Q_l^{m\ast} &=& (1+\Gamma_m) Q_{l+1}^{m\ast} + \max_{k \in \mathcal{U}_l^m} \left\{ N_0\Gamma_m / H_m^k \right\}, \nonumber \\
 &=& N_0 \Gamma_m \sum_{i=l}^L (1+\Gamma_m)^{i-l} \max_{k \in \mathcal{U}_l^m} \left\{1/H_m^k\right\}, 
 %\nonumber \\
 %&& \hspace{1.3in} 
 \;\; l=1,2, \cdots,L. 
%\\
%Q_{L+1}^{m\ast} &=& 0,\end{aligned}$$ Recall that $Q_{L+1}^{m\ast} = Q_{L+1}^{m} = 0$, the optimal power allocation for Problem OPT-Power in this case is: $$%   P_l^{i\ast}=Q_l^{i\ast}-Q_{l+1}^{i\ast}, i=m, \mbox{ for all } l.
  P_l^{m'\ast}=\left\{ \begin{array}{ll}
     Q_l^{m\ast}-Q_{l+1}^{m\ast}, & m'=m, \mbox{ for all } l \\
     0, & m' \neq m, \mbox{ for all } l.
                      \end{array} \right.$$

#### Case II–MBS and One FBS \[subsubsec:caseII\]

We next consider the case with one MBS and one FBS (i.e., $M=1$), where each user has two choices: connecting to either the FBS or the MBS.

Recall that $\mathcal{U}_l^0$ and $\mathcal{U}_l^1$ are the subset of users who connected to the MBS and the FBS, respectively, and who request the level $l$ packet. Examining (\[eq:FoldTerm\]), we find that the total power of BS $m$ can be significantly reduced if one or more levels are not transmitted, since the exponent $c_l^m$ will not be increased in this case. Furthermore, consider the two choices: (i) not transmitting level $l$, and (ii) not transmitting level $l'>l$ from BS $m$. The first choice 