## 3 Robustness Analysis Problem Formulation in CPS
Therefore, \(\mu_{j}(t)\) can be set to zero for each transition \(t\in T_{j}\) because is always a lower bound of transition \(t\) by using any resource. Figure 1 shows two examples of task subnets, \(GJ_{1}\) and \(GJ_{2}\). The firing time of transitions in \(GJ_{1}\) can be set as \(\mu_{1}(t_{1})=\mu_{1}(t_{2})=\mu_{1}(t_{3})=\mu_{1}(t_{4})=0\) if the lower bound of the firing time of each transition in \(T_{1}\) is unknown. Similarly, if the lower bound of the firing time of each transition in \(T_{2}\) is unknown, the firing time of transitions in \(GJ_{2}\) can be set as \(\mu_{2}(t_{5})=\mu_{2}(t_{6})=\mu_{2}(t_{7})=\mu_{2}(t_{8})=\mu_{2}(t_{9})=\mu_ {2}(t_{10})=0\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol/Variable** & **Meaning** \\ \hline \(J\) & The number of different types of tasks \\ \(J\) & The set of indices of all different types of tasks, \(J=\{1,2,3,\ldots,J\}\) \\ \(j\) & A type of task \(j\in J\) \\ \(GJ_{j}\) & A task subnet, \(GJ_{j}=(P_{j},T_{j},F_{j},m_{j0},\mu_{j})\) \\ \(N_{j}\) & The number of operations in \(GJ_{j}\) \\ \(R\) & The number of different types of resources \\ \(R\) & \(R=\{1,2,3,\ldots,R\}\), the set of different types of resources in the system \\ \(r\) & The index of a type of resources, \(r\in R=\{1,2,3,\ldots,R\}\) \\ \(k\) & The \(k\)-th operation performed by a type \(r\) resource, where \(r\in R\) \\ \(G_{r}^{jk}\) & A circuit (a DTPN model) that represents the activity of the \(k\)-th operation in type \(j\) task to be \\  & performed by a type \(r\) resource \\ \(\Omega_{r}^{j}\) & The set of indices of circuits of type \(r\) resources involved in type \(j\) task, \(\Omega_{r}^{j}=\left\{1,2,\ldots,K_{r}^{j}\right\}\), where \(K_{r}^{j}\) is the number of circuits involved. \\ \(\|\) & An operator to merge Petri nets \\ \(GR_{r}\) & A type \(r\) resource subnet, \(GR_{r}=(P_{r},\,T_{r},\,F_{r},m_{r0},\,\mu_{r})=\|\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\! To facilitate modeling, the operator \" \(\parallel\)\" is defined as follows to merge two DTPNs through common transitions, places and arcs. **Definition 2**. : _Given two discrete timed PNs, \(G_{1}\) = (\(P_{1}\), \(T_{1}\), \(F_{1}\), \(m_{10}\), \(\mu_{1}\)) and \(G_{2}\) = (\(P_{2}\), \(T_{2}\), \(F_{2}\), \(m_{20}\), \(\mu_{2}\)), the operator \"\(\parallel\)\" to combine \(G_{1}\) and \(G_{2}\) is defined as follows: \(G_{1}\|G_{2}\) = (\(P\), \(T\), \(F\), \(W\), \(m_{0}\), \(\mu\)), where \(P\) = \(P_{1}\cup P_{2}\), \(T\) = \(T_{1}\cup T_{2}\), \(F(p,t)\) = \(\left\{ \begin{array}{cc}F_{1}(p,t)\ if\ p\in P_{1}\ and\ t\in T_{1}\\ F_{2}(p,t)\ if\ p\in P_{2}\ and\ t\in T_{2}\end{array} \right.\), \(F(t,p)\) = \(\left\{ \begin{array}{cc}F_{1}(t,p)\ if\ p\in P_{1}\ and\ t\in T_{1}\\ F_{2}(t,p)\ if\ p\in P_{2}\ and\ t\in T_{2}\end{array} \right.\), \(m_{0}(p)=\left\{ \begin{array}{cc}m_{10}(p)\ if\ p\in P_{1}\\ m_{20}(p)\ if\ p\in P_{2}\end{array} \right.\) and \(\mu(t)=\left\{ \begin{array}{cc}\max(\mu_{1}(t),\mu_{2}(t))\ if\ t\in T_{1} \cap\ T_{2}\\ \mu_{1}(t),\ if\ t\in T_{1}\backslash T_{2}\\ \mu_{2}(t),\ if\ t\in T_{2}\backslash T_{1}\end{array} \right..\)  
An operation is performed by a resource. In this study, the number of different types of resources is denoted by \(R\). We use \(R\) = \(\{1,2,3,\ldots,R\}\) to denote the set of different types of resources in the system. We use \(r\) to refer to a type of resources, where \(r\in R\). The \(k\)-th operation performed by type \(r\) resource is represented by a circuit, \(G_{r}^{jk}\), which is a discrete timed PN. Each circuit \(G_{r}^{jk}\) has an idle place \(r\), a busy place, one start transition and one end transition. Figure 2a-e shows examples of resource circuits. Let \(\Omega_{r}^{j}\) = \(\left\{1,2,\ldots,K_{r}^{j}\right\}\) denote the set of indices of circuits of type \(r\) resources involved in type \(j\) task. The capabilities of a resource type \(r\) is described by a resource subnet defined below. **Definition 3**.