# Hidden Markov Model-Based Dynamic Hard Shoulders Running Strategy in Hybrid Network Environments
## 2 Hybrid Network Expressway Scene Modeling
### Hybrid Network Traffic Flow Modeling  
HVs are often not very capable of obtaining road instructions. This has been reflected in many studies on the acquisition rate of expressway road instruction [41; 46]. While connected autonomous vehicles are frequently equipped with on-board units (OBUs) that can be integrated into any car or used as a vehicle simulator in the absence of actual traffic, allowing real-time information exchange between the vehicle and the infrastructure [47] and ensuring full access to road indication. Therefore, in the subsequent work, we assumed that CAVs can obtain all road information indicated by variable message signs (VMSs). Considering that hard shoulder opening messages are often displayed on VMSs, which are similar to expressway routing instructions, the information acquisition rate of human-driving vehicles on expressway hard shoulders conforms to the Logit utility model.  
\[Pc(w_{1},w_{2})=\frac{exp(a+bw_{1}+cw_{2})}{1+exp(a+bw_{1}+cw_{2})}, \tag{1}\]  
\(Pc\) is the probability that HV accepts VMS information. This is affected by two factors: \(w_{1}\) is the driver's driving experience, and \(w_{2}\) is the driver's personality. \(a\), \(b\), and \(c\) are the fitting parameters, respectively.  
In a mixed traffic flow randomly composed of CAVs and HVs, we defined \(\rho_{CAV}\) (\(0\leq\rho_{CAV}\leq 1\)) as the penetration rate of CAVs. When the front vehicle of the CAV has a vehicle-to-vehicle communication function, these CAVs will adopt collaborative adaptive cruise control (CACC) mode, known as CACC vehicles [48; 49; 50]; otherwise, they will follow with adaptive cruise control (ACC), known as ACC vehicles, as shown in Figure 2[51; 52]. Currently, the most common forms of car-following are:  
Figure 1: An expressway with hard shoulder opening applies.  
Figure 2: Changes in the CAV follower model.  
HDV follows HDV.  
When the vehicles are both human-driven cars, and the driving behavior or speed of the front vehicle suddenly changes, the rear car needs a certain amount of reaction time to make a judgment and then adjust its driving behavior. Since the proportion of HDV is \(1-\rho\), the theoretical proportion of this following mode is \(P_{1}\).  
\[P_{1}=(1-\rho_{CAV})(1-\rho_{CAV}) \tag{2}\]  
HDV follows CAV.  
\(P_{2}\) is when the front automobile is connected and automated and the car behind is human-driven. When the status of the front vehicle changes, the vehicle behind still requires some reaction time to adjust. Therefore, it is the same as the first following mode with a proportion of \(P_{2}\).  
\[P_{2}=\rho_{CAV}(1-\rho_{CAV}) \tag{3}\]  
CAV follows HDV.  
When the front car is human-driven and the car behind is connected and automated, the car behind is outfitted with relevant sensing equipment (such as cameras, etc.). Therefore, when the lead car abruptly accelerates or decelerates, the on-board sensors of the following car precisely capture the behavior in front and make quick decisions. Compared with Formulas (1) and (2), the expected headway is shorter. The proportion of this following mode is \(P_{3}\).  
\[P_{3}=(1-\rho_{CAV})\rho_{CAV} \tag{4}\]  
CAV follows CAV.  
When the vehicles are both connected and automated, they can share real-time road information and driving behaviors, achieving \"vehicle-to-vehicle communication\". For instance, if the front vehicle decides to speed up and slow down, the following vehicle can receive that at the same moment, allowing the corresponding decision to be made promptly. In this car-following mode, the front and rear vehicles can be treated as a whole, and the expected headway is the shortest. According to the analysis, the proportion of this car-following mode is \(P_{4}\).  
\[P_{4}=\rho_{CAV}{}^{2} \tag{5}\]