## 4 Optimality Criteria for Load Balancing Problem Solutions  
The calculation of the control variables responsible for routing (1), load balancing based on the principles of traffic engineering (5), and limiting traffic at the network edge (3) \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Meaning** \\ \hline \(c\) & Weighting coefficient responsible for the load balancing in the network \\ \hline \(PR^{k}\) & \(k\)th packet flow priority \\ \hline \(\gamma\) & Threshold for flow rate limiting that enters the network via border routers \\ \hline \(w\) & Weighting coefficient (\(w>1\)), which determines how many times the threshold \(\gamma\) is more important than the utilization upper bound \(\alpha\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._is proposed in the process of solving the optimization problem, as it is carried out in studies [21; 22]. However, the quality of applied solutions traditionally depends on the form and content of the selected optimality criterion. In this research, various optimality criteria are proposed and investigated, considering the relation between control variables and packet flows parameters (priorities and intensities) in different ways. The first type of optimality criterion is based on the linear objective function minimum as follows:  
\[J=\sum_{k\in K}v^{k}\cdot\beta^{k}+c\cdot\alpha\rightarrow\mathbf{min}, \tag{6}\]  
where \(v^{k}\) and \(c\) are weighting coefficients that should consider the packet flows' parameters routed and limited at the network edge. For traffic limitation at the network edge to occur only after the bound reaches its maximum value, i.e., at \(\alpha=\alpha_{TH}\), the next conditions should be met when choosing weighting coefficients:  
\[v^{k}>c\ (k\in K). \tag{7}\]  
The solution, which is based on the use of criterion (6) and models (1)-(5), will be called _\"Traffic Engineering routing, linear limitation model\"_ (**TER-LLM**) later in the study. This study will consider several options for forming the weighting coefficients \(v_{k}\). The _first option_ is based on the use of such weighting coefficients:  
\[v^{k}=PR^{k}+1\ (k\in K), \tag{8}\]  
where \(PR^{k}\) is the \(k\)th packet flow priority. In an IP network using the 3 bits of IP precedence in the packet header for prioritization, the priority value ranges from 0 to 7. In contrast, for the DSCP (differentiated services code point), the priorities vary from 0 to 63 [29]. Let us consider in more detail the physical sense of the optimality criterion (6) when coefficients (8) are presented in the following form:  
\[v^{k}=\frac{(PR^{k}+1)\cdot\lambda^{k}}{\lambda^{k}}.\]  
Then, the optimality criterion (6) can be rewritten as follows:  
\[J=\sum_{k\in K}\frac{(PR^{k}+1)\cdot\lambda^{k}\cdot\beta^{k}}{\lambda^{k}}+c \cdot\alpha\rightarrow\mathbf{min}.\]  
Given that \(b^{k}=\lambda^{k}\cdot\beta^{k}\), we have the following:  
\[J=\sum_{k\in K}\frac{(PR^{k}+1)\cdot b^{k}}{\lambda^{k}}+c\cdot\alpha \rightarrow\mathbf{min}.\]  
Therefore, when using (6) and (8), flow limitation will be carried out according to their intensity (\(b^{k}\)) weighted directly and proportionally relative to the packet flows' priorities (\(PR^{k}\)), and inversely proportional relative to the \(k\)th flow intensity (\(\lambda^{k}\)). On the other hand, we can say that the flow limitation will be carried out according to its fraction (\(\beta^{k}\)) weighted in direct proportion to the packet flow priorities (\(PR^{k}\)). Within the scope of the _second option_, weighting coefficients of the following type are used:  
\[v^{k}=(PR^{k}+1)\cdot\lambda^{k}\ (k\in K). \tag{9}\]  
From (9), it becomes clear that, in this case, the limitation in the set of flows will be carried out according to their intensity (\(b^{k}\)) weighted in direct proportion to the priorities of the packet flows (\(PR^{k}\)). That is, priority flows should be limited less intensively.