## 3 System Model
### User Type Probability  
Normally, the user type of a typical UE \(l\) can be decided by the relationship between the biased RSRP from its nearest MBS and its nearest PBS as follow:  
\[l=\left\{ \begin{array}{l}pm,\text{if }\rho P_{m}hr_{m}^{-\alpha}>B_{p}P_{p}hr_{p}^{- \alpha}\\ pp,\text{if }P_{p}hr_{p}^{-\alpha}<P_{m}hr_{m}^{-\alpha}<B_{p}P_{p}hr_{p}^{- \alpha}\\ um,\text{if }\rho D_{m}hr_{m}^{-\alpha}<B_{p}P_{p}hr_{p}^{-\alpha}<P_{m}hr_{m}^{- \alpha}\\ up,\text{if }P_{m}hr_{m}^{-\alpha}<P_{p}hr_{p}^{-\alpha}\end{array} \right., \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(\lambda_{m},\lambda_{p},\lambda_{u}\) & Density of MBS, PBS and UE \\ \(\textbf{L},l\) & Set of user types, indication of the user type \\ W & Total spectrum bandwidth \\ \(h\) & Channel fast fading gain \\ \(\alpha\) & Large-scale path loss exponent \\ \(\sigma^{2}\) & Thermal noise \\ \(P_{m},P_{p}\) & Maximum transmission power of MBS and PBS \\ \(\rho\) & Power reduction factor \\ \(\theta\) & PSF ratio \\ \(B_{p}\) & Pico CRE bias \\ \(\gamma_{l}\) & SINR of a typical UE \(l\) \\ \(I_{m},I_{p}\) & Full power aggregate interference from macro tier and pico tier \\ \(r_{m},r_{p}\) & Distance from a UE to its nearest MBS and its nearest PBS \\ \(A_{l}\) & Probability of a typical UE belongs to the user type \(l\) \\ \(k_{c},k_{c},k_{p}\) & Factor of macro-cell center region, pico CRE region and pico-cell original coverage region \\ \(r_{l}\) & Distance from a typical UE \(l\) and its serving BS \\ \(f_{l}(r)\) & PDF of the distance between a UE and its serving BS \\ \(\lambda_{l}\) & Density of BSs associated with user type \(l\) \\ \(R_{l}\) & Mean achievable downlink data rate of a typical UE \(l\) \\ \(W_{l}\) & Spectrum bandwidth allocated to a typical UE \(l\) \\ \(N_{l},N_{l}^{total}\) & Mean number of UEs with user type \(l\) in a Voronoi cell, total number of UEs with user type \(l\) \\ \(P_{m,s},P_{p,s}\) & Static power of MBS and PBS \\ \(\hat{P}_{m}\) & Proportion between maximum transmission power of MBS and that of PBS \\ \(\hat{P}_{p}\) & Proportion between maximum transmission power of PBS and that of MBS \\ \(\lambda_{p,m}\) & Proportion between PBS density and MBS density \\ \(Rotal\) & Total network throughput \\ \(Pototal\) & Total network power consumption \\ \(\rho^{\prime}\) & Approximate value of power reduction factor \\ \(\hat{B}_{p^{\prime}}^{*},\hat{\lambda}_{p}^{*}\) & Near-optimal value of pico CRE bias, power reduction factor and PBS density \\ \(B_{p^{\prime}}^{*},\hat{\lambda}_{p}^{*},EE^{*}\) & Optimal value of pico CRE bias, power reduction factor, PBS density and EE \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations summary. where \(B_{p}\) is the pico CRE bias. In Equation (2), the conditions for determining user type can be further translated from biased RSRP based inequation into distance based inequation, which is shown as Equation (3). \[l=\left\{ \begin{array}{l}pm,\text{if }k_{c}r_{m}<r_{p}\\ pp,\text{if }k_{p}r_{m}<r_{p}<k_{c}r_{m}\\ um,\text{if }k_{c}r_{m}<r_{p}<k_{c}r_{m}\\ up,\text{if }k_{p}r_{m}>r_{p}\end{array} \right., \tag{3}\]  
where \(k_{c}=\left[B_{p}P_{p}/\left(P_{m}\rho\right)\right]^{1/\alpha}\), \(k_{c}=\left(B_{p}P_{p}/P_{m}\right)^{1/\alpha}\) and \(k_{p}=\left(P_{p}/P_{m}\right)^{1/\alpha}\) are defined as macro-cell center region factor, pico CRE region factor and pico-cell original coverage region factor, respectively [8], determining the coverage bound of the macro-cell center region, the pico CRE coverage region and the pico-cell original coverage region, respectively. In order to obtain the probabilities of four user types, the following lemma is proposed. **Lemma 1**. : _Due to the locations of MBSs and PBSs follow two iid. PPPs, given two arbitrary coefficient values of \(n_{a}\) and \(n_{b}\), the probability of \(n_{a}r_{m}<r_{p}<n_{b}r_{m}\) can be expressed as:_  
\[