## 2 System Model
### Modeling 3D Clusters of Millimeter-Wave Multiband Small Cells
#### 2.2.2 Modeling of 28 GHz Interference
Hence, to find an optimal minimum distance in both intra-floor and inter-floor levels, such as in [9], we also consider separating the problem into two sub-problems--sub-problem 1 for intra-floor levels and sub-problem 2 for inter-floor levels, as follows. **Sub-Problem 1**: \[ \begin{array}{ll}\min&d_{\rm CCl,intra}\\ \text{subject to}&\alpha^{\rm nor,total}_{\rm CCl,intra}\leq I_{\rm optimal,intra }\\ &P_{\rm SC}\leq P_{\rm SC,max}\end{array} \] (2)
**Sub-Problem 2**: \[ \begin{array}{ll}\min&d_{\rm CCl,inter}\\ \text{subject to}&\alpha^{\rm nor,total}_{\rm CCl,inter}\leq I_{\rm optimal, inter}\\ &P_{\rm SC}\leq P_{\rm SC,max}\end{array} \] (3)  
where \(P_{\rm SC,max}\) denotes the maximum transmission power of a small-cell transceiver operating in the 28 GHz mmWave band. Hence, solving the above two sub-problems for an optimal minimum distance of \(d_{\rm CCl,intra}\)\({}^{*}\) in the intra-floor level and \(d_{\rm CCl,inter}\)\({}^{*}\) in the inter-floor level gives an optimal minimum cluster size \(\Theta_{\rm 3D}\)\({}^{*}\). (a) Intra-Floor Interference  
The received co-channel interference power in the intra-floor level for small-cell UE can be expressed as follows:  
\[\alpha_{\rm CCl,intra}[\text{dB}]=P_{t}-PL[\text{dB}]\] \[\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})[\text{dB}]=19-\left( 61.38+17.97\log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)\] \[\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})[\text{dB}]=-\left(42.3 8+17.97\log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)\] \[10\times\log_{10}(\alpha_{\rm CCl,intra}(d_{\rm CCl,intra}))=- \left(42.38+17.97\log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)\] \[\log_{10}(\alpha_{\rm CCl,intra}(d_{\rm CCl,intra}))=-\left(42.3 8+17.97\log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)/10\] \[\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})=10^{-\left(42.38+17.97 \log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)/10}\] \[\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})=10^{-\left(42.38+17.97 \log_{10}(d_{\rm CCl,intra})+X_{\Delta}/10\right)}\]  
Since mmWave signals are highly attenuated by internal walls and the transmission power of a small cell is typically low, considering only the first tier of co-channel interferers is sufficient to incorporate the co-channel interference effect for small-cell UE. Considering the worst case scenario, the maximum co-channel interference is experienced by UE when the co-channel interferer is located at distance \(d_{\rm CCl,intra}=d_{\rm min}\). Hence, the maximum value of co-channel interference can be expressed as follows:  
\[\alpha^{\rm max}_{\rm CCl,intra}(d_{\rm CCl,intra})=10^{-\left(4.238+1.797 \log_{10}(d_{\rm min})+X_{\Delta}/10\right)}\]  
The normalized value of intra-floor interference from a co-channel interferer for small-cell UE can be expressed as:  
\[\alpha^{\rm nor}_{\rm CCl,intra}(d_{\rm CCl,intra}) =\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})/\alpha^{\rm max}_{ \rm CCl,intra}(d_{\rm CCl,intra})\] \[\alpha^{\rm nor}_{\rm CCl,intra}(d_{\rm CCl,intra}) =10^{\left(-4.238-1.797\log_{10}(d_{\rm CCl,intra})-X_{\Delta}/10 +4.238+1.797\log_{10}(d_{\rm min})+X_{\Delta}/10\right)}\] \[\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})[\text{dB}]=19-\left( 61.38+17.97\log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)\]0\[\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})[\text{dB}]=19-\left( 61.38+17.97\log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)\]1 \[\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})[\text{dB}]=19-\left( 61.38+17.97\log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)\]2 \[\alpha_{\rm CCl,intra}(d_{\rm CCl,intra})[\text{dB}]=19-\left( 61.38+17.97\log_{10}(d_{\rm CCl,intra})+X_{\Delta}\right)\]3  
(b) Inter-Floor Interference  
The expression for inter-floor interference can be derived similarly, except that additional floor penetration loss needs to be accounted for.