## 3 Problem Formulation
### Licensed Shared Access  
Denote \(M_{\text{SPS}}\) as the number of RBs in the space-satellite system (SPS) spectrum. Since the transceiver 2 of all SBSs of MNO 1 per building operate at the SPS spectrum bandwidth at CoRS during non-ABSs \(f^{\text{SPS}}_{\text{non-ABS}}\in\mathbf{T}\big{|}\mathbf{\tau}^{\text{SPS }}_{\text{ABS}}\), the aggregate capacity from the SPS spectrum is then for a single SBS is given by:  
\[\sigma^{\text{LSA}}_{s}=\sum_{t=f^{\text{SPS}}_{\text{non-ABS}}\in\mathbf{T} \backslash\mathbf{\tau}^{\text{SPS}}_{\text{ABS}}}\sum_{i=1}^{M_{\text{SPS}}} \sigma_{t,i}(\rho_{t,i}) \tag{35}\]  
Hence, for \(S_{\text{F}}\) small cells, the aggregate capacity per 3D building is given by:  
\[\sigma^{\text{SC}}_{\text{LSA,Tran\,2}}=\sum_{s=1}^{S_{\text{F}}}\sigma^{ \text{LSA}}_{s} \tag{36}\]  
Since the transceiver 1 of each SBS operates at the MNO 1 spectrum, the capacity is given by (6) as follows:  
\[\sigma^{\text{SC}}_{\text{LSA,Tran\,1}}=\sigma^{\text{SC,DedA}}_{\text{MNO,1}}= \sum_{s=1}^{S_{\text{F}}}\sigma_{s,\text{DedA}} \tag{37}\]  
Hence, the overall aggregate capacity served by all SBSs enabled with multiband in a 3D building is given by:  
\[\sigma^{\text{SC}}_{\text{LSA,MB}}=\sigma^{\text{SC}}_{\text{LSA,Tran\,1}}+ \sigma^{\text{SC}}_{\text{LSA,Tran\,2}} \tag{38}\]Conversely, using (4) the system level capacity of MNO 1 for the multiband enabled SBSs with LSA is given by:  
\[\sigma_{\text{LSA,MB}}^{\text{SYS}}=\sigma_{\text{MNO,1}}^{\text{MC,WIM}}+\sigma_{ \text{LSA,MB}}^{\text{SC}} \tag{39}\]  
Now, following the consideration mentioned in Remark 2 the average system-level spectral efficiency of MNO 1 after sharing the incumbent satellite spectrum with its SBSs is given by:  
\[\sigma_{\text{MNO,1}}^{\text{LSA,SE}}=\sigma_{\text{LSA,MB}}^{\text{SYS}}/(M_{ \text{MNO,1}}\times Q) \tag{40}\]  
Similarly, the average system-level energy efficiency of MNO 1 after sharing satellite and MNO 1 spectrums with its SBSs per building in Joules/bit (J/b) is given by:  
\[\sigma_{\text{MNO,1}}^{\text{LSA,EE}}=\left(\begin{array}{l}\left(S_{\text{F} }\times\left(\left|\mathbf{T_{\text{non-ABS}}^{\text{MNO,1}}}\right|/\left|\mathbf{T }\right|\right)\times P_{\text{SC,1}}\right)+\\ \left(S_{\text{F}}\times\left(\left|\mathbf{T_{\text{non-ABS}}^{\text{PSPS}}} \right|/\left|\mathbf{T}\right|\right)\times P_{\text{SC,2}}\right)+\\ +\left(S_{\text{P}}\times P_{\text{PC}}\right)+\left(S_{\text{M}}\times P_{ \text{MC}}\right)\end{array}\right)/\left(\sigma_{\text{MNO,1}}^{\text{LSA, SYS}}/Q\right) \tag{41}\]  
For \(L>1\), applying the assumption for (10), the average aggregate capacity of MNO 1 is roughly given by for LSA:  
\[\sigma_{\text{LSA,MB,}L}^{\text{SYS}}=\sigma_{\text{MNO,1}}^{\text{MC,WIM}}+ \left(L\times\sigma_{\text{LSA,MB}}^{\text{SC}}\right) \tag{42}\]  
Now, the spectral efficiency for \(L\) buildings is given by:  
\[\sigma_{\text{LSA,MB,}L}^{\text{SE}}=\sigma_{\text{LSA,MB,}L}^{\text{SYS}}/(M_ {\text{MNO,1}}\times Q) \tag{43}\]  
Similarly, the energy efficiency for \(L\) buildings is given by:  
\[\sigma_{\text{LSA,MB,}L}^{\text{EE}}=\left(\begin{array}{l}\left(L\times S_{ \text{F}}\times\left(\left|\mathbf{T_{\text{non-ABS}}^{\text{MNO,1}}}\right|/ \left|\mathbf{T}\right|\right)\times P_{\text{SC,1}}\right)+\\ \left(L\times S_{\text{F}}\times\left(\left|\mathbf{T_{\text{non-ABS}}^{\text{PSPS}} }\right|/\left|\mathbf{T}\right|\right)\times P_{\text{SC,2}}\right)+\\ +\left(S_{\text{P}}\times P_{\text{PC}}\right)+\left(S_{\text{M}}\times P_{ \text{MC}}\right)\end{array}\right)/\left(\sigma_{\text{LSA,MB,}L}^{\text{ SYS}}/Q\right) \tag{44}\]