# 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  
Quality-of-Service requirements are usually set in terms of bandwidth and average packet delay for each flow circulating in the network. Then the refined conditions for providing QoS on the average end-to-end packet delay for each specific \(k\)th flow in a multidomain network take the following form:  
\[\tau^{k}_{MP}\leq\tau^{k}_{req}. \tag{6}\]  
The value \(\tau^{k}_{MP}\) is measured between the input router of the source domain and the output router receiving the packets of the \(k\)th flow. For the network structure example shown in Figure 1, the value \(\tau^{k}_{req}\) determines the maximum allowable value for the average end-to-end delay of packets \(\tau^{k}_{MP}\) transmitted in the multidomain network between routers \(R^{1}_{1}\) and \(R^{2}_{4}\). Conditions (6), taking into account the multidomain network structure and the additive nature of time metrics \(\tau^{k}_{MP}=\sum_{p=1}^{N}\tau^{p,k}_{MP}\), can be represented in the following form:  
\[\sum_{p=1}^{N}\tau^{p,k}_{MP}\leq\tau^{k}_{req}, \tag{7}\]  
where \(\tau^{p,k}_{MP}\) is the value of the average packet delay of the \(k\)th flow in the \(p\)th network domain, which is measured depending on the type of the \(p\)th domain, as follows:  
* If the \(p\)th domain is the source of the \(k\)th packet flow, then \(\tau^{p,k}_{MP}\) determines the value of the average delay of packets transmitted by one or more paths from the source router of this flow to an arbitrary border router from the subset \(B^{p,k}_{out}\);
* If the \(p\)th domain is the receiver of the \(k\)th packet flow, then \(\tau^{p,k}_{MP}\) characterizes the average delay of packets transmitted by one or more paths from any border router from the subset \(B^{p,k}_{in}\) to the receiver router of the same domain;
* If the \(p\)th domain is transit for the \(k\)th packet flow, then \(\tau^{p,k}_{MP}\) determines the average delay of packets transmitted by one or more paths between any pair of routers, the first of which belongs to the subset \(B^{p,k}_{in}\) and the second one is from the \(B^{p,k}_{out}\). In supporting the multidomain network structure, the provision of required values of end-to-end QoS indicators is often implemented based on their preliminary normalizing [22]. Let \(\tau^{p,k}_{req}\) be the normalized requirements for the average packet delay of the \(k\)th flow in the \(p\)th network domain. That is, for each of the flows' \(k\in K\), the requirements  
Figure 2: Network structure under investigation. for the average end-to-end packet delay, \(\tau^{k}_{req}\), are distributed (normalized) in some way between individual domains with the following condition:  
\[\sum_{p=1}^{N}\tau^{p,k}_{req}\leq\tau^{k}_{req}, \tag{8}\]  
when  
\[\tau^{p,k}_{MP}\leq\tau^{p,k}_{req}. \tag{9}\]  
Therefore, in the case of ensuring the normalized QoS in a multidomain network, in addition to the value \(\tau^{k}_{req}\), the values \(\tau^{p,k}_{req}\) that must be taken into account when calculating both intra-domain and inter-domain routes are also set in advance. For an example of the network structure, as shown in Figure 1, the value \(\tau^{1,1}_{req}\) determines the requirements for the average packet delay of the first flow transmitted in the first domain between the routers \(R^{1}_{1}\) and \(R^{1}_{3}\) or \(R^{1}_{1}\) and \(R^{1}_{4}\). The value \(\tau^{2,1}_{req}\)characterizes the average delay requirements of the first packet flow transmitted in the second domain between the routers \(R^{2}_{1}\) and \(R^{2}_{4}\) or \(R^{2}_{2}\) and \(R^{2}_{4}\). Within the framework of this study, it is considered that \(\tau^{k}_{req}\) and \(\tau^{p,k}_{req}\) are predetermined. This raises the problem of formulating the conditions of normalized QoS on the network in terms of calculating expressions for \(\tau^{p,k}\). It is necessary to use the functionality of tensor analysis of networks to obtain such conditions.