## 2 System Model
### Modeling 3D Clusters of Millimeter-Wave Multiband Small Cells
#### 2.2.2 Modeling of 28 GHz Interference  
To model 28 GHz mmWave path loss, we consider the omnidirectional, multi-frequency, combined polarization, close-in free space (CIF) reference distance with the large-scale, frequency-dependent path loss exponent model, as given below. \[PL[\mathrm{dB}]=10\log_{10}(4\Pi d_{0}f_{\mathrm{c}}/c)^{2}+10n(1+b(f-f_{0}/f_{ 0}))\log_{10}(d/d_{0})+X_{\Delta}\]  
where \(n\) denotes the path loss exponent, \(b\) represents the slope of the linear frequency dependence of the path loss,\(f_{0}\) represents a fixed reference frequency serving as the balancing point of the linear frequency dependence of \(n\), and \(X_{\Delta}\) is the Gaussian random variable, with standard deviation \(\Lambda\) representing large-scale signal variation about the mean path loss due to shadowing. For line-of-sight (LOS) cases, \(n=2.1\), \(b=0.32\), \(\Lambda=9.9\) dB, and \(f_{0}=51\) GHz. Putting these values in the above equation, for \(f=f_{\mathrm{c}}=28\) GHz, the following can be obtained. \[PL[\mathrm{dB}]=61.38+\left(10\times 2.1\times(1+(0.32\times((28-51)/51))) \log_{10}(d)\right)+X_{\Delta}\]  
\[PL[\mathrm{dB}]=61.38+17.97\log_{10}(d)+X_{\Delta}\]  
2.3 Modeling Intra-floor and Inter-floor Co-Channel Interference Effects and 3D Clusters of Small Cells  
Previously, we presented a model for microwave spectra in [9] to find the size of a 3D cluster, which we adopt in this paper to extend its applicability to mmWave spectra. In line with this, the co-channel interference effect experienced by small-cell UE is given by the total co-channel interference effect from both intra-floor and inter-floor levels, i.e., \(\alpha^{\mathrm{nor,total}}_{\mathrm{CCI}}=\alpha^{\mathrm{nor,total}}_{ \mathrm{CCI,intra}}+\alpha^{\mathrm{nor,total}}_{\mathrm{CCI,inter}}\) subject to \(\alpha^{\mathrm{nor,total}}_{\mathrm{CCI}}\leq I_{\mathrm{optimal,total}}\). Here, \(\alpha^{\mathrm{nor,total}}_{\mathrm{CCI,intra}}\) and \(\alpha^{\mathrm{nor,total}}_{\mathrm{CCI,inter}}\) denote the total interference effect that small-cell UE experience in intra-floor and inter-floor levels, respectively. Here, \(I_{\mathrm{optimal,total}}=I_{\mathrm{optimal,intra}}+I_{\mathrm{optimal,inter}}\) denotes the total optimal value of co-channel interference in both intra-floor and inter-floor levels set by the MNO, while \(I_{\mathrm{optimal,intra}}\) and \(I_{\mathrm{optimal,inter}}\) denote the optimal values of co-channel interference in intra-floor and inter-floor levels, respectively. Note that a reduction in the size of a 3D cluster corresponds to an increase in reuse of any spectrum for a given number of small cells within a building. Hence, the optimal size of a 3D cluster requires the size of the cluster to be minimized. The minimum size can be found by solving the following minimization problem:  
\[ \begin{array}{ll}\mathrm{minimize}&\Theta_{\mathrm{3D}}\\ \mathrm{subject\ to}&\mathrm{(a)}&\alpha^{\mathrm{nor,total}}_{\mathrm{CCI}} \leq I_{\mathrm{optimal,total}}=\left(I_{\mathrm{optimal,intra}}+I_{\mathrm{ optimal,inter}}\right)\\ &\mathrm{(b)}&\alpha^{\mathrm{nor,total}}_{\mathrm{CCI}}=\alpha^{\mathrm{nor,total}}_{\mathrm{CCI,intra}}+\alpha^{\mathrm{nor,total}}_{\mathrm{CCI,inter}}\\ &\mathrm{(c)}&\alpha^{\mathrm{nor,total}}_{\mathrm{CCI,intra}}\leq I_{ \mathrm{optimal,intra}}\\ &\mathrm{(d)}&\alpha^{\mathrm{nor,total}}_{\mathrm{CCI,inter}}\leq I_{ \mathrm{optimal,inter}}\end{array}  \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Obstacle** & **Penetration Loss (dB)** \\ \hline Floor (reinforced concrete) 1 & 55 \\ Internal wall 2 & 6.84 \\ External wall (brick) 2 & 28 \\ \hline \hline \end{tabular} Note: 1 taken from [13; 14; 15]; 2 taken from [16].  
\end{table} 
Table 1: Floor penetration loss in the 28 GHz mmWave spectrum band. However, finding the minimum size of a 3D cluster of small cells requires finding the minimum distance between co-channel interferers in both intra-floor and inter-floor levels.