## 5 Analysis of Impact of Resource Failures on CPS
This indicates that the growth of computational time with respect to number of tasks influenced due to resource failures is polynomial. For the second type of experiment, to study the scalability of the proposed algorithm, we conduct experiments by varying the number of transitions in the processes from 8 to 40. The time horizon, \(\Pi\), is set to 722 in this type of experiment. The results are shown in Figure 7a. This indicates that the growth of computational time with respect to number of transitions in the task subnet is polynomial. Figure 6: (**a**) The rerouted flows due to resource failure represented in \(RSTN_{1}(V_{1},A_{1})\); (**b**) One solution represented in \(STN_{1}(V_{1},A_{1})\) obtained after resource failure. ## 7 Discussion

Although CPS provides a paradigm to manage, monitor and control manufacturing resources in enterprises based on the real-time information from the IoT and ICT infrastructure, an effective scheme to deal with unexpected events must be developed to enhance robustness of CPS. Robust design of CPS becomes an important issue to achieve stable and secure CPS. It has attracted a lot of attention in the CPS research community. A comprehensive survey of the concept and strategies for robust design of CPS can be found in [5]. Although many studies have addressed the robustness issues on stability, security and systematicness in the context of CPS, robustness of CPS with respect to resource failures is less explored in the literature. This study aims to bridge this gap through development of a methodology to model and analyze robustness of CPS with respect to resource failures. CPS relies on an effective scheme to deal with changes in processes, resources and implementation technology. Model driven approach (MDA) provides a method to accommodate these changes [11]. MDA is an approach for developing software systems based on specifications described as models. This MDA approach is supported by the Unified Modeling Language(tm) [41]. Although UML is the standard for specification of software systems, it is weak in analyzing production processes with shared resources, concurrent, synchronous and asynchronous operations in CPS. Petri nets provides the capabilities to model and analyze CPS. In [8], a class of Petri nets called DTPN is proposed to model CPS. In addition, due to the lack of an efficient method to analyze DTPN, an analysis method is proposed based on transformation of the DTPN model to a network model to analyze temporal property of the DTPN model. However, the problem in [8] considers order requirements with a single type of product demand and the issue to deal with resource failures is not addressed in [8]. In terms of modeling tools and analysis methods, despite the DTPN model for CPS being proposed previously, there still lacks an analysis method to model and analyze the impact of resource failures on the performance of CPS. Although the development of this paper is based on the preliminary results in [15] to analyze the influence of resource failures on CPS, the content presented in this paper is different from [15], in that it includes the rigorous proof of robustness properties of CPS, computational complexity analysis and verification of complexity analysis results by experiments, which are not covered in [15]. In addition, the problem studied in this paper considers order requirements with multiple types of product demands whereas the problem in [8] considers order requirements with a single type of product demand. In Petri net literature, there are two ways to describe time regarding transition firing. Timed Petri nets specify a duration to represent firing time for a transition [27]. Time Petri nets describe a time interval for firing transition [42].