# Median U-Turn Intersection Critical Parameter Research and Operational Performance Evaluation
## 3 Critical Parameter Calculation of MUT Intersections  
### The Calculation of the Separation Distance of MUT Intersections  
Separation distance refers to the distance between the intersection and the median U-turn. In various unconventional intersection designs, MUT intersections with a correct separation distance usually have greater operational efficiency. Thus, determining the separation distance plays a critical role in the operational performance of MUT intersection. Figure 5 shows the specific segments of the vehicle movement trajectory at an MUT intersection. Functional distance is the distance before reaching the stop line at the entrance of an MUT intersection. It represents an important component of the intersection. Entrance lanes within the functional distance often give rise to additional conflicts [29]. Vehicles have to stop in front of the stop line of the MUT intersection during the red phase. The subsequent vehicles need to slow down in order to avoid colli  
Figure 4: Signal timing scheme diagram of MUT intersection. Figure 5: The division of different segments in the vehicle movement trajectory in an MUT intersection. queuing ahead. The excessive functional distance may lead to road congestion when traffic is at peak traffic volume. To ensure the operational efficiency of an MUT intersection, the separation distance, \(L\), must be greater than the functional distance, \(L_{f}\). This study divides the functional distance, \(L_{f}\), into three sections: the queuing length of vehicles, \(L_{1}\); the deceleration distance of the vehicle, \(L_{2}\); and the vehicle's perception-reaction distance, \(L_{3}\). The functional distance, \(L_{f}\), can be calculated as follows:  
\[L_{f}=L_{1}+L_{2}+L_{3}=\frac{\beta_{i}\times\rho^{2}}{1-\rho}N\times l_{p}+ \frac{v^{2}}{2a}+v\times\tau, \tag{1}\]  
\[\rho=\frac{\lambda}{\mu\times N}. \tag{2}\]  
where \(\beta_{i}\) is the proportion of vehicles on each lane to the total number of vehicles at the entrance, \(\rho\) is the service intensity of the intersection, \(N\) is the number of lanes at the entrance of the intersection, \(l_{p}\) is the average length of the vehicle (m), \(v\) is the speed of the vehicle (m/s), \(a\) is the constant deceleration of the vehicle (m/s\({}^{2}\)), \(\tau\) is the reaction time of the driver within the function area, \(\lambda\) is the average arrival rate of vehicles on the major road (pch/h), and \(\mu\) is the average service rate of the intersection (pch/h). Because of the unconventional geometric design of the MUT, the traffic on the minor road is bound to first detour onto the major road. It is inevitable that the detoured traffic will combine with the through-straight traffic on the exits of the MUT intersection [30]. The calculation formula for the weaving length, \(L_{w}\), based on HCM 2010 is as follows:  
\[L_{w}=\frac{(V_{m}+V_{r})\times w}{S\times(1-P_{w})\times n_{w}}, \tag{3}\]  
where \(V_{m}\) is the traffic volume of the traffic on the major road entering the intersection exit (pcu/h), \(V_{r}\) is the traffic volume of the traffic on the minor road entering the intersection exit (pcu/h), \(w\) is the width of a single lane (m), \(P_{w}\) is the ratio between the weaving traffic volume and total traffic volume at the intersection exit, \(S\) is the lane service level (pcu/h/lane), and \(n_{w}\) is the number of the lane with weaving traffic. The turn-around process of the detoured vehicles from the minor road can be considered a queuing model when it enters the major road via a median U-turn. When the gap between each vehicle in the traffic on the major road is large enough for one vehicle to enter the it, a queuing vehicle can merge directly onto the major road. Otherwise, the queuing vehicles will have to wait outside the median U-turn. The arrival of the detoured vehicles at the median U-turn can be regarded to follow the Poisson distribution. The gap between each vehicle in the traffic on the major road can be regarded as following the Erlang distribution.