## 5 Analysis of Impact of Resource Failures on CPS
This advantage is due to the polynomial growth of the STN structure with respect to problem size parameters, compact representation of solutions and an effective way to search for solutions in STN. In this paper, we limit our scope and aim to study the robustness of CPS for manufacturing systems. Depending on the domains in which CPS are applied, the challenges vary. Therefore, robustness of CPS need to be defined and studied in the context of a specific problem domain. Although the scope of this paper is target at robustness of CPS for manufacturing systems, it is promising to extend the proposed method to apply to other temporal analysis problems in CPS for manufacturing systems modelled with timed Petri nets. In addition to the issue to analyze the impact of resource failures addressed in this study, the widespread adoption of CPS raises other challenging research issues. Safety of CPS is another important issue that has attracted researchers' attention. For example, the study [48] focuses on the safety property in which multiple CPS collaborate. Manysafety properties are relevant to temporal analysis. An interesting issue to be studied is the feasibility of applying the proposed method to analyze the safety property of CPS. ## 8 Conclusions

With the maturity and adoption of the IoT and ICT infrastructure in CPS, availability of the real-time information from the shop floor enhances visibility of production status for manufacturers. The real-time information provided by the IoT and ICT infrastructure enables the development of an effective strategy to deal with undesirable events in CPS, such as resource failures. Resource failures may take place unexpectedly and have negative effects on the operations of CPS. Assessing the impact of resource failures on CPS is an important issue. Development of a systematic method to support analysis of the impact of resource failures on CPS is needed. This paper aims to propose a method to evaluate the effects of resource failures on operations and temporal properties of CPS by constructing residual spatial-temporal networks obtained based on transformation of a class of discrete timed Petri net models for CPS and the system state information. To evaluate the impact due to resource failures, we formulate a problem based on the residual spatial-temporal networks to check whether the order deadline is feasible after resource failures to efficiently assess the system. The proposed method is illustrated by examples. To assess practicality of the proposed method in terms of computational feasibility, we analyzed the computational complexity of the proposed algorithm and conducted experiments to study the growth of computational time required with respect to the number of tasks influenced due to resource failures and number of transitions in the processes. The analysis indicates that the lower bound on the complexity of the algorithm is polynomial with respect to problem size parameters. The numerical results of the experiments also indicate that the CPU time grows polynomially with respect to the number of tasks influenced due to resource failures and number of transitions in the processes. The numerical results are consistent with the complexity analysis and indicate that the proposed method can be applied to solve real problems. Formal models such as timed Petri nets have been widely accepted as tools for the specifying, modelling and simulation of systems. However, analysis methods of timed Petri nets are still limited. The method proposed in this paper attempts to analyze the temporal property of timed Petri nets based on transformation of the reachability problem into a problem relevant to the scheduling problem in the literature. Such transformation is nontrivial and provides an alternative approach to the analysis of timed Petri nets. One future research direction is to extend the proposed method to other types of processes. Although this study focuses on robustness of temporal property of CPS with respect to resource failures, this is not the only robustness research issue.