## 3 Problem Formulation and Algorithm Development  
### Problem Formulation  
Let \(S_{\text{F}}\) denotes the maximum number of small cells per 3D cluster, such that \(s\in\{1,2,\ldots,S_{\text{F}}\}\) and \(S_{\text{F,total}}\) per building, meaning that \(S_{\text{F,total}}=\varepsilon_{\text{RF}}\times S_{\text{F}}\). Assume that \(L\) denotes the number of buildingsper macrocell coverage, such that \(l\in\{1,2,\ldots,L\}\) for the MNO. Assume that there are \(S_{\text{M}}\) macrocells in the MNO system and \(S_{\text{P}}\) picocells per macrocell. Let \(M_{\text{2-GHz}}\), \(M_{\text{28-GHz}}\), and \(M_{\text{60-GHz}}\) denote the number of resource blocks (RBs) in the 2 GHz microwave spectrum, 28 GHz licensed mmWave spectrum, and 60 GHz unlicensed mmWave spectrum, respectively, where a RB is equal to 180 kHz. Also assume that transceivers 1 and 2 in each small cell operate at the transmission powers of \(P_{\text{28-GHz}}\) and \(P_{\text{60-GHz}}\), respectively; whereas the transmission powers of macrocells and picocells are denoted as \(P_{\text{2-GHz,MC}}\) and \(P_{\text{2-GHz,PC}}\), respectively. The downlink received signal-to-interference-plus-noise ratio for UE at RB = \(i\) in the transmission time interval (TTI) = \(t\) can be expressed as:  
\[\rho_{t,i}=\left(P_{t,i}/\left(N_{t,i}^{\text{s}}+I_{t,i}\right)\right)\times H _{t,i} \tag{11}\]  
where \(P_{t,i}\) is the transmission power, \(N_{t,i}^{\text{s}}\) is the noise power, and \(I_{t,i}\) is the total interference signal power. Here, \(H_{t,i}\) is the link loss for a link between UE and a base station at RB = \(i\) in TTI = \(t\), and can be expressed in dB as:  
\[H_{t,i}(\text{dB})=\left(G_{\text{t}}+G_{\text{r}}\right)-\left(L_{\text{F}}+ PL_{t,i}\right)+\left(LS_{t,i}+SS_{t,i}\right) \tag{12}\]  
where \(\left(G_{\text{t}}+G_{\text{r}}\right)\) and \(L_{\text{F}}\) are the total antenna gain and connector loss, respectively. Here, \(LS_{t,i}\), \(SS_{t,i}\), and \(PL_{t,i}\) denote the large-scale shadowing effect, small-scale Rayleigh or Rician fading, and distance-dependent path loss, respectively, between a base station and a piece of UE at RB = \(i\) in TTI = \(t\). Using [PERSON]'s capacity formula, a link throughput at RB = \(i\) in TTI = \(t\) in bps per Hz is given by [17]:  
\[\sigma_{t,i}(\rho_{t,i})=\left\{ \begin{array}{ll}0,&\rho_{t,i}<-10\text{ dB}\\ \beta\log_{2}\!\left(1+10^{\left(\rho_{t,i}(\text{dB})/10\right)}\right),&-10 \text{ dB}\leq\rho_{t,i}\leq 22\text{ dB}\\ 4.4,&\rho_{t,i}>22\text{ dB}\end{array} \right\} \tag{13}\]  
where \(\beta\) denotes the implementation loss factor. The total capacity of all macro UE serving in the 2 GHz microwave spectrum of MNO can be expressed as:  
\[\sigma_{\text{2-GHz}}=\sum\
olimits_{t=1}^{Q}\sum\
olimits_{i=1}^{M_{\text{ 2-GHz}}}\sigma_{t,i}\!\left(\rho_{t,i}\right) \tag{14}\]  
where \(\sigma\) and \(\rho\) are responses over \(M_{\text{2-GHz}}\) RBs of all macro UE in \(t\)\(\in\)\(T\). Recall that transceiver 1 of an SBS operates in the 28 GHz spectrum, such that the capacity served by transceiver 1 of an SBS is given by:  
\[\sigma_{\text{28-GHz,s}}^{\text{Transceiver 1}}=\sum\
olimits_{t\in T}\sum \
olimits_{i=1}^{M_{\text{28-GHz}}}\sigma_{t,i}\!\left(\rho_{t,i}\right) \tag{15}\]  
Note that the same 28 GHz spectrum is used for each 3D cluster of small cells.