\text{ if }R_{i}^{p}\
eq R_{in}^{p},R_{out}^{p};\\ \sum\limits_{w_{i,j}^{p,k}\in W^{p}}x_{i,j}^{p,k}-\sum\limits_{w_{j,i}^{p,k} \in W^{p}}x_{j,i}^{p,k}=-1,\text{ if }R_{i}^{p}=R_{out}^{p}.\end{array} \right. \tag{10}\]  
According to the methodology of network tensor modeling [22; 23; 24; 32; 33], the domain structure determines the anisotropic space formed by coordinate paths. Network edges (links), circuits, interpolar paths, and node pairs can serve as coordinate paths where the network poles are the routers \(R_{in}^{p}\) and \(R_{out}^{p}\). Here, the space dimension is determined by the total number of edges in the network and is equal to \(n_{p}^{\sim}\)[22].