## 5 Analysis of Impact of Resource Failures on CPS
Let \(\textbf{J}(\Delta(m))\) denote the set of task types influenced by resource failures \(\delta\). We propose Algorithm 3 to iteratively check the robustness of \(G=G\|GR\) with respect to \(\delta\) and \(\Delta(m)\). The algorithm first applies the Procedure I to compute residual capacity \(\widetilde{C}_{rt}\), construct \(RSTN_{j}(V_{j},A_{j})\) according to \(STN_{j}(V_{j},A_{j})\), \(f_{2}\), \(\Pi\), \(\theta\), \(\Delta(m)\) and set the capacity constraints of the FPRSTN based on \(RSTN_{j}(V_{j},A_{j})\) according to the residual capacity \(\widetilde{C}_{rt}\). Finally, it finds the solution \(f_{2}^{\prime}\) for the FPRSTN for \(RSTN_{j}(V_{j},A_{j})\). If the solution \(f_{j2}^{\prime}\) found for each \(j\in\textbf{J}\) satisfies the condition of Property 1, the \(Robustness\_Indicator\) will be set to \"true\" to indicate the solution is robust with respect to resource failures \(\delta\) and \(\Delta(m)\). ```
Input: \(STN_{j}(V,A)\), \(f_{j}\), \(j\in\textbf{J}\), \(\Pi\), \(\theta\), \(\delta\), \(\Delta(m)=(m,\,\delta,\,\omega)\). Output: \(Robustness\_Indicator\), \(f_{j2}^{\prime}\) Step 1: Find \(\textbf{J}(\Delta(m))\) Step 2: Apply Procedure I to compute residual capacity \(\widetilde{C}_{rt}\) for \(r\in\textbf{R}\ \forall t\in\{1,2,\ldots,\Pi\}\) and \(\mathbb{Q}_{j}^{s}\) for each \(j\in\textbf{J}\) Step 3: While \(\textbf{J}(\Delta(m))\
eq\Phi\) Select \(j=j_{k}\in\textbf{J}(\Delta(m))\) Construct \(RSTN_{j}(V_{j},A_{j})\) according to \(STN_{j}(V_{j},A_{j})\) and \(\widetilde{C}_{rt}\) Set the capacity constraints of the FPRSTN defined by \(RSTN_{j}(V_{j},A_{j})\) according to the residual capacity \(\widetilde{C}_{rt}\) Find the solution \(f_{j2}^{\prime}\) for the FPRSTN defined by \(RSTN_{j}(V_{j},A_{j})\) with residual capacity constraint \(\widetilde{C}_{rt}\) Update \(f^{\prime}\) based on \(f_{j2}^{\prime}\) Update residual capacity \(\widetilde{C}_{rt}\) by deducting the capacity assigned to \(f_{j2}^{\prime}\) from \(\widetilde{C}_{rt}\):\(\widetilde{C}_{rt}\leftarrow\widetilde{C}_{rt}-\sum\limits_{(u,w)\in A_{jrt}}f_{2} ^{\prime}(u,w)\) If the solution \(f_{j2}^{\prime}\) satisfies the condition of Property 1  Set \(Robustness\_Indicator_{j}\) = true Else Set \(Robustness\_Indicator_{j}\) = false End If \(\textbf{J}(\Delta(m))\leftarrow\textbf{J}(\Delta(m))\backslash\{j\}\) End While \(Robustness\_Indicator\gets true\) For \(j\in\textbf{J}(\Delta(m)\) \(Robustness\_Indicator\gets Robustness\_Indicator_{j}\) End For Return \(Robustness\_Indicator\)
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**Algorithm 3**Check Robustness with respect to Uncertainty \(\Delta(m)\)

**Property 3**. : _If the Robustness_Indicator is \"true\" by applying the Algorithm 3 to Check Robustness of \(G=G\|GR\) with respect to \(\Delta(m)\), \(G=G\|GR\) is robust with respect to \(\Delta(m)\)._

Proof of Property 3: Please refer to Appendix B.

Computational complexity of the Algorithm 3 to check robustness is analyzed as follows. Note that the Algorithm 3 to check robustness invokes the construction of \(RSTN_{j}(V_{j},A_{j})\) based on the nominal solution and \(STN_{j}(V_{j},A_{j})\) for each \(j\in I(\Delta(m))\). To construct \(RSTN_{j}(V_{j},A_{j})\), Procedure I to is called to compute the residual capacity \(\widetilde{C}_{rt}=C_{rt}-\sum\limits_{j\in I\;\mathbf{a}=(u,w)\in A_{jrt}}f_{j1}(u, w)-C_{rt}^{\delta}\) for \(r\in\mathbf{R},\;t\in\{1,2,\ldots,\Pi\}\). As the number of arcs in the set \(A_{jrt}\) is no greater than \(R\Pi\), the number of arithmetic operations involved is bounded by \(J\Pi\) for each \(r\in\mathbf{R},\;t\in\{1,2,\ldots,\Pi\}\). Therefore, the overall computational complexity to compute the residual capacity is bounded by \(J\!R^{2}\Pi^{2}\). Therefore, the overall computational complexity to compute the residual capacity is \(O(J\!R^{2}\Pi^{2})\). Note that the number of nodes constructed in the spatial-temporal network \(STN_{j}(V_{j},A_{j})\) for type \(j\) task, where \(j\in\mathbf{J}\) is \(N_{j}\Pi+2\).