# Advanced Tensor Solution to the Problem of Inter-Domain Routing with Normalized Quality of Service
## 2 Research Background  
Among the conventional hierarchical multidomain routing protocols, well-known OSPF, Integrated IS-IS, and PNNI are traditionally used [1; 2; 3; 11; 12]. Nevertheless, these protocols have significant drawbacks, such as using combinatorial algorithms for route calculation and the absence of explicit QoS guaranteeing mechanisms, while routing metrics are mainly based on the bandwidth of the links only. Here the need arises for implementing novel solutions that consider network hierarchy and QoS guarantees in terms of the average end-to-end packet delay for every distinct flow.  
Under analysis, some research results concern the multidomain SDN routing and specifics of softwarized architectures [13; 14; 15; 16; 17; 18; 19; 20; 21]. The most promising solutions are explained in Table 1.  
An analysis of existing works regarding hierarchical routing shows that the most promising approaches are grounded on flow-based models and optimization methods [15; 16; 17; 22; 23; 24]. Particular attention should be paid to solutions related to principles and postulates of the hierarchical multilevel systems theory [25; 26]. The main feature of such an approach lies in the decision-making hierarchy in the routing process. Moreover, the decomposition within the framework of a network mathematical model is used. In some works, it can be implemented by a system of differential equations [27]. However, in hierarchical routing, upper-level coordination is required, and it concerns the routing decisions of the lower levels.  
Regarding QoS routing [19; 22; 28; 29; 30; 31; 32; 33], the tensor methodology of network research seems to be relevant [31; 32; 33]. This class of solutions allows us to obtain the analytical  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Reference** & **Contribution** \\ \hline
[13] & The inter-domain communication in Time-Sensitive Networks (TSNs) to achieve \\  & end-to-end bounded latency has been proposed. \\  & Type of solution: heuristic. \\ \hline
[14] & The Inter-Domain Routing Brokering Plane (IDRBP) aimed at optimal scalability \\  & and resource management has been proposed in SDN-based architectures. \\  & Type of solution: heuristic. \\ \hline
[15] & The multidomain SDN end-to-end routing mechanism for mobility management has \\  & been specified. The optimization-based solution considers bandwidth, number of \\  & domains, and flow operations for mobility. \\  & Type of solution: optimization. \\ \hline
[16] & The software-defined internetworking (SDI) model has been formulated to extend \\  & SDN from intra-domain to inter-domain control. The corresponding prototype for \\  & the optimization algorithm has been developed and verified. \\  & Type of solution: optimization. \\ \hline
[17] & The new SDN-based Traffic Engineering (TE) method has been developed for large \\  & networks to solve intra-domain and inter-domain resource allocation jointly. \\  & Type of solution: optimization. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Existing solutions on SDN inter-domain routing.  
conditions for Quality of Service guaranteed under several indicators, such as average end-to-end packet delay, bandwidth, and packet loss probability.  
Therefore, in the present paper, we propose the advanced tensor solution to the problem of inter-domain routing with normalized Quality of Service, with its further application in Software-Defined Networks. The central concept of the approach is providing normalized QoS when the requirements for the end-to-end QoS are set for each domain in the form of corresponding norms. The presented solution further develops the [22; 23; 24] results.