## 3 Cross-Layer Controller Tasking Scheme
### Learning Network Process  
The learning network used for both the mappings differs based on the conditions \(Fgd\
eq 0\), \(\dalmathbb{C}_{Lyc}=(n-Fgd)\dalmathbb{C}_{tsd}\) and \(\exists\). An edge device is available in the task's grouped time if it is 1 or else 0. The output of the mediate layer in the first task grouping \(\mathit{Task}_{N}::TM\) generates a linear output, whereas \((\mathit{Task}_{N},Fgd)::TM\) extracts the output of \(n\) from \(N\) with \(Fgd\
eq 0\). In Equations (7) and (8), the \(\epsilon_{O}\) and \(\vartheta_{O}\) for \(\mathit{Task}_{N}::TM\) are grouped. The computations are performed for both the estimation of \(\mathit{fail}_{\mathit{kg}}\) and \(\mathit{fail}_{\mathit{ctrl}}\) for the conditional assessment of either \(\exists=1\) or \(\exists=0\) in \(i\). Therefore, the results are required from all the resources allocated and utilize time interval \(TM\). In the above task grouping process, \(Fgd\) serves as an input after the detection of \(\mathit{fail}_{\mathit{kg}}\) in \(\mathit{Task}_{N}::TM\) grouping is represented as \[ \begin{array}{c}\epsilon_{O_{1}}=\dal\Upar_{Lyc_{1}}\times TM_{1}+\mu_{G}. \exists\\ \epsilon_{O_{2}}=\dal\Upar_{Lyc_{2}}\times TM_{2}-f\al_{Lyc_{1}}+\mu_{G}.\exists\\ \epsilon_{O_{3}}=\dal\Upar_{Lyc_{3}}\times TM_{3}-f\al_{Lyc_{2}}+\mu_{G}. \exists\\ \vdots\\ \epsilon_{O_{i}}=\dal\Upar_{Lyc_{i}}\times TM_{i}-f\al_{Lyc_{i}}+\mu_{G}. \exists\end{array}  \tag{7}\]  
\[ \begin{array}{c}\phi_{O_{1}}=\epsilon_{O_{1}}\\ \theta_{O_{2}}=\epsilon_{O_{2}}-fail_{ctrl_{1}}\times\Delta_{1}\\ \theta_{O_{3}}=\epsilon_{O_{3}}-fail_{ctrl_{2}}\times\Delta_{3}\\ \vdots\\ \theta_{O_{i}}=\epsilon_{O_{i}}-fail_{ctrl_{i}}\times\Delta_{i-1}\end{array} \begin{array}{c}\phi_{O_{1}}=\dal\Upar_{Lyc_{1}}\times TM_{1}+\mu_{G}. \exists-fail_{ctrl}\times\Delta_{1}\\ \theta_{O_{2}}=\dal\Upar_{Lyc_{2}}\times TM_{2}-f\al_{Lyc_{1}}+\mu_{G}.\exists -fail_{ctrl_{1}}\times\Delta_{2}\\ \theta_{O_{3}}=\dal\Upar_{Lyc_{3}}\times TM_{3}-f\al_{Lyc_{2}}+\mu_{G}.\exists -fail_{ctrl_{2}}\times\Delta_{3}\\ \vdots\\ \theta_{O_{i}}=\epsilon_{O_{i}}-fail_{ctrl_{i}}\times\Delta_{i-1}\end{array}  \tag{8}\]  
In (7) and (8) above, the linear output is formulated as \(\phi_{O_{i}}=\dal\Upar_{Lyc_{i}}\times TM_{i}-f\al_{Lyc_{i}}+\mu_{G}.\exists- fail_{ctrl_{i-1}}\times\Delta_{i-1}\), and if \(Fgd=0\), then \(\exists=1\) and \(\dal\Upar_{Lyc_{i}}=N\dal\Upar_{Lyc_{i}}\). Therefore, the network is recurrently trained for the inverse outcome of the first phase to prevent TAE, and the second phase trains the learning network based on task migration between the controllers as the optimal output and \(R_{n}=1\). The linear deep learning network initialization is presented in Figure 4.  
The linear learning representation for \(R_{n}\) classification and \(v_{oi}\) estimation is illustrated in Figure 4 above. The change in \((\mu_{G}.\exists)\) to \(\dal\Upar_{Lyc}\) is validated throughout the hidden layer for \(R_{n}\) extraction. The satisfying condition is \(\dal\Upar_{Lyc_{i}}=N\dal\Upar_{Lyc_{i}}\) such that \(R_{n}\) is maximum. However, the change in synchronous (based on \(n\)) completion varies this process, and therefore, \(v_{o_{i}}\) is required. This learning phase requires \(f_{ctrl}\) and \(\Delta\) changeovers to identify both phase controls. Therefore, the output from \(i=1\) to \(TM\) resurrects the \(v_{o_{i}}\) for TAE reduction (referenced in Figure 4). Hence, the reputation of such edge devices/machinery in smart industries is retained as 1. The IIoT stores \((R_{n},\ N)\) for each \(TM\), and this graph structure determines the cross-layer controller for the edge devices. Instead, \((Task_{N},Fgd)::TM\) for \(\epsilon_{O}\) and \(\theta_{O}\) is computed using Equations (9) and (10), respectively. \[