# On Mathematical Modelling of Automated Coverage Optimization in Wireless 5G and beyond Deployments
## 1 Introduction  
The deployments of 5G wireless networks have been rapidly growing these days. They provide not only low latency networks but also high throughput for different kinds of devices and services (e.g., Virtual Reality (VR), Augmented Reality (AR), drones and connected cars). Further, these networks serve also the huge variety of customer and industrial Internet of Things (IoT) applications such as connected security systems, intelligent lights in smart homes, detection sensors, smart grids and others [1; 2; 3; 4]. Due to this significant increase in the range of devices connected to the network, authors in [5] expect that, in 2023, there will be 29 billion wireless devices connected to the Internet. It is evident that not all of these wireless devices will be connected using 5G networks, however, it is confirmed by the following studies [6] that 5G networks will be the dominant technology on the market. For that reason, the operators need to consider a large number of simultaneously connected devices that lead to important tasks for the effective deployment and redesign of network infrastructure since, according to [5], 5G networks will generate nearly 3 times more traffic than 4G systems. Here, the challenge for network operators is to efficiently find the optimal placement of gNodeB (gNodeB is the 5G term for network equipment that transmits and receives wireless communications between the user equipment and a mobile network) nodes to deploy or optimizethe cellular network infrastructure [7]. Having a solution to this task is especially important since it provides benefits mainly in Total Cost of Ownership (TCO) as it avoids the over-provisioning since the final network infrastructure contains only the needed number of gNodeB nodes. Currently, the most widely-used tool for network planning in the industry is Atoll Radio Planning Software (ARPS) [8]. The network designer in this tool inserts a specification for a gNodeB node and then places the gNodeB node on a map manually verifying its coverage based on the provided heatmap. This planning process is shown in Figure 1[9; 10]. Here, the first two steps, that is, radio network definition and propagation analysis, require a lot of additional manual work that can be reduced by solving the problem of automated searching for the optimal gNodeB location. In ARPS, there is a possibility to improve this manual planning using a module called Automatic Cell Planning (ACP). Based on the documentation [10], it could improve the existing networks by tuning parameters that can easily be changed remotely such as antenna electrical tilt and power. Further, this module can optimize a network planning phase by (i) selecting antennas; (ii) modifying the antenna azimuth; (iii) setting the mechanical downtilt of the antenna; (iv) changing the antenna height; and (v) choosing sites from a list of candidate sites [10]. The variant provided by ARPS considers selecting sites (gNodeB locations), without the consideration of advanced aspects as is the expected customer throughput in the defined areas to be covered. The whole idea of deploying or optimizing the network is shown in Figure 2. The provided solution is fully automated. That is an important advantage over the existing software solutions. Figure 1: Conventional approach to cellular network planning. The proposed model has four aspects: (i) radio transmission (the transmission range, bidirectional/unidirectional, frequencies, etc. ); (ii) resource allocation (the budget represented for the deployment); (iii) system architecture (types of gNodeB to use in the deployment, restricted places, etc. ); and (iv) mobile subscriber (the expected number of users in the selected area and the expected average user traffic). Taking into account the parameters, we can compute the optimal gNodeB placement for the given area. To solve such a task, we need to develop a mathematical model (see Section 2) and suitable implementation that will fit the above-mentioned automatic network design in an optimal way.