## 3 Problem Formulation
### Co-primary Shared Access  
#### 3.3.1 Dynamic Spectrum Pooling Technique  
If \(M_{\text{MNO,2}}\) denotes the total number of RBs in MNO 2 spectrum the total spectrum due to spectrum pooling at CoRS is given by:  
\[M_{\text{MNO,T}}=M_{\text{MNO,1}}+M_{\text{MNO,2}} \tag{13}\]  
Since the transceiver 2 of all SBSs of MNO 1 per building operate at the spectrum of MNO 2 at CoRS during non-ABSs \(t_{\text{non-ABS}}\in\mathbf{T}\mathbf{T_{ABS}^{MNO,2}}\), the capacity served by an SBS at MNO 2 spectrum is then given by:  
\[\sigma_{s}^{\text{DySP}}=\sum_{t=t_{\text{non-ABS}}\in\mathbf{T}}\sum_{ \begin{subarray}{c}\mathbf{T_{ABS}^{MNO,2}}\\ \mathbf{T_{ABS}}\end{subarray} }^{M_{\text{MNO,2}}}\sigma_{t,i}(\rho_{t,i}) \tag{14}\]  
Now following (6), the aggregate capacity for all SBSs per 3D building due to MNO 2 spectrum is given by:  
\[\sigma_{\text{DySP,Trans 2}}^{\text{SC}}=\sum_{s=1}^{S_{\text{F}}}\sigma_{s}^{ \text{DySP}} \tag{15}\]  
Recall that the transceiver 1 of each SBS operates at the MNO 1 spectrum, which contributed the capacity as given by (6) as follows:  
\[\sigma_{\text{DySP,Trans 1}}^{\text{SC}}=\sigma_{\text{MNO,1}}^{\text{SC,DedA}}= \sum_{s=1}^{S_{\text{F}}}\sigma_{s,\text{DedA}} \tag{16}\]  
Now using (15) and (16), the aggregate capacity of all SBSs by per building enabled with both spectrums of MNO 1 and MNO 2 for the DySP technique is given by:  
\[\sigma_{\text{DySP,MB}}^{\text{SC}}=\sigma_{\text{DySP,Trans 1}}^{\text{SC}}+ \sigma_{\text{DySP,Trans 2}}^{\text{SC}} \tag{17}\]  
So, using (4) and (17), the aggregate capacity of MNO 1 for all dual-band enabled SBSs per building is given by:  
\[\sigma_{\text{DySP,MB}}^{\text{SYS}}=\sigma_{\text{MNO,1}}^{\text{MC,WIM}}+ \sigma_{\text{DySP,MB}}^{\text{SC}} \tag{18}\]  
**Remark 2**. : _Note that, in estimating the spectral efficiency, we consider only the licensed spectrum of an MNO, not the shared or reused spectrum from other MNOs or systems. Hence, the capacity that is achieved by the spectrum of MNO 2 using SBSs for MNO 1 can be interpreted as the capacity achieved because of sharing the same MNO 2 spectrum with MNO 1 [26] such that the effective spectrum of MNO 1 is its licensed spectrum of \(M_{\text{MNO,1}}\) RBs only._  
Now, after sharing the spectrum with its SBSs, the average spectral efficiency of MNO 1 is given by:  
\[\sigma_{\text{DySP,MB}}^{\text{SE}}=\sigma_{\text{DySP,MB}}^{\text{SYS}}/(M_ {\text{MNO,1}}\times Q) \tag{19}\]Similarly, the average system-level energy efficiency of MNO 1 after sharing spectrums with its SBSs per building in Joules/bit (J/b) is given by [13]:  
\[\sigma^{\rm EE}_{\rm DySP,MB}=\left( \begin{array}{l}S_{\rm F}\times\left(|T^{ \mathbf{MNO},1}_{\mathbf{non-ABS}}|/|\mathbf{T}|\right)\times P_{\rm SC,1}\\ \left(S_{\rm F}\times\left(|T^{\mathbf{MNO},2}_{\mathbf{non-ABS}}|/|\mathbf{ T}|\right)\times P_{\rm SC,2}\right)+\\ +\left(S_{\rm P}\times P_{\rm PC}\right)+\left(S_{\rm M}\times P_{\rm MC} \right)\end{array} \right)/\left(\sigma^{\rm SYS}_{\rm DySP,MB}/Q\right) \tag{20}\]  
where \(T^{\mathbf{MNO},1}_{\mathbf{non-ABS}}\) denotes the set of non-ABSs when SBSs operate at the MNO 1 spectrum to serve small cell UEs and \(T^{\mathbf{MNO},2}_{\mathbf{non-ABS}}\) denotes the set of non-ABSs when SBSs serve small cell UEs at the MNO 2 spectrum. For \(L>1\), applying the assumption for (10), the average aggregate capacity of MNO 1 is roughly given by for DySP:  
\[\sigma^{\rm SYS}_{\rm DySP,MB,L}=\sigma^{\rm MC,WIM}_{\rm MNO,1}+\left(L \times\sigma^{\rm SC}_{\rm DySP,MB}\right) \tag{21}\]  
Now, the spectral efficiency is given by:  
\[\sigma^{\rm SE}_{\rm DySP,MB,L}=\sigma^{\rm SYS}_{\rm DySP,MB,L}/\left(M_{ \rm MNO,1}\times Q\right) \tag{22}\]  
Similarly, the energy efficiency is given by:  
\[\sigma^{\rm EE}_{\rm DySP,MB,L}=\left(