## 3 Robustness Analysis Problem Formulation in CPS
: _A resource subnet \(GR_{r}\) for a type \(r\) resource is a discrete timed Petri net \(GR_{r}\) = (\(P_{r}\), \(T_{r}\), \(F_{r}\), \(m_{r0}\), \(\mu_{r}\)) obtained by merging the circuits of type \(r\) resource, where \(r\in R\) and the function \(\mu_{r}\) is used to specify the firing time of each transition in \(T_{r}\). That is, \(GR_{r}\) = (\(P_{r}\), \(T_{r}\), \(F_{r}\), \(m_{r0}\), \(\mu_{r}\)) = \(\left\|\begin{array}{cc}\underset{j\in I_{R}\in\Omega_{r}^{j}}{\parallel}&G_{ r}^{jk}\). A resource subnet \(GR_{r}\) has an idle place denoted by \(r\)._Figure 2f shows the resource subnet \(GR_{r_{1}}\) obtained by merging \(GR_{r_{1}}^{11}\) in Figure 2a with \(GR_{r_{1}}^{21}\) in Figure 2c. \(GR_{r_{2}}\), \(GR_{r_{3}}\) and \(GR_{r_{4}}\) can be obtained similarly. The firing time of transitions in \(GR_{r_{1}}\) is set as \(\mu_{r_{1}}(t_{1})=1\), \(\mu_{r_{1}}(t_{2})=0\), \(\mu_{r_{1}}(t_{5})=2\), \(\mu_{r_{1}}(t_{6})=0\). The firing time of transitions in \(GR_{r_{3}}\) is set as \(\mu_{r_{2}}(t_{3})=2\), \(\mu_{r_{2}}(t_{4})=0\). The firing time of transitions in \(GR_{r_{3}}\) is set as \(\mu_{r_{3}}(t_{7})=1\), \(\mu_{r_{3}}(t_{8})=0\). The firing time of transitions in \(GR_{r_{4}}\) is set as \(\mu_{r_{4}}(t_{9})=1\), \(\mu_{r_{4}}(t_{10})=0\). Operations in manufacturing systems are performed by resources to process parts. The proposed model captures synchronization between parts and resources by applying the merging operator \" \(\|\)\" to a common transition in a task subnet and a resource subnet. The operator \" \(\|\)\" also merges relevant common places and arcs involved between a task subnet and the associated resource subnet. Note that as the firing time for a transition in the task subnet is just a lower bound, which is always smaller than the firing time specified in the resource subnet, the exact time for firing a transition depends on the resource involved. Therefore, after applying the merging operator, the firing time of the merged transition is defined as the maximum of the transition firing time specified in the task subnet and the resource subnet. Consider a task subnet \(GJ_{j}=(P_{j},T_{j},F_{j},m_{j0},\mu_{j})\) and a resource subnet \(GR_{r}=(P_{r},T_{r},F_{r},\)\(m_{r0},\mu_{r})\). For the model \(GJ_{j}\|GR_{r}\) obtained by applying the \"\(\|\)\" operator to combine a task subnet \(GJ_{j}=(P_{j},T_{j},F_{j},m_{j0},\mu_{j})\) with a resource subnet \(GR_{r}=(P_{r},T_{r},F_{r},m_{r0},\mu_{r})\), the firing time of transition \(t\in T_{j}\cap\ T_{r}\) in \(GJ_{j}\|GR_{r}\) is \(\mu(t)=\max(\mu_{j}(t),\mu_{r}(t))=\mu_{r}(t)\) as \(\mu_{j}(t)\) is a lower bound of \(\mu_{r}(t)\). That is, the firing time \(\mu(t)\) of transition \(t\in T_{j}\cap\ T_{r}\) in \(GJ_{j}\|GR_{r}\) is determined by the firing time \(\mu_{r}(t)\) specified in \(GR_{r}\). Therefore, \(\mu_{j}(t)\) can be set to zero for each transition \(t\in T_{j}\) because \(\mu_{j}(t)\) is a lower bound of \(\mu_{r}(t)\). To capture cooperation and interaction between different types of tasks and resources in the CPS, we merge the task subnets with the resource subnets to construct the discrete timed Petri net model \(G\) for the nominal CPS defined \(G\) as follows. **Definition 4**. : _A nominal model is described by a discrete timed Petri net \(G=(P,T,F,m_{0},\mu)=GJ\|GR\), where \(GJ=\underset{j\in I}{\|}GJ_{j}\) and \(GR=\underset{r\in R}{\|}GR_{r}=(P_{R},\)\(T_{R},\)\(F_{R},\)\(m_{R0},\)\(\mu_{R})\), where \(m_{0}:P\to Z^{|P|}\) and \(\mu:T\to Z^{+}\cup\left\{0\right\}.\) The state of \(G\) is called a marking represented by a vector \(m\in Z^{|P|}\)._  
In this paper, it is assumed that the timed Petri net models work under infinite server policy. As we focus on the deterministic and aim to determine the feasible firing sequences to meet the deadline, the firing of transitions is determined by the algorithm. Note that operations of the DTPN models are based on the firing sequences found by the proposed algorithm. Therefore, firing policy and memory policy are not required in the proposed method.