## 2 Problem Statement and Background  
In this paper, we focus on the minimization of total losses. The telecommunication network is presented in the form of a directed graph, the parameters and keys of which are given below:  
* number of network nodes, \(N\);
* channel bandwidth capacity, \(\mu_{ij}\), from the \(i\)th node to the \(j\)th one;
* channel buffer size, \(N_{ij}\), from the \(i\)th node to the \(j\)th one. Let us assume that \(\mu_{ij}=0\) if there is no communication channel between these nodes, and \(\mu_{ij}=-1\) if the probability of information loss in the corresponding channel is 0. Therefore, e.g., if each node is connected to itself, there is no information loss. The above-mentioned parameters characterize the network. Notion \"Target request\" is introduced to describe the network. Let the \"Target request\" be marked by \(g\) denoting the intensity of traffic transmitted from one node to another. Thus, e.g., writing \(g=[16;7;200]\) means that one should deliver 200 units of information per second from the 16 th to the 7 th node. We use the following notation:  
* number of target requests, \(Y\);
* matrix of target requests, \(G=[g_{i}]\);
* a specific target request, \(g_{i}=[start_{i};finish_{i};Q_{i}]\), where \(start_{i}\) is the source node, \(finish_{i}\) is the recipient node, and \(Q_{i}\) is the traffic intensity transmitted from the source node to the recipient node. The intensity of losses when transmitting information over each channel can be determined using the following equation:  
\[e_{ijk}=x_{ijk}p(x_{ij},\mu_{ij},N_{ij}). \tag{1}\]  
The physical meaning of the variables in (1) is as follows: \(e_{ijk}\) is the intensity of losses in the channel from the \(i\)th node to the \(j\)th one for the \(k\)th target request, \(x_{ijk}\) is the intensity of information flow in the channel from the \(i\)th node to the \(j\)th one for the \(k\)th source, \(x_{ij}\) is the total intensity of information flow in the corresponding channel, and \(p(x_{ij},\mu_{ij},N_{ij})\) is the probability of information loss for the channel from the \(i\)th node to the \(j\)th node, which is usually a strictly increasing function due to physical fundamentals. In this work, the objective function is defined as information losses when the required amount of information is delivered according to the target request matrices. Currently, there are two possible approaches to formulating the traffic loss optimization problem [5; 6; 14; 15; 16; 17; 18; 19; 20]. According to the first approach, the information densities, \(x_{ijk}\), are considered as variables. In this regard, we obtain a system of restrictions:  
\[e_{jk}=\sum_{i=1}^{n}\left[x_{ijk}\frac{\left(1-\frac{\sum_{i}x_{iji}}{\mu_{ ij}}\right)}{1-\left(\frac{\sum_{i}x_{iji}}{\mu_{ij}}\right)^{N_{ij}+1}} \left(\frac{\sum_{t}x_{ijk}}{\mu_{ij}}\right)^{N_{ij}}\right]=\sum_{i=1}^{n}x_ {ijk}-\sum_{i=1}^{n}x_{ijk}-c_{kj}+inp_{jk}. \tag{2}\]  
where \(c_{kj}\) is the total information rate provided when delivering data from the \(k\)th node to the \(j\)th one, \(inp_{jk}\) are constants determined by the speed of information received from the \(j\)th source, \(e_{jk}\) is the intensity of losses in the channel from the \(j\)th node to the \(k\)th one, and \(x_{ijt}\) is the intensity of information flow in the channel from the \(i\)th node to the \(j\)th node for the \(t\)th source. Obviously, the intensity of the information flow cannot be negative:  
\[x_{ijk}\geq 0. \tag{3}\]  
On the one hand, the advantages of this approach are its simplicity and a relatively small number of variables, which can be approximately estimated as \(n^{3}-n^{2}+n\) for a fully connected network. On the other hand, the large number of restrictions in the form of equality, which can similarly be approximately estimated as \(n^{2}\), is its major disadvantage. When the number of network nodes is \(n\leq 5\), the use of standard program libraries of conditional optimization algorithms makes it possible to solve simple problems with an acceptable runtime.