# On Developing Techniques for Sharing Satellite Spectrum with Indoor Small Cells in 5G +
## 3 Mathematical Analysis
### Estimation of an Optimal Number of Almost Blank Subframes (ABSs)
Likewise, since the FCC proposed to deploy SBSs in the 3.5 GHz satellite band (i.e., 3550 MHz-3650 MHz frequency band used for the satellite system globally [17]), we also assumed that the satellite system operates at 3.5 GHz (i.e., S-band) with a total system bandwidth of 40 GHz. Furthermore, the performance results were generated by a simulator built using the computational tool MATLAB R2012b. ### Defining Cluster Size for nOSS

To define a 3D cluster size in a building, using Table 1 and employing (A4) and (A5) in Appendix B, we can find that \(d_{\text{planer}}{}^{\star}\geq 12\) m and \(d_{\text{linear}}{}^{\star}\geq 7.07\) m or equivalently \(d_{\text{ver}}{}^{\star}\geq 5\) m (Figure 11). Allowing these minimum distances between co-channel SBSs requires SBSs on both the intra-floor and inter-floor levels to stay apart from one another by at least 1-tier, resulting in each 3D cluster (Figure 11) consisting of eight apartments (i.e., four apartments per floor multiplied by two floors). Since we assumed that each building consisted of 32 apartments, there were four 3D clusters in total per building (i.e., \(\xi_{\text{reuse}}=4\)), so the same satellite spectrum can be reused four times per building. ### Estimating the Number of Apartments Containing SBSs and the Amount of Satellite Spectrum

Recall that SBSs and CGC stations can be deployed in four ways within an apartment, as shown in Figure 2, and the probability of occurrence of a deployment option is equally likely with others so that the occurrence of a deployment option comprises of one-fourth of the total number of apartments per building. Hence, each of the four subsets of apartments corresponding to any deployment option consists of eight apartments per building. Since RBs of the satellite spectrum are allocated orthogonally to nodes per building in any TTI, and the occurrence of any deployment option is equally likely, for the fairness and simplicity in modeling, we assumed that the whole satellite spectrum was allocated equally to each subset of apartments so that the spectrum allocated to each subset of apartments was one-fourth of the total satellite spectrum (i.e., 10 MHz per subset of apartments for the satellite spectrum of 40 MHz). As mentioned earlier, out of the four subsets, only two subsets corresponded to include at least one SBS so only 50% of the total satellite spectrum can be utilized per building when considering an orthogonal allocation of spectrum resources. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters and Assumptions** & **Value** \\ \hline Type of SBSs & Closed subscriber group \\ \hline Channel state information & Ideal \\ \hline \(\lambda_{\text{SU}}\) and \(\lambda_{\text{SC}}\) & 2/8, 8/8 \\ \hline TTI \({}^{1}\) and scheduler time constant (\(t_{\text{c}}\)), and \(T_{\text{APP}}\) & 1 ms, 100 ms, and 8 ms \\ \hline Parameters and assumptions for nOSS technique & \(y_{\text{max,planer}}=8\); \(y_{\text{max,linear}}=2\), \(d_{\text{min}}=5\)m, \(\alpha_{\text{thr,planer}}=0.578\), \\  & \(\alpha_{\text{thr,linear}}=0.044\), \(d_{\text{ver}}=3\)m \\ \hline Taken from \({}^{1}\)[30] and \({}^{2}\)[31]. \\ \end{tabular}
\end{table} 
Table 1: _Cont._

## 6 Performance Result and Comparison

### Performance Result

#### 6.1.1 Capacity, Spectral Efficiency, and Energy Efficiency Performances

We considered two cases to evaluate the performances of OSS and nOSS. Case 1 considered four floors per building, whereas case 2 considered 20 floors per building, with a maximum of eight apartments per floor per building in each case. Recall that for \({d_{\mathrm{planet}}}^{*}\geq 12\) m and \({d_{\mathrm{linear}}}^{*}\geq 7.07\) m or equivalently \({d_{\mathrm{ver}}}^{*}\geq 5\) m, each 3D cluster (Figure 11) consists of eight apartments with four apartments per floor. Since in case 1 we assumed that each building consisted of four floors (i.e., 32 apartments), there were in total four 3D clusters of SBSs per building (i.e., \(\xi_{\mathrm{reuse}}=4\)).