# On Mathematical Modelling of Automated Coverage Optimization in Wireless 5G and beyond Deployments
## 3 Computational Concept
### Employment of Developed Models  
Based on the selected propagation model, we construct the initial matrix representing the relation between service centres and customer locations. This initial matrix is represented as a matrix (rather than linked list) since it brings the benefit of a fast access (\(\mathcal{O}(1)\)) to particular rows and columns. The process of data representation and manipulation is shown in Example 1. In Example 1, we define a \(P_{opt}\) as \(P_{opt}=\{P_{opt}\in\mathbb{R}\,\mid\,P_{opt}>-90\) dBm}. It is designed based on the specification of Reference Signal Received Power Categories (RSRPC) for Sub-6 GHz 5G cellular service mode that is briefly shown in Table 4[76; 77]. Here, we see that, if the signal power is \(-80\) dBm or higher, then the signal is excellent, while when the signal power drops to \(-100\) dBm and lower, it becomes impossible to establish or keep the wireless connection. **Example 1**. : _The following matrix expresses the relationship between the service centres and customer locations. Rows represent service centres and columns are related to the customer locations._  
\[ \begin{array}{cccccccc}&&\omit\span\omit\span\omit\span\omit\span\omit \span\omit\span\omit\span\omit\span\omit\span\omit\span\omit\span\omit\span \omit\span\omit\span\omit\span\omit\span\omit\span\omit\span\omit\span\omit \span\omit\  
Depending on \(P_{opt}\), the reachability matrix would be as follows:  
\[\begin{array}{cccccccc}1&2&3&4&5&6\\ 1&0&0&1&1&0\\ 2&0&1&1&0&1&1\\ 3&0&0&1&0&0&0\\ 4&1&1&0&1&1&1\\ 5&1&0&1&0&0&1\end{array}\end{array} \]  
Based on the reachability matrix, the matrix including customer locations capacities would be as follows (the number in round brackets of \(S\) means the maximal capacity of the service centre and the number in a particular column means the requested capacity for the customer location):  
\[ \begin{array}{cccccccc}S1(30,000)\\ S2(30,000)\\ S3(30,000)\\ S4(30,000)\\ S5(30,000)\\ \end{array}\begin{pmatrix}44&0&0&29&17&0\\ 0&26&28&0&17&51\\ 0&0&28&0&0&0\\ 44&26&0&29&17&51\\ 44&0&28&0&0&51\end{pmatrix} \]  
Here, we generally use non-unitary variables, but you can take this number to correspond to the network throughput demanded by a given customer location, or it can be a combination of multiple parameters. If we need to consider the existing services from models described in Section 2.3, we can do this in the same way as we did in [65] for the necessary services. This refers to adding dummy columns or directly setting \(x_{i}\) to 1. The whole concept of producing the optimal solution is presented in Algorithm 1. The computational concept for the developed models is shown in Figure 4.  
The decision node for an initial matrix represents the condition that if the initial matrix size is larger than \(X\cdot Y\), where \(X\) is the number of rows and \(Y\) is the number of columns (the value depends on the computation environment, where the run will be invoked), we should take into account parallel processing being used. For example, if the matrix has a dimension of 5000 \(\times\) 15,000, the application needs to calculate the prediction model 75,000,000 times. The parallel processing for that can be achieved by spliting the dataset into separate threads. Then the process waits for each thread to finish its computation to merge the results into the initial matrix. The next key decision is to consider the number of rows (\(X\)). In that case, we know from previous sections that this problem has at least the complexity of \(\mathcal{O}(2^{n})\). For that reason, for more than 60 rows (\(X>60\)) we can expect that the duration of computation with exact methods, on a typical desktop computer with 3 GHz processor, will be in the range of years [65]. The common exact methods used for calculations with \(X<60\) are: (i) branch-and-bound [78; 79; 80; 81]; and (ii) removal algorithm [82]. These exact methods are commonly applied together with relaxations algorithms that reduce the size of SCP-based problems.