# On Mathematical Modelling of Automated Coverage Optimization in Wireless 5G and beyond Deployments
## 1 Introduction
### Literature Review
#### 1.1.1 Location Covering Problems
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Authors** & **Model** & **Description** & **Published** \\ \hline [PERSON] [27] & SCP & The base covering model for all other models. Finding the minimal number of services to cover all the demands. & 1957 \\ \hline [PERSON] [13] & Location Set Covering Problem & Extension of SCP considering services and demands locations. & 1970 \\  & (LSCP) & & \\ \hline [PERSON] et al. [28] & Maximal Covering Location Problem (MLCP) & Describes the situation of a restricted number of services with the best coverage sought. & 1974 \\ \hline Plane and & Existing Service System & Extension of LSCP addressing the need to consider the existing services. & 1977 \\ [PERSON] [29] & Location Set Covering Problem (ESS-LSCP) & & \\ \hline [PERSON] [30] & Location Set Covering Problem & Extension of LSCP addressing the need of specific demand to be covered & 1979 \\  & with Facility Types (LSCP-FT) & by specific services. & \\ \hline [PERSON] [31] & Site Quality Maximal Covering & Takes into account the quality of the service provided for each demand. & 1986 \\  & Location Problem (SQ-MCLP) & & \\ \hline Current and & Capacitated Location Set & The first article proposing capacities in LSCP. This model could not be used in network deployment since it does not force the entire demand at a particular node being assigned to the same facility. It assumes that the demand is split among facilities (see Equation (13)). & 1988 \\ \hline [PERSON] and & Maximal Availability Location & This model is characterized by an intention to maximize the availability & 1989 \\ [PERSON] [33] & Problem (MALP) & Provided by a-reliable coverage & \\ \hline [PERSON] and & Capacitated Location Set & Models considering services and capacities to satisfy the demands using the closest service. & 1996 \\ [PERSON] [34] & Covering Problem with Closest Assignment (CLSCP-CA) & & \\ \hline [PERSON] and & Generalized Maximal Covering & Generalized variant of MLCP distinguishing the value of benefits & 2002 \\ [PERSON] [35] & Location Problem (CMCLP) & received over a series of coverage ranges. & \\ \hline [PERSON] and & CLSCP-EA & Models considering services and capacities to equally satisfy & 2009 \\ [PERSON] [12] & & the demands. & \\ \hline [PERSON] and [PERSON] [56] & LSCP and MLCP with Threshold & This model considers a threshold meaning that each service needs some portion of demand to be viable for each type of service. & 2016 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: An overview of core mathematical models for covering problems. service centres and the third one is about service centre capacities that assign the fragmented capacities to the customer locations. In this fashion, an overview of related mathematical models for covering problems is shown in Table 2. For further mathematical model details, see the following publications [32; 37; 38; 39; 40; 41; 42; 43; 44; 45]. Even if these models are dealing with capacities for location-based SCP, they are not applicable in 5G and beyond deployment since they do not take into account the fact that the user device requested capacity (bandwidth) could not be split over more than one gNodeB at the same time. Further, these models do not take into account interferences in any way. In our use-case, we need to consider the aspects such as wireless interference, direct (capacity) assignment of customer locations to a service centre and existing services in a given area. From the above-mentioned review, we deduce that the already proposed modifications of the SCP are not suitable for the use-case of optimization of network infrastructures. For that reason, the main contribution of this paper lies in the development of new mathematical models to provide a solution to the emerging scenario of 5G and beyond wireless deployments.