FOR $m=0$ TO $M$
  4:    $\;\;\;\;\;\;$ $\Delta_l^m=\Gamma_m(1+\Gamma_m)^{c_l^m}W_m(\mathcal{S}_l^m)$;
  5:    $\;\;\;$ END FOR
  6:    $\;\;\;$ Set $\Omega=\{1,\cdots,M\}$ and $\Psi=\emptyset$;
  7:    $\;\;\;$ WHILE ($\Omega \neq \emptyset$)
  8:    $\;\;\;\;\;\;$ $m'=\arg\min_{m\in\Omega} \Delta_l^m$;
  9:    $\;\;\;\;\;\;$ Compute $\Delta'=\Gamma_0(1+\Gamma_0)^{c_l^0}W_0(\bigcup_{m\in\overline{\Psi\cup m'}}\mathcal{S}_l^m)$;
  10:   $\;\;\;\;\;\;$ IF ($(\sum_{m\in\Psi\cup m'}\Delta_l^m + \Delta') < \Delta_0$)
  11:   $\;\;\;\;\;\;\;\;\;$ Add $m'$ to $\Psi$;
  12:   $\;\;\;\;\;\;\;\;\;$ $\Delta_0=\sum_{m\in\Psi}\Delta_l^m + \Delta'$;
  13:   $\;\;\;\;\;\;$ END IF
  14:   $\;\;\;\;\;\;$ Remove $m'$ from $\Omega$;
  15:   $\;\;\;$ END WHILE
  16:   $\;\;\;$ IF ($\Psi = \emptyset$)
  17:   $\;\;\;\;\;\;$ $\mathcal{U}_l^0=\mathcal{S}_l^0$;
  18:   $\;\;\;\;\;\;$ $c_l^0=c_l^0+1$;
  19:   $\;\;\;\;\;\;$ Set $\mathcal{U}_l^m = \emptyset$, for all $m \neq 0$;
  20:   $\;\;\;$ ELSE
  21:   $\;\;\;\;\;\;$ $\mathcal{U}_l^0=\bigcup_{m\in\overline{\Psi}}\mathcal{S}_l^m$;
  22:   $\;\;\;\;\;\;$ IF ($|\Psi|<M$)
  23:   $\;\;\;\;\;\;\;\;\;$ $c_l^0=c_l^0+1$;
  24:   $\;\;\;\;\;\;$ END IF
  25:   $\;\;\;\;\;\;$ FOR $m \in \Psi$
  26:   $\;\;\;\;\;\;\;\;\;$ $c_l^m=c_l^m+1$;
  27:   $\;\;\;\;\;\;\;\;\;$ $\mathcal{U}_l^m=\mathcal{S}_l^m$;
  28:   $\;\;\;\;\;\;$ END FOR
  29:   $\;\;\;$ END IF
  30:   END FOR
  31:   FOR $l=L$ TO $1$
  32:   $\;\;\;$ FOR $m=0$ TO $M$
  33:   $\;\;\;\;\;\;$ $Q_l^m=F_m(Q_{l+1}^m,\mathcal{U}_l^m)$ and $P_l^m=Q_l^m-Q_{l+1}^m$;
  34:   $\;\;\;$ END FOR
  35:   END FOR
  ----- ------------------------------------------------------------------------------------------------------------------------

  : Power Allocation Algorithm For Case III

\[tab:Case3Algo\]

Performance Evaluation \[sec:sim3\]
-----------------------------------

We evaluate the performance of the proposed power allocation algorithms using MATLAB^TM^. Three scenarios corresponding to the three cases in Section \[sec:alg\] are simulated: (i) Case I: a single MBS; (ii) Case II: one 