\Gamma_m I_m^k, 
 \nonumber \\ 
 && \hspace{0.7in} 
 \mbox{ for all } k \in \mathcal{U}_l, l=1, \cdots, L \label{eq:cntrate2} \\
 && \hspace{-0.2in} Q_l^m \ge Q_{l+1}^m, l=1, \cdots, L \\
 && \hspace{-0.2in} \sum_{m=0}^M I_m^k=1, \mbox{ for all } k. \end{aligned}$$

For $l\le L$, constraint (\[eq:cntrate2\]) can be rewritten as: $$\begin{aligned}
\label{eq:QmlIneq}
 Q_l^m I_m^k \ge \left[ N_0 \Gamma_m / H_m^k + (1 + \Gamma_m) Q_{l+1}^m \right] I_m^k.\end{aligned}$$ Let $\mathcal{U}_l^m$ be the subset of users connecting to BS $m$ in $\mathcal{U}_l$. Since $Q_l^m \ge Q_{l+1}^m$, (\[eq:QmlIneq\]) can be rewritten as, $$\label{eq:QmlEqu}
  Q_l^m = \max \left\{ Q_{l+1}^m, \max_{k \in \mathcal{U}_l^m} \left[ N_0 \Gamma_m / H_m^k + (1 + \Gamma_m) Q_{l+1}^m \right] \right\}. 
%  Q_l^m = \max \left\{ Q_{l+1}^m, \max_{k \in \mathcal{U}_l^m} \left[ N_0 \frac{\Gamma_m}{H_m^k} + (1 + \Gamma_m) Q_{l+1}^m \right] \right\}.$$ From (\[eq:QmlEqu\]), we define a function $Q_l^m = F_m(Q_{l+1}^m,\mathcal{U}_l^m)$ as: $$\begin{aligned}
 \label{eq:FmDef}
%&& \hspace{-0.1in} 
F_m(Q_{l+1}^m,\mathcal{U}_l^m) 
%\label{eq:FmDef} \\
%&=&\hspace{-0.1in} 
= \left\{\begin{array}{l l}
 Q_{l+1}^m,& \mathcal{U}_l^m=\emptyset \\
 \max_{k\in\mathcal{U}_l^m} \left\{ \frac{N_0\Gamma_m}{H_m^k}+(1+\Gamma_m)Q_{l+1}^m \right\}, & \hspace{-0.05in} \mathcal{U}_l^m \neq \emptyset.
\end{array}\right.  %\nonumber\end{aligned}$$ Obviously, $F_m(Q_{l+1}^m,\mathcal{U}_l^m)$ is non-decreasing with respect to $Q_{l+1}^m$. It follows that $$\begin{aligned}
\label{eq:ObjFun3}
Q_1^m &=& F_m(Q_2^m, \mathcal{U}_1^m) \; = \; F_m(F_m(Q_3^m,\mathcal{U}_2^m),\mathcal{U}_1^m) \nonumber \\
&=& F_m(\cdots (F_m(Q_{L+1}^m, \mathcal{U}_L^m), \mathcal{U}_{L-1}^m), \cdots, \mathcal{U}_1^m) \nonumber \\
&=& F_m(\cdots (F_m(0, \mathcal{U}_L^m), \mathcal{U}_{L-1}^m), \cdots, \mathcal{U}_1^m).\end{aligned}$$

If 