# Transitioning to Society 5.0 in Africa: Tools to Support ICT Infrastructure Sharing
## 3 Mathematical Modelling and Problem Definition
### Signal Strength and Coverage Models  
It must be remembered, however, that several factors in addition to economic and emissions costs must be considered for telecommunications infrastructure sharing. If a load is reassigned (transferred) from one mast \(j\) to another mast \(i\), then there are implications in terms of network coverage, network operability and potentially network reliability issues that must be considered, some of which may even require human 'expert' judgement to be assessed. One way in which coverage and operability issues could be considered is to define a number \(R\) of receiver locations that are located at strategic areas of a map, which defines the total coverage area under consideration. Such locations could be critical areas of dense urban population, specific points of interest on the map such as the extremal edges or the center of a small town. Alternatively, a grid placed over the entire required coverage area could be employed. As elaborated below, as the number of locations increases, the accuracy of the discrete representation of the coverage area considered in the optimization problem increases, along with the complexity of the optimization problem itself; as with use of coverage simulation tools (e.g., Atoll) human judgement can be employed as needed to dimension the problem accordingly. Suppose that the distance from receiver location \(k\) to mast location \(i\) is measured at \(d_{k,i}\) meters, and mast \(i\) is operational. Then, the receiver signal strength \(r_{k,i}\) (in \(dB\)) from receiver location \(k\) to mast location \(i\), assuming a mast base radius of \(1\ m\), can be very well approximated as [5]:  
\[r_{k,i}=dBm_{i}-10.0\gamma_{k,i}\log_{10}(d_{k,i}) \tag{4}\]  
where \(dBm_{i}\) is the equivalent received signal strength at the base of the tower \(i\) (which can be taken as \(-43\) dB in the absence of other information) and \(\gamma_{k,i}\) is the path loss exponent of the channel between mast \(i\) and receiver \(k\) (which can be taken as \(\cong 4.0\) in the absence of other information). Both the received signal level at the base of the mast and path loss exponent of the channel can be adjusted as required for a specific network configuration [5]. Clearly, network simulation tools in addition to physical measurements can be deployed to obtain more accurate estimates of receiver power and configuration at these specified locations than raw application of Equation (4) and are encouraged in most cases. In the default (initial) condition for the network, the coverage of the network configuration is given by the indication of certain (assumed acceptable) levels of receiver signal strength at each of the receiver locations given some initial assignment of \(T\) transmitters to the \(N\) mass. The reassignment of transmitter \(j\) to mast \(i\) clearly has an impact on the distance, directionality, height and terrain of the signal channel and hence affects receiver signal strength through expression (4). Thus, returning to the previous example, although it may hold that \(f_{1}(l_{1}+l_{2})+f_{2}(0)<\)\(f_{1}(l_{1})+f_{2}(l_{2})\), it may well be that for receiver \(1\), \(r_{1,1}<\)\(<r_{1,2}\), and there is a loss of coverage indicated by reduction of signal strength at this location. It may well be the case that the assignment must be discarded due to not meeting performance requirements, as signal strength does not meet some minimum level \(r_{k,min}\). The discussions above may now be formalized into an optimization problem.