# 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)
Let \(\lambda_{\text{SC}}\) and \(\lambda_{\text{SU}}\) denote the average rate of arrivals of UEs to the CGC stations and SBSs over a certain duration of time T so that the solution of an optimal number of TTIs allocated to SBSs can be found by solving the following problem, subject to the above concept as the condition for optimality. We considered that the UEs of SBSs operate during non-ABSs per APP, whereas the UEs of CGC stations operate during ABSs, so that in favor of the SBSs (i.e., aTMS), the problem can be formulated as follows. \[ \begin{array}{ll}\min&T_{\text{ABS}}\\ \text{subject to :}&\text{(a)}\ \lambda_{\text{SC}}/\lambda_{\text{SU}}=T_{ \text{nABS}}/T_{\text{APP}}\\ &\text{(b)}\ \{S_{\text{SBS}},S_{\text{CGC}}\}\in\{\mathbf{S_{\text{SBS}}},\mathbf{S_{ \text{CGC}}}\}\\ &\text{(c)}\ \forall T_{\text{APP}}\left(\begin{array}{c}T_{\text{ABS}}+T_{ \text{nABS}}=T_{\text{APP}}:\\ T_{\text{ABS}}=_{0}\ \Lambda T_{\text{nABS}}=_{0}\end{array}\right)\\ &\text{(d)}\ P_{\text{T,SBS}}=P_{\text{T,CGC}}\leq P_{\text{T,SBS}}^{\text{max}} \end{array}  \tag{23}\]

An optimal value in favor of small cell UEs is given by

\[T_{\text{nABS}}^{*}=[\lambda_{\text{SC}}/(\lambda_{\text{SU}}+\lambda_{\text {SC}})]\times T_{\text{APP}}.\]

**Proof. ** Using constraint (23) (a) and (23) (c), and allowing a favor to the UEs of SBSs (i.e., aTMS), the optimal value of \(T_{\text{nABS}}\) can be given by

\[ \begin{array}{ll}\lambda_{\text{SC}}/\lambda_{\text{SU}}=T_{\text{nABS}}/T_ {\text{ABS}}\\ \Rightarrow\lambda_{\text{SC}}/\lambda_{\text{SU}}=(T_{\text{APP}}-T_{\text{ ABS}})/T_{\text{ABS}}\\ \Rightarrow(\lambda_{\text{SC}}/\lambda_{\text{SU}})\times T_{\text{ABS}}=T_{ \text{APP}}-T_{\text{ABS}}\end{array} \]

\[ \begin{array}{ll}\Rightarrow(\lambda_{\text{SC}}/\lambda_{\text{SU}}+1) \times T_{\text{ABS}}=T_{\text{APP}}\\ \Rightarrow T_{\text{ABS}}=1/(\lambda_{\text{SC}}/\lambda_{\text{SU}}+1) \times T_{\text{APP}}\\ \Rightarrow T_{\text{ABS}}=\lambda_{\text{SU}}/(\lambda_{\text{SC}}+\lambda_{ \text{SU}})\times T_{\text{APP}}\\ \Rightarrow T_{\text{APP}}-T_{\text{nABS}}=\lambda_{\text{SU}}/(\lambda_{ \text{SC}}+\lambda_{\text{SU}})\times T_{\text{APP}}\\ \Rightarrow T_{\text{nABS}}=(1-\lambda_{\text{SU}}/(\lambda_{\text{SC}}+\lambda _{\text{SU}}))\times T_{\text{APP}}\\ \Rightarrow T_{\text{nABS}}^{*}=[(\lambda_{\text{SC}}/(\lambda_{\text{SC}}+ \lambda_{\text{SU}}))\times T_{\text{APP}}]\\ \Rightarrow T_{\text{nABS}}^{*}=[(\lambda_{\text{SC}}/(\lambda_{\text{SC}}+ \lambda_{\text{SU}}))\times T_{\text{APP}}]\\ \end{array} \]

Then, \(T_{\text{ABS}}^{*}=(1-T_{\text{nABS}}^{*})\times T_{\text{APP}}\). \(\square\)

In other words, the number of TTIs allocated to SBSs in those apartments where they are collocated with CGC stations per building or per 3D cluster is directly proportional to the ratio of the average rate of arrival of the UEs to SBSs to the sum of the average rates of arrival of UEs to both the SBSs and CGC stations. ## 5 Parameters, Assumptions, and Preliminary Estimations

### Default Parameters and Assumptions

Following [22], Table 1 shows the default parameters and assumptions considered for the performance evaluation of both orthogonal and non-orthogonal spectrum sharing techniques. Since we limited evaluating the performance of aTMS only in a hybrid iSMS/aTMS, no channel link models for the satellite system (i.e., iSMS) are given explicitly in Table 1. However, we explicitly modeled the terrestrial channel models depending on the type of BSs (e.g., macrocell BS, picocell BS, or SBS) and the environmental profiles (e.g., indoor or outdoor). The performance evaluation and analysis carried out are completely independent of the satellite channel models and hence are not affected by the satellite channel models.