## 2 Materials and Methods
### Petri Nets
However, as \(k\) increases, new tokens become increasingly fine-grained, leading to smoother firing that resembles continuous flow. When \(k\) tends to infinity, the set of reachable markings becomes infinite, and a marking can no longer be expressed in terms of tokens since its components can become infinitely large. Instead, a marking is represented by a vector of real numbers, \(m=(\alpha_{1},\ldots,\alpha_{n})\), to ensure continuity, while rational numbers can still be used for arc weights since they have concrete values. Consequently, the transition \(t_{j}\) is enabled in the marking \(m\) if and only if \(\alpha_{i}>0\) for \(p_{i}\in\,^{\circ}t_{j}\) and its degree of enabling is determined by \(enab(t_{j},m)=\min_{p_{i}\in\,^{\circ}t_{j}}\frac{\alpha_{i}}{Pre(p_{i},t_{j})}\). Starting from marking \(m\), firing \(t_{j}\) by an amount \(\beta\) (or \(\beta\) times simultaneously in one go), denoted by \([t_{j}]^{\beta}\), where \(0<\beta\leq enab(t_{j},m)\), results in a new marking \(m^{\prime}=m+\beta\cdot C[P,t_{j}]\), where \(C[P,t_{j}]\) represents the column corresponding to \(t_{j}\) in the incidence matrix. An HPN is a 6-tuple \(R=(P,T,h,Pre,Post,M_{0})\), where \(P\), the set of places, is composed of subset of \(D\) places, \(P^{D}\), and subset of \(C\) places, \(P^{C}\), and \(P\
eq\emptyset\), \(P=P^{D}\cup P^{C}\), \(P^{D}\cap P^{C}=\emptyset\); \(T\), the set of transitions, consists of subset of \(D\) transitions, \(T^{D}\), and subset of \(C\) transitions, \(T^{C}\), and \(T\
eq\emptyset\), \(T=T^{D}\cup T^{C}\), \(T^{D}\cap T^{C}=\emptyset\); \(h\), hybrid function that discriminates between \(D\) and \(C\) components, is a rule from \(P\cup T\) to \(\{D,C\}\), \(Pre\) is a function from \(P\times T\) to \(\mathbb{Q}^{+}\); \(Post\) is a function from \(P\times T\) to \(\mathbb{N}\); and \(M_{0}\) is same to that in the definition of CPN, \(Pre\) and \(Post\), the weight functions, set the weight of the arc \((p,t)\) and \((t,p)\), respectively, to a positive rational number if \(p\in P^{C}\), and to a natural number if \(p\in P^{D}\). An SPN is a 6-tuple \(R=(P,T,I,O,M_{0},\Lambda)\) where \(P,T,I,O\), and \(M_{0}\) are the same as those in the definition of the classical untimed Petri net, and \(\Lambda=(\lambda_{1},\ldots,\lambda_{m})\) is a vector of real numbers, where \(\lambda_{k}\) represents the firing rate of \(t_{k}\), \(1\leq k\leq m\). We apply SPNs to model the association between random time delays and the occurrence of transitions. Within an SPN framework, the time interval between the enabling of a transition and its subsequent occurrence is characterized by a random variable following a negative exponential probability distribution function as follows:  
\[F(t)=1-e^{-\lambda t}\text{ if }t\geq 0\text{ and }F(t)=0\text{ otherwise, } \lambda>0.\]  
In the use of extended Petri nets, we sometimes need to introduce additional arc types beyond the standard arcs defined earlier. Two commonly used nonstandard arcs in this research are the inhibitor arc and the read (or test) arc. An inhibitor arc imposes a condition that a transition can only fire if the connected place is empty. A read arc, on the other hand, models a scenario in which a resource (such as an enzyme in a chemical reaction) is required but not consumed during firing. A read arc is represented as an arc with a solid circle and an inhibitor arc with a hollow circle [14].