Algorithm 1. The optimal mode selection

1: Initialize system parameters such as T, $f_{\mathrm{op}}$, $R_{\mathrm{th}}$, K, B, $P_{\mathrm{AP}}$, $F_0$, $\alpha$, $M_c$, $N_0$, $\xi$,

2: Calculate optimal ${\tau }_{\mathrm{ipt}\left(\mathrm{local}\right)}^{*}$ according to Lemma 1,

3: Calculate optimal ${\rho }_{\left(\mathrm{local}\right)}^{*}$ and ${\tau }_{\mathrm{cpt}}^{*}$ according to Theorem 1,

4: Calculate the minimal energy requirement (i.e., $E_{\mathrm{u}\left(\mathrm{local}\right)}$) by substituting ${\tau }_{\mathrm{ipt}\left(\mathrm{local}\right)}^{*}$, ${\rho }_{\left(\mathrm{local}\right)}^{*}$, and ${\tau }_{\mathrm{cpt}}^{*}$ into the

objective function of Problem $P_1$,

5: Calculate optimal ${\tau }_{\mathrm{ipt}\left(\mathrm{offload}\right)}^{*}$ according to Theorem 2 by using the bisection method,

6: Calculate optimal ${\rho }_{\left(\mathrm{offload}\right)}^{*}$, ${\tau }_{\mathrm{u}-\mathrm{f}}^{*}$, and $P_{\mathrm{u}-\mathrm{f}}^{*}$ based on ${\tau }_{\mathrm{ipt}\left(\mathrm{offload}\right)}^{*}$ according to Theorem 1,

7: Calculate the minimal energy requirement (i.e., $E_{\mathrm{u}\left(\mathrm{offload}\right)}$) by substituting ${\tau }_{\mathrm{ipt}\left(\mathrm{offload}\right)}^{*}$, ${\rho }_{\left(\mathrm{offload}\right)}^{*}$, ${\tau }_{\mathrm{u}-\mathrm{f}}^{*}$, and $P_{\mathrm{u}-\mathrm{f}}^{*}$

into the objective function of Problem $P_2$,

8: Compare $E_{\mathrm{u}\left(\mathrm{local}\right)}$ with $E_{\mathrm{u}\left(\mathrm{offload}\right)}$

If $E_{\mathrm{u}\left(\mathrm{local}\right)}\le E_{\mathrm{u}\left(\mathrm{offload}\right)}$, then trigger the local computing mode,

Otherwise, trigger the fog computing mode.