# 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
 \right|\right|, \tag{16}\]  
where \(E_{\gamma\varepsilon}^{(1)}\) is the square \(\kappa_{p}^{\sim}\times\kappa_{p}^{\sim}\) submatrix, \(E_{\gamma\varepsilon}^{(4)}\) is the square \(\theta_{p}^{\sim}\times\theta_{p}^{\sim}\) submatrix, \(E_{\gamma\varepsilon}^{(2)}\) is the \(\kappa_{p}^{\sim}\times\theta_{p}^{\sim}\) submatrix and \(E_{\gamma\varepsilon}^{(3)}\) is the \(\theta_{p}^{\sim}\times\kappa_{p}^{\sim}\) submatrix. Ensuring compliance with QoS conditions (7) or (8), (9) is associated with the need for analytical calculation of delay values \(\tau_{MP}^{p,k}\) in each network domain. The solution of this problem under conditions (2) is based on the use of the results obtained by the following [22]:  
\[\tau_{MP}^{p,k}=\frac{\Lambda_{\gamma}^{t}E_{\gamma\varepsilon}^{(1)}\Lambda _{\gamma}}{\lambda_{req}^{k}}, \tag{17}\]  
where the components of the numerator in terms of geometrization and metrization of the network tensor model are tied, firstly, to the structure of the \(p\)th network domain and its links parameters, and secondly, to the characteristics of the \(k\)th packet flow. Then the conditions for Quality of Service for the average end-to-end delay of the \(k\)th packet flow and bandwidth in the \(p\)th network domain (9) can be written in more detail:  
\[\frac{\Lambda_{\gamma}^{t}E_{\gamma\varepsilon}^{(1)}\Lambda_{\gamma}}{ \lambda_{req}^{k}}\leq\tau_{req}^{p,k}. \tag{18}\]  
However, Equation (17) is valid for the case of a bipolar \(p\)th network domain structure, i.e., where the packets of the \(k\)th flow are transmitted from one border router from the set \(B_{in}^{p,k}\) (source router) to another border router from the set \(B_{out}^{p,k}\) (receiver router). Therefore, depending on the specifics of the structural construction and relation of individual network  
From the technological point of view, minimizing the number of iterations of the procedure (22) when obtaining the desired optimal solution aims to reduce the amount of service traffic transmitted between hierarchical levels about the results of calculations at each iteration and to decrease the total time of solving the problem of inter-domain routing in the network as a whole [22; 23; 24]. Numerical Research of the Proposed Method of Hierarchical-Coordination Inter-Domain Routing in Communication Network under Provision of Normalized QoS  
We studied the proposed hierarchical coordination inter-domain routing method in the communication network to confirm the efficiency and adequacy of the obtained numerical results. Within the calculation example, we analyzed the nature of the solutions to the hierarchical coordination inter-domain QoS routing for the variant of the network structure, which is shown in Figure 2. The network consists of two domains (Figure 2). The first and second domains contain four routers:  
\[R^{1}=\left\{R_{1}^{1},R_{2}^{1},R_{3}^{1},R_{4}^{1}\right\},\ R^{2}=\left\{R_ {1}^{2},R_{2}^{2},R_{3}^{2},R_{4}^{2}\right\}.\]  
The interaction between domains is carried out via routers \(R_{3}^{1}\left(R_{1}^{2}\right)\) and \(R_{4}^{1}\left(R_{2}^{2}\right)\). Let the packet source node be the router \(R_{1}^{1}\) in the first domain and the receiver node \(R_{4}^{2}\) in the second domain, respectively. The gaps in communication links (Figure 1) show their  
Figure 4: Two-level method architecture. bandwidth (pps). The matrix's main diagonal \(H_{p}^{k}\) (19) is the values \(\frac{10^{8}}{\sigma_{v,i}^{2}}\) by analogy with the values of routing metrics used in the OSPF protocol [1; 2]. The operation of each router interface was simulated by using the queueing system M/M/1. For example, consider a single flow case when the following parameters set the requirements for the QoS level in a multidomain network: \(\lambda_{req}^{1}=350\) pps and \(\tau_{req}^{1}=80\) ms.