# Hidden Markov Model-Based Dynamic Hard Shoulders Running Strategy in Hybrid Network Environments
## 4 Numerical Results and Discussion
### Scene Introduction
#### 4.1.2 Validation of a Hidden Markov Model-Based Traffic Breakdown Prediction Model  
After obtaining the traffic breakdown critical speed, the historical data were combined to calculate the hidden state transition probability matrix A and the two-state corresponding probability matrix B. We defined the state of a road section as 1 when it was in the traffic breakdown state, and 0 otherwise, as shown in Figure 9. Since the \(\rho\) of the CAV parameter is taken into account in the hybrid grid-connected environment, the value within the hidden state transition probability matrix is a variable that varies with \(\rho\). As seen in Figure 9, as the penetration rate \(\rho\) increases, there are progressively fewer transfers of road segment states from 0-0, replaced by an increase in 1-1, which indicates an increase in the probability of road segment collapse as the penetration rate increases. For the convenience of presentation, this paper only showed the hidden state transition probability matrix when the penetration rate \(\rho\) was 0 in Table 1. Table 2 shows the state of the collapse state corresponding to \(K(t)\), \(\rho(t)\), where the density of the road section is processed, 18 veh/km was selected as the interval, and K was divided into \(Z_{max}\) segments equally from 0 to \(K_{max}\), and the number of data points in each interval was taken as the value of the interval.  
Figure 8: Breakdown speed characteristics: (**a**) Relationship between average speed and the standard deviation \(\sigma\) of the corresponding speed; (**b**) \(V_{c}\) under different \(\rho\).  
Figure 9: The event probability with different \(\rho\).  
After obtaining the above traffic breakdown prediction parameters, the accuracy of the prediction model was tested by running the simulated road network, and at the same time, the traffic breakdown prediction model based on the logistic model was used for comparison (denoted as HMMD and LMD, respectively). The relationship between the probability of the occurrence of traffic breakdown and the traffic density of the upstream road section under different penetration rates was finally obtained, as shown in Figure 10. In order to explore the difference in critical density under HMMD and LMD more intuitively, the traffic density interval when no traffic breakdown occurs was defined as the stable traffic density area, which is shown in Figure 11. With \(\rho\) in the range of 0.1-0.6, HMMD can detect a traffic breakdown phenomenon at a much lower traffic density. At other penetration rates, the LMD model will determine that a lower traffic density will produce traffic breakdown.  
In addition, HMMD and LMD were applied to predict the collapse state of the road network, respectively, and the prediction results are shown in Table 3. In predicting the number of traffic breakdowns occurring at 1763 moments of simulation, HMMD and LMD predicted that traffic breakdowns would occur 1235 and 1311 times, respectively, whereas 1212 traffic breakdowns occurred in reality. Therefore, it can be obtained that the prediction accuracies of HMMD and LMD were 0.9870 and 0.9438, respectively, and the prediction accuracy of HMM was 4.32% higher than that of the LM prediction model.  
\begin{table}
\begin{tabular}{c c c} \hline  & **0** & **1** \\ \hline
0 & 0.6939 & 0.0170 \\
1 & 0.0170 & 0.2721 \\ \hline \end{tabular}
\end{table}
Table 1: Hidden state transition probability matrix.  
Figure 10: The relationship of traffic density and traffic break under different \(\rho\): (**a**) HMMD prediction method; (**b**) LMD prediction method.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **(0,0)** & **(1,0)** & **(2,0)** & **—** & **(Zmax,1)** \\ \hline
0 & 0 & 0 & 0 &... & 0.0034 \\
1 & 0 & 0 & 0 &... & 0 \\ \hline \end{tabular}
\end{table}
Table 2: Two-state correspondence probability matrix.