## 3 Timed Petri Net and Retention-Free Petri Net  
Petri nets are extended by assigning a delay time of the firing to each transition to facilitate a system-level understanding through means of a simulation. Such an extended Petri net is called a timed Petri net. Figure 4: An enzyme place in a Petri net model. **Definition 5**. : _Timed Petri net TPNis defined by \(TPN=(PN,D)\), where \(D\) is a set of positive numbers expressing the firing delay times (or delay time for short) of transitions in \(T\). The firing rule of a TPN is as follows:_  
1. _initially, all of the tokens are in a non-reserved state; once a transition_ \(t_{i}\) _is decided to have fired, the tokens required for firing are changed from the non-reserved state to the reserved state;_
2. _when the delay time_ \(d_{i}\) _of_ \(t_{i}\) _has passed,_ \(t_{i}\) _fires to remove the reserved tokens from the input places of_ \(t_{i}\) _and put non-reserved tokens into the output places of_ \(t_{i}\)_._ _In a timed Petri net, the firing times of a transition_ \(t_{i}\) _per unit of time are called firing frequency_ \(f_{i}\)_._ \(\overline{f_{i}}\) _represents the maximum firing frequency of_ \(t_{i}\)_. The delay time_ \(d_{i}\) _of_ \(t_{i}\) _is given by the reciprocal of_ \(\overline{f_{i}}\)_._ _Note that firing frequencies indicate reaction speed data in a biological pathway._  
We show here the method of conflict resolution using stochastic decision rules. To resolve the conflict firing problem, there are three methods that can be adopted: priority, probabilistic choice and alternate firing [25]. The proposed firing rule given in this paper is to combine a probabilistic choice and alternate firing. We introduce a stochastic approach to determine the firings for a series of transitions in conflict (Figure 5), as defined in the following definition. **Definition 6**. : _Suppose a place \(p\) possesses output transitions, \(t_{O_{1}},t_{O_{2}},\cdots,t_{O_{n}}\), as shown in Figure 5. Then, the firing rule is as follows [23]:_  
1. _Each unreserved token deposited to input place_ \(p\) _is assigned to be reserved by the transition_ \(t_{O_{j}}\) _that satisfies the following expression:_ \[j=\underset{1\leq i\leq n}{\text{arg min}}\ \delta(i),\] (1) _where_ \(\delta(i)\) _is decided by the following condition:_ \[\delta(i)=|\frac{(\lfloor\frac{c_{i}}{a_{i}}\rfloor)}{\sum_{k=1}^{n}(\lfloor \frac{c_{k}}{a_{k}}\rfloor)}-s_{i}\ |;\] (2)
2. _When the number of reserved tokens of_ \(t_{O_{j}}\) _is not less than the required token number for the firing, the firing of_ \(t_{O_{j}}\) _is decided;_
3. _After the delay time_ \(d_{O_{j}}\) _of_ \(t_{O_{j}}\) _has passed,_ \(t_{O_{j}}\) _fires to remove the reserved tokens from the input place of_ \(t_{O_{j}}\) _and deposits unreserved tokens into the output places of_ \(t_{O_{j}}\)  
Figure 5: An example illustrating a conflict state. In the above expression (2), \(\alpha_{j}\) is the arc weight of \(e(p,t_{O_{j}})\); \(s_{j}\) is the firing probability of transition \(t_{O_{j}}\), which represents the proportion of the firing frequency of each transition within the total firing frequency of the transitions in conflict. Probability \(s_{j}\) is assigned to the corresponding transition \(t_{O_{j}}\), which is given as a constant in advance according to the event. Variable \(c_{j}\) is an accumulated number of tokens that \(t_{O_{j}}\) has reserved so far, and thus, \(\lfloor\frac{c_{j}}{a_{j}}\rfloor\) represents the number of firing times of transition \(t_{O_{j}}\) from the beginning. Expression (2) is designed to reserve the token to such a transition \(t_{i}\) that has the largest difference between the calculated firing probability \(\lfloor\frac{c_{j}}{a_{j}}\rfloor/\sum_{k=1}^{n}\lfloor\frac{c_{k}}{a_{k}}\rfloor\) and the given firing probability \(s_{j}\) among all transitions in conflict. Definition 7: (23) With the firing of transition \(t_{1}\), token amounts flowing into place \(p\) per unit of time are called \"input token-flow\" and are denoted by \(TF_{t_{1},p}\).