# SPN-Based Dynamic Risk Modeling of Fire Incidents in a Smart City
## 3 Modeling Methodology and Analysis
### SPN Model and Isomorphic MC Chain
Table 5 illustrates the identifier set of the SPN model, where \"M\" (as defined in Section 2.1) represents the marking vector that describes the current state of the Petri net. Each marking M\({}_{i}\) corresponds to a specific configuration of tokens across the places normal state (p\({}_{0}\)) to end of emergency rescue information (p\({}_{16}\)) in the model. The identifier set is essential for understanding the reachability and state transitions of the SPN model, as it provides a systematic way to track the evolution of the system during the rescue process. By visualizing the identifier set, we can better comprehend the dynamic behavior of the model and its alignment with the actual fire rescue scenario. \begin{table}
\begin{tabular}{c c c} \hline
**Place** & **Meaning** & **Transition** & **Meaning** \\ \hline p2 & Fire information & t2 & False alarm removal \\ p3 & False alarm removal information & t3 & Return to normal state \\ p4 & Expert decision information & t4 & Expert decision \\ p5 & Emergency plan upgrade & t5 & Emergency plan upgrade \\ p6 & Medical rescue support information & t6 & Upgrade plan rescue feedback \\ p7 & Emergency communication support information & t7 & Activate emergency plan \\ p8 & Road condition isolation support information & t8 & Rescue personnel in place \\ p9 & Emergency resource support information & t9 & Open emergency communication \\ p10 & Rescue end information & t10 & Road condition isolation completed \\ p11 & Communication restoration information & t11 & Allocate rescue materials \\ p12 & Road condition isolation end information & t12 & Emergency rescue start feedback \\ p13 & Emergency resource allocation end information & t13 & Emergency rescue feedback \\ p14 & Rescue information & t14 & Expert adjustment information feedback \\ p15 & Rescue feedback information & t15 & End of emergency rescue action \\ p16 & End of emergency rescue information & \\ \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline  & **P\({}_{0}\)** & **P\({}_{1}\)** & **P\({}_{2}\)** & **P\({}_{3}\)** & **P\({}_{4}\)** & **P\({}_{5}\)** & **P\({}_{6}\)** & **P\({}_{7}\)** & **P\(  
The isomorphic MC of the SPN can be obtained according to the reachability identity set of the SPN model for the rescue process, as shown in Figure 3.  
Using the Petri net performance simulation and analysis software PIPE 4.3.0, the SPN model was rigorously evaluated for boundedness, liveness, and reachability. The boundedness analysis confirmed that the model operates within predefined resource limits, ensuring that no place exceeds its token capacity and preventing resource exhaustion. The liveness analysis demonstrated that all transitions can eventually fire, indicating that the system is free from deadlocks and can continuously progress through the rescue process. The reachability analysis verified that all possible states of the system are accessible from the initial marking, ensuring that the model can adapt to various scenarios during the fire rescue mission. These analyses collectively confirm that the SPN model is safe, bounded, and free from bottlenecks, with efficient allocation and utilization of resources. The absence of deadlocks further ensures the continuity and efficiency of the rescue process. Based on  
Figure 3: Markov chain for urban fire emergency response processes. \begin{table}
\begin{tabular}{c c c c c c c c c  
these findings, it can be concluded that the SPN model is theoretically feasible and effective, providing reliable support and guidance for urban fire rescue operations.