is another UE (still indexed by $k$ without loss of generality) reusing the same pilot symbol. This is modeled through a binary stochastic variable $\chi_{0kj} \in \{0,1\}$, where $\chi_{0kj}=1$ means that UE $k$ in cell $j$ uses the same pilot as the typical UE and thus causes pilot contamination. This occurs with probability $K/B$. Similarly, $\chi_{0kj}=0$ means that there is currently no pilot contamination from cell $j$, and it occurs with probability $1-K/B=1-1/\beta$. Therefore, the typical UE $k$ transmits a single pilot symbol (e.g., $s_{0k}=1$) and $\mathrm{BS}_0$ receives the following signal: $$\label{eq:received-pilot}
\begin{split}
{\mathbf{z}}_{0k} &= \sqrt{p_{0k}} (\sqrt{1-\epsilon^2} + e_{0k}) {\mathbf{h}}_{00k} \\ &+ \sum_{j \in \Psi_{\lambda} } \chi_{0kj} \sqrt{p_{jk}} (\sqrt{1-\epsilon^2} + e_{jk}) {\mathbf{h}}_{0jk} + {\bf{n}}_{0}.
\end{split}$$ We compute the minimum mean-squared error (MMSE) estimate of ${\mathbf{h}}_{00k}$ from the observation ${\mathbf{z}}_{0k}$.

$$\begin{aligned}
 \tag{8}  \label{eq:effective-SINR}
\frac{ M(1-\epsilon^2)^2   }{  \!\!\!\!\Bigg( K + {{\underset{{j \in \Psi_{\lambda}}}{\sum}}} {\overset{K}{\underset{i=1}{\sum}}} 
\frac{d_{jji}^{\alpha}}{ d_{0ji}^{\alpha} } + \frac{\sigma^2}{  \rho}  \Bigg) \!\!\Bigg(  1   + \frac{1}{\beta} {{\underset{{l \in \Psi_{\lambda} }}{\sum}}}   \frac{d_{llk}^{\alpha}}{ d_{0lk}^{\alpha} } + \frac{\sigma^2}{\rho} \Bigg) 
+ M (1-\epsilon^2) \Bigg( \frac{1}{\beta} {{\underset{{j \in \Psi_{\lambda} }}{\sum}}}  \!\!
\left( \frac{ d_{jjk}^{\alpha} }{ d_{0jk}^{\alpha}} \right)^2  + \epsilon^2 \Bigg)  }\end{aligned}$$

$$\begin{aligned}
 \tag{10} \label{eq:average-SINR}
\frac{ M (1-\epsilon^2)^2  }{ 
\Big( K + \frac{\sigma^2}{  \rho}  \Big) \! \Big(  1   + 