## 5 Analysis of Impact of Resource Failures on CPS
From the perspective of theoretical development, the timed Petri net model considered in this study is different from [43; 44] in that the DTPN model used in this study specifies transition firing time delay whereas the

Figure 7: (**a**) Computational time with respect to the number of tasks influenced, \(Q_{j}^{\delta}\), due to resource failures; (**b**) Computational time with respect to number of transitions in task subnets. time Petri net model used in [43] specifies time intervals for transition firing. The transition firing time delay in DTPN is discrete, which makes it possible to analyze the robustness property of the system with RSTN. In [42], [PERSON] et al. defined a model to be robust if the set of discrete behaviors is preserved under arbitrarily small perturbations in the firing intervals of transitions. [PERSON] et al. showed that TPNs are not robust in general and that the problem to check robustness with respect to boundedness and safety properties is undecidable. In [45], robustness is defined as a property to measure the allowed variability of the timing delays in their neighborhood. The model used in [45] is also TPN which specifies time intervals for transition firing. The robustness property studied in this paper is not robustness with respect to firing time variation. Instead, we study the robustness with respect to perturbation in markings, which corresponds to changes in the number of resources in the system. Therefore, the study in this paper is different from the papers mentioned above. In this study, we focus on the development of an algorithm to analyze the impact of resource failures on the operations and temporal properties of CPS, based on transformation of the DTPN model of CPS into relevant spatial-temporal network models. We illustrate the proposed algorithm by applying it to two application scenarios. In the first scenario, only one type of production process is influenced by the resource failure. The results indicate that the influenced workflows are rerouted properly to reflect the unavailability of the failed resource. In the second scenario, two types of production processes are influenced due to the resource failure. The results also show that the influenced workflows of the two production processes are rerouted properly to reflect the unavailability of the failed resource. In addition to the two application scenarios above, we also perform experiments to study how the computational time grows with respect to number of tasks influenced due to resource failures and how the computational time grows with respect to number of transitions in the processes. The results of the computational time experiments indicate that the proposed is computationally feasible and is consistent with the characteristics of the lower bound on the polynomial complexity obtained in our analysis. This study provides a foundation for the analysis of temporal properties in CPS. Complexity analysis provides a formal and widely accepted way to assess the computationally feasibility of different algorithms in the literature. We also follow this approach in this paper to compare our approach with classical reachability analysis methods. Classical reachability analysis methods suffer from the exponential state explosion problem even for untimed Petri nets. In the literature, complexity of the reachability problem of untimed Petri nets is exponential space and time although it is decidable [46; 47]. For timed Petri nets, the complexity is higher than untimed Petri nets due to consideration of time factor. The complexity of classical reachability analysis methods for timed Petri nets also suffers from the state explosion problem. For the STN based approach proposed in this paper, the lower bound of the computational complexity is polynomial with respect to problem size parameters. Our experimental results are consistent with this lower bound. The proposed STN based approach has advantage over classical reachability analysis methods in terms of computational feasibility to deal with real problems.