Algorithm 1 Overall DT-Assisted Edge Computing System with Freshness Awareness

Require:  $M, T, \Delta t, C_{\mathrm{backhaul}}, \{f_t^m, f_{\mathrm{edge}}^{m,j}, c_{\mathrm{edge}}^{m,j}\}$, $E_{\mathrm{init},m}, \eta , \rho , \kappa , {ϵ}_{\mathrm{stale}}$ Ensure:  Decisions $\left\{({\beta }_i^{t,m},{\gamma }_i^{t,m},{\delta }_i^{t,m})\right\}$   1:Initialize $E_{1,m}\leftarrow E_{\mathrm{init},m},\forall m$   2:Initialize ${\mathbf{x}}_{1,m}\leftarrow \mathbf{0},\forall m$, ${\tau }_1\leftarrow 1$   3:${\mathbf{X}}_1\leftarrow [{\mathbf{x}}_{1,1},\dots ,{\mathbf{x}}_{1,M}]$   4:for $t=1$ to T do   5:      Observe incoming tasks $\{D_i^{t,m},T_{P,i}^{t,m},T_{D,i}^{t,m}\}$   6:      Compute freshness: ${\mathcal{F}}_t\leftarrow t-{\tau }_t$   7:      Compute staleness: ${\mathcal{S}}_t\leftarrow 1-\exp (-\kappa {\mathcal{F}}_t)$   8:      if ${\mathcal{S}}_t>{ϵ}_{\mathrm{stale}}$ then   9:           Synchronize DT with physical system 10:           ${\tau }_t\leftarrow t$ 11:           Update ${\mathbf{X}}_t$ 12:      end if 13:      ${\mathbf{z}}_t\leftarrow {\mathcal{F}}_t\mathsf{\Psi }\left({\mathbf{X}}_t\right)$                ▷ Freshness-weighted DT analytics 14:      for each task i at server m do 15:           Compute local latency $T_{i,\mathrm{loc}}^{t,m}$ using (4) 16:           Compute migration latency $T_{i,\mathrm{mig}}^{t,m}(n,j)$ using (5) 17:           ${\tilde{T}}_i^{t,m}\leftarrow \min \{T_{i,\mathrm{loc}}^{t,m},{\min }_{n\ne m,j}{T}_{i,\mathrm{mig}}^{t,m}(n,j)\}$ 18:           if ${\tilde{T}}_i^{t,m}>T_{D,i}^{t,m}$ then 19:                ${\delta }_i^{t,m}\leftarrow 1$, ${\beta }_i^{t,m}\leftarrow 0$, ${\gamma }_i^{t,m}\leftarrow 0$ 20:                continue 21:           end if 22:           if ${\mathrm{Trigger}}_{\mathrm{offload}}^{t,m}=1$ then 23:                ProposedMILP(${\mathbf{z}}_t,{\mathcal{S}}_t,i,m$) 24:           else 25:                ${\beta }_i^{t,m}\leftarrow 1$ if $U_{\mathrm{CPU},m}^t<{\overline{U}}_{\mathrm{CPU}}^t-\sigma$ 26:           end if 27:      end for 28:      Compute total latency $T_i^{t,m}$ using (6) 29:       Update energy: $$E_t^m\leftarrow \sum _i\left({\beta }_i^{t,m}{E}_{i,\mathrm{loc}}^{t,m}+{\gamma }_i^{t,m}{E}_{i,\mathrm{mig}}^{t,m}\right)$$ 30:      $E_{t+1,m}\leftarrow \max (E_{t,m}-E_t^m,\rho E_{\mathrm{init},m}),\forall m$ 31:       Update fairness: $${\overline{U}}_{\mathrm{CPU}}^t\leftarrow {\displaystyle \frac{1}{M}}\sum _m{U}_{\mathrm{CPU},m}^t$$ 32:      for each server m do 33:            Extract features: $${\mathbf{y}}_{t,m}\leftarrow \varphi \left((\beta +\gamma )D,T_P,T\right)$$ 34:            Update DT state: $${\mathbf{x}}_{t+1,m}\leftarrow (1-\eta ){\mathbf{x}}_{t,m}+\eta {\mathbf{y}}_{t,m}+{\mathit{\xi }}_{t,m}$$ 35:      end for 36:      ${\mathbf{X}}_{t+1}\leftarrow [{\mathbf{x}}_{t+1,1},\dots ,{\mathbf{x}}_{t+1,M}]$ 37:end for 38:return all scheduling decisions