# Advanced Tensor Solution to the Problem of Inter-Domain Routing with Normalized Quality of Service
## 8 Conclusions  
In summary, the work proposed and investigated the method of hierarchical-coordination inter-domain routing in a Software-Defined Network, with the provision of normalized Quality of Service. The study of several numerical examples confirmed the efficiency of the proposed method. This method provides an adaptive nature of packet flow routing under changing requirements (norms) for the Quality-of-Service level in terms of the average end-to-end packet delay, bandwidth in the network, and its distinct domains. Within the framework of the proposed method, the problem of inter-domain QoS routing was presented in an optimization form with a quadratic optimality criterion (19). The principle of goal coordination from the theory of hierarchical multilevel control systems is used to solve the optimization problem. This mathematical apparatus allowed us to correctly formulate and ensure the solution of problems that are assigned to two hierarchical levels. The lower level (level of SDN domain controllers) was responsible for calculating intra-domain routes based on minimizing expression (21) according to constraints (3), (4), (8), (10), and (18). The upper level (network SDN controller level) was responsible for coordinating the lower-level solutions by fulfilling the conditions of inter-domain interaction (5) to ensure the consistency of inter-domain routes within the gradient procedure (22). Figure 11: Dependence of the coordination procedure number of iterations on the network domains number. Coordination of routing solutions was completed in the case of approaching the gradient (22) to zero. The study of the proposed method of inter-domain QoS-routing on several computational examples confirmed its efficiency in ensuring normalized Quality of Service. It is experimentally established that the method converged to the optimal solution for a finite number of iterations (22). For the network structure selected as a test (Figure 2), the number of iterations of the coordination procedure (22) provided the appropriate setting of the gradient search and did not exceed three. Reducing the number of such iterations helps decrease the amount of service traffic transmitted on the network between routers and SDN controllers at different levels and minimize the overall time to solve the problem of inter-domain QoS routing. The method provided the high scalability of routing solutions, as routing tasks in individual domains were solved independently based on local information about the status of communication links and routers of these domains that positively affects the efficiency of network management and overhead. In ensuring the normalized Quality of Service in multidomain SDN, the main problem was coordinating the order of routing at the domain border, which passed through more than one router. The studies confirmed the convergence of the method to the desired optimal solutions for a finite number of iterations, on average, for two or three iterations. The number of such iterations could increase critically (to 6-8) only if the Quality-of-Service level norms in the domains met these domains' limits. Future research is concerned with improving the presented method by enhancing its flexibility and providing end-to-end Quality of Service without prior normalizing. In addition, modifications will be made to the mathematical model by using other types of coordination, such as predicting and evaluating interactions. Conceptualization, O.L., M.Y. and J.P.; software, O.L. and O.Y. ; validation, O.L. and O.Y. ; formal analysis, O.L., O.Y., M.Y. and J.P.; investigation, M.Y. ; resources, M.Y. ; data curation, O.L. ; writing--original draft preparation, O.L., J.P. and O.Y. ; writing--review and editing, J.P. and O.Y. ; visualization, O.L. and J.P.; supervision, M.Y. and J.P.; project administration, J.P. and O.Y. ; funding acquisition, J.P. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable.