## 3 Problem Formulation and Algorithm Development
### Problem Formulation
If all SBSs in each multistory building serve simultaneously in \(t\)\(\in\)\(\mathbf{T}\), then the aggregate capacity served by transceiver 1 for all SBSs per 3D cluster, as well as per building, is given by:  
\[\sigma_{\text{28-GHz, 3D cluster}}^{\text{Transceiver 1}}=\sum \
olimits_{s=1}^{S_{\text{F}}}\sigma_{\text{28-GHz, $s$}}^{\text{Transceiver 1}} \tag{16}\]  
\[\sigma_{\text{28-GHz, $L=1$}}^{\text{Transceiver 1}}=\epsilon_{\text{RF}} \times\sigma_{\text{28-GHz, 3D cluster}}^{\text{Transceiver 1}}\]  
Similarly, transceiver 2 for all SBSs per building operates in the 60 GHz spectrum, such that the capacity served by transceiver 2 for all SBSs per building is given by:  
\[\sigma_{\text{28-GHz, $s$}}^{\text{Transceiver 1}}=\sum\
olimits_{t\in T}\sum \
olimits_{i=1}^{M_{\text{28-GHz}}}\sigma_{t,i}\!\left(\rho_{t,i}\right) \tag{17}\]If all SBSs in each multistory building serves simultaneously in \(t\)\(\in\)T, the aggregate capacity served by transceiver \(1\) for all SBS per 3D cluster, as well as per building, is given by:  
\[\sigma_{\text{60-GHZ, 3D cluster}}^{\text{Transceiver 2}}=\sum\
olimits_{s=1}^{S_{ \text{F}}}\sigma_{\text{60-GHZ, $s$}}^{\text{Transceiver 2}} \tag{18}\]  
\[\sigma_{\text{60-GHZ, $L=1$}}^{\text{Transceiver 2}}=\varepsilon_{\text{RF}} \times\sigma_{\text{60-GHZ, 3D cluster}}^{\text{Transceiver 2}}\]  
where \(\sigma_{\text{60-GHZ, $s$}}^{\text{Transceiver 2}}=\sum\limits_{t\in T}\sum \
olimits_{i=1}^{M_{\text{60-GHZ}}}\sigma_{t,i}\Big{(}\rho_{t,i}\Big{)}\). Then, the total aggregate capacity served by transceivers 1 and 2 for all SBSs per building is given by:  
\[\sigma_{\text{MB, $L=1$}}=\sigma_{\text{28-GHZ, $L=1$}}^{\text{Transceiver 1}}+\sigma_{\text{60-GHZ, $L=1$}}^{\text{Transceiver 2}} \tag{19}\]  
\[\sigma_{\text{MB, $L=1$}}=\varepsilon_{\text{RF}}\times\big{(}\sigma_{\text{28-GHZ, 3D cluster}}^{\text{Transceiver 1}}+\sigma_{\text{60-GHZ, 3D cluster}}^{\text{Transceiver 2}}\big{)}\]  
Due to the short distance between the small-cell UE and its SBS, along with the low transmission power of SBSs, we assume similar indoor signal propagation characteristics for all \(L\) buildings for each macrocell. Then, by linear approximation, the system-level average aggregate capacity of the MNO for \(L>1\) is given by:  
\[\sigma_{\text{28-GHz,s}}^{\text{Transceiver 1}}=\sum\
olimits_{t\in T}\sum \
olimits_{i=1}^{M_{\text{28-GHz}}}\sigma_{t,i}\!\left(\rho_{t,i}\right) \tag{15}\]0  
\[\sigma_{\text{28-GHz,s}}^{\text{Transceiver 1}}=\sum\
olimits_{t\in T}\sum \
olimits_{i=1}^{M_{\text{28-GHz}}}\sigma_{t,i}\!\left(\rho_{t,i}\right) \tag{15}\]1  
The spectral efficiency (SE) for \(L\) buildings is then given by:  
\[\sigma_{\text{28-GHz,s}}^{\text{Transceiver 1}}=\sum\
olimits_{t\in T}\sum \
olimits_{i=1}^{M_{\text{28-GHz}}}\sigma_{t,i}\!\left(\rho_{t,i}\right) \tag{15}\]2  
Similarly, the energy efficiency (EE) for \(L\) buildings is given by:  
\[ \begin{array}{l}\sigma_{\text{EE}}^{\text{sys}}(L)=\\ \left(\begin{array}{l}(L\times S_{\text{F}}\times(P_{\text{28-GHZ}}+P_{\text {60-GHZ}}))+\\ (S_{\text{P}}\times P_{\text{2-GHZ,PC}})+(S_{\text{M}}\times P_{\text{2-GHZ,MC}} )\end{array}\right)\bigg{/}\Big{(}\sigma_{\text{cap}}^{\text{sys}}(L)/Q\Big{)}\end{array}  \tag{23}\]  
It should be noted that for the SE estimation, only the licensed spectra (i.e., 2 and 28 GHz spectra) of the MNO are considered, due to the licensing fee being paid by the MNO to use these bands. This is why the 60 GHz unlicensed spectrum is not accounted for in the SE estimation, because there is no charge to use this band. In other words, only the licensed spectra are considered as the effective spectra for the MNO.