## 3 Problem Formulation
### Co-primary Shared Access
#### 3.3.2 Static Spectrum Renting Technique  
Let \(\rm M_{\rm SSR}\) denote the number of RBs of MNO 2 that is rented to MNO 1 such that \(M_{\rm SSR}<M_{\rm MNO,1}\). Then, the total spectrum of MNO 1 due to spectrum renting is given by:  
\[M_{\rm MNO,1,SR}=M_{\rm MNO,1}+M_{\rm SSR} \tag{24}\]  
For the StSR technique, Since the transceiver 2 of all SBSs of MNO 1 per building operate only at the shared rented spectrum, i.e., \(M_{\rm StSR}\) RBs, in all TTIs \(t\in\mathbf{T}\), then the capacity served by an SBS is then given by,  
\[\sigma^{\rm StSR}_{s}=\sum_{t\in\mathbf{T}}\sum_{i=1}^{M_{\rm SSR}}\sigma_{t,i}( \rho_{t,i}) \tag{25}\]  
For all SBSs per 3D building, if serving simultaneously in \(t\in\mathbf{T}\) at the rented spectrum of \(M_{\rm StSR}\) RBs on transceiver 2, the aggregate capacity per 3D building for static spectrum renting technique is then given by,  
\[\sigma^{\rm SC}_{\rm StSR,Trans\,2}=\sum_{s=1}^{S_{\rm F}}\sigma^{\rm SSR}_{s} \tag{26}\]  
Like DySP, in StSR, the transceiver 1 of each SBS operates at the MNO 1 spectrum, such that the aggregate capacity of all \(N\) macro UEs for \(M_{\rm MNO,1}\) RBs and \(Q\) TTIs is given by:  
\[\sigma^{\rm SC}_{\rm SSR,Trans\,1}=\sigma^{\rm SC,DedA}_{\rm MNO,1}=\sum_{s=1} ^{S_{\rm F}}\sigma_{s,\rm DedA} \tag{27}\]  
Then, the overall aggregate capacity served by all SBSs enabled both the MNO 1 spectrum and the rented MNO 2 spectrum in StSR technique is given by in a 3D building, i.e., \(L=1\):  
\[\sigma^{\rm SC}_{\rm StSR,MB}=\sigma^{\rm SC}_{\rm StSR,Trans\,1}+\sigma^{\rm SC }_{\rm StSR,Trans\,2} \tag{28}\]Conversely, using (4), the system level capacity of MNO 1 for the dual-band enabled SBSs is given by:  
\[\sigma^{\text{SYS}}_{\text{SGR,MB}}=\sigma^{\text{MC,WIM}}_{\text{MNO,1}}+\sigma^{ \text{SC}}_{\text{SGR,MB}} \tag{29}\]  
Now, the average spectral efficiency of MNO 1 after sharing the spectrum with its SBSs is given by:  
\[\sigma^{\text{SE}}_{\text{SGR,MB}}=\sigma^{\text{SYS}}_{\text{SGR,MB}}/((M_{ \text{MNO,1}}+M_{\text{SGR}})\times Q) \tag{30}\]  
Note that in estimating spectral efficiency above, we assume that the rented spectrum given to MNO 1 is with exclusive right and MNO 2 gets paid for the license of the rented spectrum such that MNO 2 cannot use the rented spectrum for a particular duration of time as negotiated between MNOs. Hence, the rented spectrum is treated as the licensed spectrum. Similarly, the average energy efficiency of MNO 1 per building in Joules/bit (J/b) is given for multiband enabled SBSs by:  
\[\sigma^{\text{EE}}_{\text{SGR,MB}}=\Bigg{(}\begin{array}{c}\left(S_{\text{F}} \times\left(\left|\mathbf{I}^{\text{MNO},1}_{\text{non-ABS}}\right|/\left| \mathbf{I}\right|\right)\times P_{\text{SC,1}}\right)+\\ \left(S_{\text{F}}\times P_{\text{SC,2}}\right)+\left(S_{\text{P}}\times P_{ \text{PC}}\right)+\left(S_{\text{M}}\times P_{\text{MC}}\right)\end{array} \Bigg{)}/\left(\sigma^{\text{SYS}}_{\text{DYP,MB}}/Q\right) \tag{31}\]  
Now, following the assumption for dynamic spectrum pooling for \(L>1\), the average aggregate capacity of MNO 1 for static spectrum renting technique is roughly given by:  
\[\sigma^{\text{SYS}}_{\text{SGR,MB},L}=\sigma^{\text{MC,WIM}}_{\text{MNO,1}}+ \left(L\times\sigma^{\text{SC}}_{\text{SGR,MB}}\right) \tag{32}\]  
Now, the spectral efficiency is given by:  
\[\sigma^{\text{SE}}_{\text{SGR,MB},L}=\sigma^{\text{SYS}}_{\text{SGR,MB},L}/((M _{\text{MNO,1}}+M_{\text{SGR}})\times Q) \tag{33}\]  
Similarly, the energy efficiency is given by:  
\[\sigma^{\text{EE}}_{\text{SGR,MB},L}=\Bigg{(}\begin{array}{c}\left(L\times S _{\text{F}}\times\left(\left|\mathbf{I}^{\text{MNO,1}}_{\text{non-ABS}} \right|/\left|\mathbf{I}\right|\right)\times P_{\text{SC,1}}\right)+\\ \left(L\times S_{\text{F}}\times P_{\text{SC,2}}\right)+\left(S_{\text{P}}\times P _{\text{PC}}\right)\\ +\left(S_{\text{M}}\times P_{\text{MC}}\right)\end{array}\Bigg{)}/\left( \sigma^{\text{SYS}}_{\text{SGR,MB},L}/Q\right) \tag{34}\]