# Advanced Tensor Solution to the Problem of Inter-Domain Routing with Normalized Quality of Service
## 3 Decomposition Model of Inter-Domain QoS Routing in Communication Network  
Suppose that the network structure is represented as an oriented graph \(\Gamma=(R,\ W)\), where \(R\) is the set of vertices that model routers and \(W\) is the set of edges that describe the network links (Table 2). In the general case, a packet flow is generated when providing a particular information service on the network. We denote the number of flows circulating in the network by \(K\). Then value \(|K|=\tilde{K}\) is the set's power, which quantitatively characterizes the total number of flows in the network. For each \(k\)th flow (\(k\in K\)), its average packet rate (intensity), \(\lambda_{req}^{k}\), is known measured in packets per second (pps) and defines the requirements for the network bandwidth allocated to that flow. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Symbol**} & \multicolumn{1}{c}{**Meaning**} \\ \hline \(\Gamma=(R,\ W)\) & Oriented graph of network structure \\ \(R\) & Set of vertices (routers)s \\ \(W\) & Set of edges (network links) \\ \(K\) & Number of flows in the network \\ \(\lambda_{req}^{k}\) & Average packet rate of the \(k\)th flow (\(k\in K\)) \\ \(N\) & Number of domains \\ \(\Gamma^{p}=(R^{p},\ W^{p})\) & Oriented subgraph of the \(p\)th domain \\ \(R^{p}\) & Set of links of the \(p\)th domain \\ \(W^{p}\) & Set of links of the \(p\)th domain \\ \(m_{p}\) & Number of routers in the \(p\)th domain \\ \(n_{p}\) & Number of links in the \(p\)th domain \\ \(B^{p}\) & Set of border routers of the \(p\)th domain (\(B^{p}\subset R^{p}\)) \\ \(B_{in}^{p,k}\) & Subset of the border routers through which the packets of the \(k\)th flow \\ \(B_{out}^{p,k}\) & income into the \(p\)th domain \\ \(B_{out}^{p,k}\) & Subset of the border routers through which the packets of the \(k\)th flow outgo \\ \(\varrho_{i,j}^{p}\) & Bandwidth of the link \(W_{i,j}^{p}\in W^{p}\) of the \(p\)th domain \\ \(x_{i,j}^{p,k}\) & Routing variables that characterize the fraction of the \(k\)th packet flow in the \\ \(\lambda_{i,j}^{p,k}\) & link \(W_{i,j}^{p}\in W^{p}\)of the \(p\)th domain \\ \(\lambda_{i,j}^{p,k}\) & Average intensity of the \(k\)th packet flow in the link \(W_{i,j}^{p}\in W^{p}\) \\ \(\overset{-k}{x}_{p}^{k}\) & Routing vector with coordinates \(\overset{-k}{x}_{p}^{k}\) for the \(k\)th flow in the \(p\)th domain \\ \(\overset{\text{ck}}{C}_{p,q}^{k}\) & Matrix of the \(p\)th and \(q\)th domains interaction \\ \(\overset{\text{t}}{Mp}\) & Average packet end-to-end delay of the \(k\)th flow \\ \(\overset{\text{t}}{r_{req}^{k}}\) & Maximum allowable value for the average packet end-to-end delay of the \(k\)th flow \\ \(\overset{p,k}{Mp}\) & Average packet delay of the \(k\)th flow in the \(p\)th network domain \\ \(\overset{R}{p\eta}\) & Router through which the \(k\)th flow incomes to the network \\ \(R_{out}^{p,k}\) & Router through which the \(k\)th flow outgoes from the network \\ \(\overset{\text{t}}{V^{p}}\) & Set of links belonging to the \(p\)th domain (network geometrization) \\ \(n_{p}^{-}\) & Number of routers in the \(p\)th domain with imaginary network elements \\ \(m_{p}^{-}\) & Number of links in the \(p\)th domain with imaginary network elements \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation summary. When developing a decomposition model of inter-domain routing, suppose that the network consists of \(N\) interconnected subnets--domains. Then let each \(p\)th individual network domain be described by the graph \(\Gamma\) subgraph \(\Gamma^{p}=(R^{p},\,W^{p})\), where \(R^{p}=\left\{R_{i}^{p};\,i=\overline{1,m_{p}}\right\}\) is the set of routers of the \(p\)th domain, \(W^{p}=\left\{W_{i,j}^{p};\,i,j=\overline{1,m_{p}},\,i\
eq j\right\}\) is the set of links connecting the routers of the \(p\)th domain, and \(m_{p}\) and \(n_{p}\) are the total numbers of the routers and communication links in the \(p\)th domain, respectively.