Algorithm 1. Optimal bandwidth allocation using the sub-gradient method.

1. Declare, instantiate, and initialize s = 0 as a counter of the number of iterations s according to (35).

2. Input the initial values of $B_k^L$, $B_k^W$, $p$, $q$, and ${\theta }_k,1\le k\le K$.

3. Initialize maximum tolerance threshold $\left(\epsilon \right)$ and $v\left(0\right)$.

4. Calculate gradients $\frac{\partial L_2}{\partial B_k^L}$$\mathrm{and} \frac{\partial L_2}{\partial B_k^W}$ according to (39) and (40).

5. Solve $\frac{\partial L_2}{\partial B_k^L}=0$ $\mathrm{and} \frac{\partial L_2}{\partial B_k^W}=0$ then calculate current optimal values $B_k^{L\ast }\left(s-1\right)$$\mathrm{and} B_k^{W\ast }\left(s-1\right)$$\mathrm{based} \mathrm{on} P_L^{\ast }$$\mathrm{and} P_W^{\ast }$, as obtained from the optimal power-allocation mechanism described by (34).

6. Calculate the current optimal effective network capacity, $E_i^{\ast }\left(s-1\right)$.

7. Increment the counter for the next iteration of the algorithm: $s=s+1$.

8. Continue solving $\frac{\partial L_2}{\partial B_k^L}=0$$\mathrm{and} \frac{\partial L_2}{\partial B_k^W}=0$ and obtaining current optimal values $B_i^{L\ast }\left(s\right)$$\mathrm{and} B_i^{W\ast }\left(s\right)$ according to optimal power allocations $P_L^{\ast }$$\mathrm{and} P_W^{\ast }$.

9. $\mathrm{Calculate} E_i^{\ast }\left(s\right)$.

10. $\mathrm{Update} p$$\mathrm{and} q$ according to the iteration formula in (42).

11. $\mathrm{Loop} \mathrm{until} \left|E_{}^{\ast }\left(s\right)-E_{}^{\ast }\left(s-1\right)\right|\le \epsilon$.

12. End.