## 3 Robustness Analysis Problem Formulation in CPS
Figure 3 shows an example of \(G=(P,T,F,m_{0},\mu)=GJ\|GR\), where \(GJ=GJ_{1}\|GJ_{2}\), \(GJ_{1}\) and \(GJ_{2}\) are shown in Figure 1 and \(GR\) is obtained by merging all the circuits in Figure 2.  
The firing time of each transition of \(G\) in Figure 3 is as follows:  
\(\mu(t_{1})=\max(\mu_{1}(t_{1}),\mu_{r_{1}}(t_{1}))=1\), \(\mu(t_{2})=\max(\mu_{1}(t_{2}),\mu_{r_{1}}(t_{2}))=0\), \(\mu(t_{3})=\max(\mu_{1}(t_{3}),\mu_{r_{2}}(t_{3}))=2\), \(\mu(t_{4})=\max(\mu_{1}(t_{4}),\mu_{r_{2}}(t_{4}))=0\)  
\(\mu(t_{5})=\max(\mu_{2}(t_{5}),\mu_{r_{1}}(t_{5}))=2\), \(\mu(t_{6})=\max(\mu_{2}(t_{6}),\mu_{r_{1}}(t_{6}))=0\)  
\(\mu(t_{7})=\max(\mu_{2}(t_{7}),\mu_{r_{3}}(t_{7}))=1\), \(\mu(t_{8})=\max(\mu_{2}(t_{8}),\mu_{r_{3}}(t_{8}))=0\)  
\(\mu(t_{9})=\max(\mu_{2}(t_{9}),\mu_{r_{4}}(t_{9}))=1\), \(\mu(t_{10})=\max(\mu_{2}(t_{10}),\mu_{r_{4}}(t_{10}))=0\)  
Note that the set \(P_{R}\) of all the places in \(GR\) can be divided into two subsets: the subset of idle places \(\{r_{s}|s\in\mathbf{R}\}\) of resources and the subset of non-idle places of resources \(P_{R}/\{r_{s}|s\in\mathbf{R}\}\). That is, \(P_{R}=(P_{R}/\{r_{s}|s\in\mathbf{R}\})\cup\{r_{s}|s\in\mathbf{R}\}\). As each non-idle places subset in \(P_{R}/\{r_{s}|s\in\mathbf{R}\}\) belongs to \(P=\underset{j\in J}{\cup}P_{j}\cup\{r_{s}|s\in\mathbf{R}\}\), the set of places in \(G\) is \(P=\underset{j\in J}{\cup}P_{j}\cup\{r_{s}|s\in\mathbf{R}\}=\underset{j\in J}{ \cup}\{p_{1}^{j},\,p_{2}^{j},\,p_{3}^{j},\,\ldots,\,p_{2N_{j}+1}^{j}\}\cup\{r_ {1},\,r_{2},\,r_{3},\,\ldots,\,r_{|\mathbf{R}|}\}\). Therefore, the dimension of \(P\) is \(|P|=\underset{j\in J}{\sum}(2N_{j}+1)+R\). A marking \(m\) is defined by  
\[m=\left[m(p_{1})\ m(p_{2})\ m(p_{3})\ \ldots\ \ \ \ \ \ m(p_{\underset{j\in J}{ \sum}(2N_{j}+1)})\ m(r_{1})\ m(r_{2})\ m(r_{3})\ \ldots\ \ \ \ \ m(r_{|\mathbf{R}|})\right]\]  
In this paper, the final state place in \(P_{j}\) is \(p_{2N_{j}+1}^{j}\). The set of all final state places in \(P\) is denoted by \(P^{F}=\{p_{2N_{j}+1}^{j},j\in J=\{1,2,3,\ldots,J\}\}\). In a make-to-order manufacturing system, an order is specified by the quantity of different types of product demands, \(Q_{j}\), where \(j\in J=\{1,2,3,\ldots,J\}\), and the deadline \(\theta\). The requirements to meet an order with product demands, \(Q_{j}\), where \(j\in J=\{1,2,3,\ldots,J\}\)  
Figure 3: The model of two processes \(G\). can be described by a marking \(m_{d}\) in the discrete timed Petri net \(G=(P,T,F,m_{0},\mu)\), where \(m_{d}(p)=\left\{ \begin{array}{c}0\ p\in P\backslash\left(P^{F}\cup\{r_{s}|s \in R\,\}\right)\\ Q_{j}\ p=p^{j}_{2N_{j}+1}\in P^{F},j\in\{1,2,3,\ldots,J\}\end{array} \right..\)  
Therefore, the problem to determine whether the order can be fulfilled by the deadline \(\theta\) can be stated as the following feasibility problem:  
Feasibility Problem for Nominal CPS (FPN):  
Given a DTPN model \(G\) under a reachable marking \(m\), a target marking \(m_{d}\) defined by \(m_{d}(p)=\left\{ \begin{array}{c}0\ p\in P\backslash\left(P^{F}\cup\{r_{s}|s \in R\,\}\right)\\ Q_{j}\ p=p^{j}_{2N_{j}+1}\in P^{F},j\in\{1,2,3,\ldots,J\}\end{array} \right.\) and a deadline \(\theta\), determine whether there exists a firing sequence that brings \(G\) from \(m\) to \(m_{d}\) by the deadline \(\theta\). Operations of manufacturing systems are usually influenced by unexpected events such as resource failures. The influence of resource failures may have impact on the operations and orders in the system. To deal with resource failures properly, the impact due to resource failures must be evaluated. To study the influence of resource failures on CPS, we must represent resource failures properly. We introduce the uncertainty model \(\Delta(m)\) based on the concept of perturbation vectors and failure time intervals as follows. A \(|P|\) dimensional perturbation vector \(\delta\) is an integer vector to specify the number of tokens lost for the set of places in \(P\) due to resource failures.