A mixed traffic flow feature is presented on urban arterials in China due to a large amount of buses. Based on field data, a macroscopic mixed platoon flow dispersion model (MPFDM) was proposed to simulate the platoon dispersion process along the road section between two adjacent intersections from the flow view. More close to field observation, truncated Gaussian mixture distribution was adopted as the speed density distribution for mixed platoon. Expectation maximum (EM) algorithm was used for parameters estimation. The relationship between the arriving flow distribution at downstream intersection and the departing flow distribution at upstream intersection was investigated using the proposed model. Comparison analysis using virtual flow data was performed between the Robertson model and the MPFDM. The results confirmed the validity of the proposed model.
Traffic flow in urban areas presents interrupted flow features. Due to the compression and splitting by signal lights, traffic flow is separated into series and moves downstream in platoons. Vehicles in platoon travel at different speeds because of the diverse behaviors of drivers and maneuvering characteristics of vehicles. While moving downstream, the platoon starts spreading in a longer segment which is called platoon dispersion. Platoon dispersion modeling is one of the key aspects in intelligent transportation system (ITS) area, which provides theoretical support for signal coordination control.
Many researchers have worked on the platoon dispersion topic. Pacey [
Robertson’s model in TRANSYT implies dispersion by the platoon dispersion factor for three external friction levels. Manar and Baass [
The literature review shows that researchers have doubted the distribution assumptions of both Pacey’s and Robertson’s models. However, due to the simplicity of Robertson’s recurrent equation, it has received the most popularity. Meanwhile, very few studies tried to develop a new dispersion model. Recent researches present a trend investigating the impact of heterogeneity, mixed flow, and internal frictions on platoon dispersion.
The traffic on urban arterials in China presents a mixed flow feature due to the large amount of buses. Typically, buses run on three types of facilities: normal lanes with mixed traffic, dedicated bus lanes, and bus rapid traffic (BRT) lanes.
Dedicated bus lanes and BRT lanes are special lanes separated from other traffic by road markings or physical barriers, which present unique operational features. However, urban arterials in China mostly belong to the first class, which present mixed traffic flow.
Generally, the percentage of buses in mixed traffic flow varies from 10% to 25% during peak periods. Mixed platoon presents special characteristics compared to car platoon because of the lesser maneuverability of buses and the running speed constrained by scheduled stops. Previous research has not been done on bus platoon dispersion modeling, and no car and bus mixed platoon dispersion model has been developed either. The investigation of the mixed platoon dispersion problem will provide theoretical support for signal coordination and bus priority control.
In Pacey’s platoon dispersion model, the speed is assumed following normal distribution ranging from negative to positive infinity, which does not properly reflect the field situation. Because vehicles with speeds
By modifying Pacey’s speed normal distribution, the proposed TGMD is shown in the following equation:
Assuming the start time of the green phase of upstream signal
During time differential
Then, after dividing by the time differential in both sides of (
Without loss of generality, there are three typical departing flow patterns in the actual world: stable linear flow, decreasing linear flow, and stable combined with decreasing linear flow as demonstrated in Figure
Three typical departing flow patterns.
The departing flow function of the stable linear flow pattern at the upstream intersection stop line
Then, the arriving flow function at downstream intersection location
when
when
Let
when
when
The departing flow function of the decreasing linear flow pattern at the upstream intersection stop line
Following the method for stable linear flow pattern, the arriving flow function at the downstream intersection location
when
when
Bases on (
when
when
The first item in (
The expanded Taylor series can be computed by integration. As the Taylor series method is an approximation method, for application requiring high computation accuracy the numerical integration method is needed which can be easily obtained with the help of a modern computer.
The departing flow function of the stable combined with decreasing linear flow pattern at the upstream intersection stop line
Because different classes of flows are addable, the arriving flow function at downstream intersection for the stable combined with decreasing linear flow pattern can be expressed by adding the arriving flow of the stable linear flow pattern with the arriving flow of the decreasing linear flow pattern by shifting time
The proposed MPFDM is developed as given in the previous section. If
Field data were collected for model development and validation. The surveyed road is a typical fourlane twoway urban arterial, Wushan Road, which normally operates at undersaturation traffic condition. Along this road, there are 14 bus lines, and the posted speed limit is 50 km/h (13.89 m/s). License plates were recorded by video cameras at two locations (650 m distance): one is right after the signals at Yuehan Road and the other is located right before the diverging points. Travel times were directly computed from video records, and the original speeds (journey speeds for buses, running speed for cars) were derived from travel time and distance. The data was collected from 7 : 45 AM to 10 : 40 AM. Three time periods were apparently identified based on different traffic volume levels. A statistics summary of the original car and mixed platoon speed data for all time periods is presented in Table
Origin platoon journey speed data.
Period 1 (7:45–8:25 am)  Period 2 (8:25–10:00 am)  Period 3 (10:00–10:40 am)  

Car  Bus  Mixed  Car  Bus  Mixed  Car  Bus  Mixed  
Length (m) 


 
Flow rate (veh/h) 


 
Bus traffic (%) 


 
Sample size 









Minimum speed (m/s) 









Maximum speed (m/s) 









Average speed (m/s) 









Standard deviation 









As shown in Table
The plots of the speed histogram and the fitted Gaussian mixture distribution curves are shown in Figure
Estimated parameter values of Gaussian mixture distribution based on EM algorithm.
1st component  2nd component  Iteration times  






 
Period 1 







Period 2 







Period 3 







Speed distribution histogram and fitted Gaussian mixture distribution curve of the study segment.
Period 1
Period 2
Period 3
Furthermore, performances of different distributions (including normal, lognormal, Weibull, and gamma) fitting for the mixed platoon speed data present KS evaluation
What is worth mentioning is that
Because the MPFDM demonstrated here assumes the speed following TGMD, the parameters used in the model need to be transferred from the Gaussian mixture distribution. The TGMD statistics of the vehicle speed data and the parameters estimated by EM algorithm of period 1 are summarized in Table
Statistics of TGMD of Time Period 1.
Symbol  Mixed platoon  

TGMD coefficient 

1.055 
Minimum speed (m/s) 

5.65 
Maximum speed (m/s) 

20.97 
Parameters of Gaussian 

0.8290 

0.1710  

13.6642  

8.9297  

3.2344  

4.0870 
To compare the performance of the proposed model with the Robertson model, virtual departing flow distributions for mixed platoon from upstream intersection are assumed for numeric analysis and are shown in Figure
Departing flow distribution at upstream intersection.
The virtual downstream intersections are assumed at
The arriving flow distribution at downstream locations at
Comparison of arriving flow distribution between the proposed model and the Robertson model.
Based on Figure
According to MPFDM, during time period
Compared to the Robertson model, vehicles at the front of the platoon reach the downstream intersection earlier and those at the rear of platoon spread in a shorter range for MPFDM. As the distance increases, the difference increases. This is because the platoon speed of MPFDM follows TGMD, which spreads in a narrower range in
Compared to the Robertson model, the peak of flow is lower and appears as a smooth hump for MPFDM, and the hump becomes flatter as the distance increases. This is due to faster vehicles presented in the Robertson model and the fact that the volume conservation rule cannot be violated.
Compared to the Robertson model, MPFDM presents the exact time the first vehicle and the last vehicle reaches the downstream intersection, which also reflects the fact in the field. However, vehicles travelling at a very small or even zero speed exist in the Robertson model.
Large percentage of bus flow in mixed flow affects the accuracy of platoon dispersion modeling in Pacey’s model or the Robertson model, which does not discriminate between bus traffic and car traffic. Through speed TGMD assumption, the mixed flow can be modeled by combining bus platoon with car platoon. Mixed platoon speed distribution will be influenced by the interaction between cars and buses, which is affected by flow rate, roadway function class, and percentage of buses. However, the interaction will eventually manifest a complicated speed distribution which cannot deal with simple distribution [
This strategy used here for mixed platoon modeling can be applied for all kinds of vehicletype combination. For buscar mixed traffic, only this mixed platoon dispersion model is needed; for multiple vehicle types, because platoon speed distribution can be fitted by adjusting the number of mixed components, the arriving mixed flow at the downstream intersection can be obtained. Therefore, the model has wide application value.
Vehicles with infinite speeds exist in both Pacey’s model and Robertson model, which violate the speed distribution limits (minimum and maximum speeds) in the actual world. The proposed truncated distribution assumption fixes the defect of those models.
This work was supported by the National Science Foundation of China under Contract no. 61174188. This support is gratefully acknowledged. The authors also thank Dr. Shen for his instructive advice and useful suggestion.