Traffic impacts analysis of bus stops near signalized intersections based on an optimal velocity model

Abstract

The aim of this study is to investigate the impact of the bus stop near the signalized intersection on the traffic flow from two aspects, traffic volume and delay. A modified Optimal Velocity model is established to analyze the dynamic traffic flow near a signalized intersection with curbside bus stops, and two cases including bus stops placed upstream and downstream the intersection are simulated to compare the impacts. The influencing factors, including the entering probability and the distance between a bus stop and its neighboring signalized intersection, are considered in this study. The simulation results indicate that the impact of the bus stop on the traffic flow is significant when entering probability is above the critical value, and gradually disappears as the distance increases. With respect to the volume, the downstream bus stop is greatly superior to the upstream one when the distance is less than 70 m, and slightly inferior to the upstream one when the distance ranges from 70 m to 200 m. With regard to the vehicle delay, the upstream bus stop is better than the downstream one. We hope these findings will be helpful to offer scientific guidance for the public transport planning and management.

Keywords

Bus stop signalized intersection simulation model traffic volume delay

Introduction

In the past few decades, traffic flow theories have been attracting the attention of many researchers. Traffic simulation, as a method of traffic flow theory research, is a technique of reproducing the dynamic process of traffic flow by means of computer mathematics models. In recent years, several traffic simulation models have been proposed, such as the stimulus–response model, safety distance model, optimal velocity (OV) model, artificial intelligence–based model, and others. The OV mode based on the cellular automata, in particular, has attracted a lot of attention. This model can simulate the spatiotemporal evolution of complex systems with discretization time, space, and state variables, so it is considered to be an idealization of a physical system,¹ such as traffic system.

Signalized intersections play an important role in management of the urban traffic system. With the rapid development of Chinese urbanization and motorization, traffic jam and traffic accidents in cities become more and more serious, especially at the intersections.^2,3 Of course, a coordinated signal control system is very useful to alleviate congestion.⁴ Many Chinese cities also adopted some car ownership policies to limit the number of cars on the road.⁵ In addition, public transport priority policies are also regarded as one of the effective means to solve traffic congestion. However, in order to give full play to the priority role of public transport, it is very important to set up a reasonable bus stop. Considering the convenience and safety of passengers traveling and transferring between the perpendicular bus lines,^6,7 bus stops are commonly located near the signalized intersections, which can be placed upstream or downstream of the neighboring intersection. Due to the tight constraints on streets in cities, buses always occupy a travel lane while dwelling at the stop to load and unload passengers.⁸ The dwelling bus would form a bottleneck, which may have a serious impact on the traffic flow near the intersection and constrain the performance of the intersection. Consequently, the study of the location of a bus stop near a signalized intersection is important to both the public traffic management and the traffic theory development.

Numerous studies have focused on the impacts of bus stops close to the signalized intersection.^7,9–11 In these studies, different analytical models are established to evaluate the impacts of bus stops on the performance of its neighboring intersection, especially variations of the traffic volume near the intersection. Several studies have established traffic simulation models to investigate the impacts.^8,12–15 For example, Wong et al.⁸ built a simulation model to analyze the impact of a bus stop upstream of a signalized intersection in a single-lane traffic system, where the lane-change behavior was not allowed. But real traffic is always a multilane system, and the following vehicles will change to a better driving condition lane when a leading bus dwells at the bus stop. Zhao et al.¹³ used a cellular automata model to study the impacts of bus stops which reside upstream of its neighboring intersection or downstream of it. Then they built another cellular automata model to study the effect of a bus stop between two nearby intersections.¹⁴ These two works assumed that the dwelling time of buses is constant. However, the facts of the situation are not always like this. According to a previous study, the dwelling time is always distributed randomly.¹¹ Moura et al.¹⁵ used a micro-simulation model to find the optimal location of a bus stop near a signalized intersection. However, the changes of the traffic volume near the intersection have not been elaborated.

Moreover, debates on optimal location about a bus stop relative to its neighboring intersection have not been settled. Terry and Thomas¹⁶ found that the downstream bus stop was better. Otherwise, Fitzpatrick et al.¹⁷ draw the opposite conclusion. This can be attributed to different factors being considered, such as the distance between the bus stop and the intersection, traffic volume, bus dwell time, and so on.⁸ Therefore, recent studies investigating the optimal location of bus stops close to the intersection considered some different influence factors. Zhao et al.¹³ found that the upstream bus stop was superior to the downstream one when it was not far away from the intersection. Gu et al.⁷ discovered that the downstream bus stop was better when the intersection approaches saturation, or when the bus dwell time was too long. Up to now, how to reside the bus stop near a signalized intersection under different circumstances has yet to be settled. Therefore, it is necessary to perform further analysis.

Due to the advantage of traffic flow simulation, this study attempts to build a simulation model to reproduce the dynamic process of traffic flow near the intersection with curbside bus stops in a two-lane traffic system. Because of the simplicity and few parameters, an extended OV model with lane-changing rule is established to investigate the impacts of a bus stop on traffic flow from both the traffic volume and vehicle delay aspects around its neighboring intersection. Therefore, an OV model for a signalized intersection with curbside bus stops is presented in section “The model.” In section “Simulation and results,” the simulation results are analyzed and the traffic impacts of bus stops from two aspects, traffic volume and vehicle delay, are shown. Finally, the conclusions are given in section “Conclusion.”

The model

The car-following model

In this study, the symmetric two-lane traffic system is considered and Jiang’s full velocity difference (FVD) model is used as the basic car-following regulation in which the velocity of each vehicle is adjusted instantly in the light of the distance and the velocity difference between vehicle n and its leading vehicle $n + 1$ .¹⁸ The acceleration equation in this model is written as

\frac{d v_{n} (t)}{dt} = κ [V (Δ h_{n} (t)) - v_{n} (t)] + λ Δ v_{n}

(1)

where the parameter $κ$ denotes the sensitivity of a driver’s reaction; $Δ h_{n} (t)$ is the headway between vehicle n and its leading vehicle $n + 1$ on the same lane; $v_{n} (t)$ is the velocity of vehicle n; $λ$ is a constant parameter; $Δ v_{n}$ denotes the velocity difference between vehicle n and its leading vehicle $n + 1$ , which corresponds to $(v_{n + 1} (t) - v_{n} (t))$ ; and $V (Δ h_{n} (t))$ is the OV function given as

V (Δ h_{n} (t)) = V_{1} + V_{2} \tanh [C_{1} (Δ h_{n} (t) - l_{c}) - C_{2}]

(2)

According to formula (2), the minimum expected velocity corresponds to $(V_{1} - V_{2})$ and the maximal expected velocity equals $(V_{1} + V_{2})$ . $l_{c}$ is the length of a vehicle. $C_{1}$ and $C_{2}$ are two calibration parameters.

Besides $V_{1}$ and $V_{2}$ , the values of other parameters were referred from previous studies.^1,18 Consequently, the value of the sensitive parameter $κ$ is set to be 0.41 s⁻¹, the constant parameter $λ$ corresponds to 0.5 s⁻¹, and the parameters $C_{1}$ and $C_{2}$ are calibrated as 0.13 and 1.57, respectively.

Similar to the study of Wong et al.,⁸ in this two-lane model with a signalized intersection, when approaching the intersection, each vehicle moves on the road in a car-following state. If there is no leading vehicle on the approach, the vehicle proceeds at its desired velocity and checks the signal continuously. When the signal is green, the vehicle moves forward and passes the stop line. When the signal becomes red, the vehicle calculates the minimum stopping distance $D_{min}$ according to formula (3)

D_{min} = \frac{v_{n} (t) \times v_{n} (t)}{2 \times dc e_{n} (t)}

(3)

where $v_{n} (t)$ is the instantaneous velocity of vehicle n and $dc e_{n} (t)$ denotes the deceleration of vehicle n. Considering the heterogeneity in the braking performance of drivers, in this two-lane model with a signalized intersection, $dc e_{n} (t)$ is assumed to follow a uniform distribution, $dc e_{n} (t) ~ U (a 1, b 1)$ . Here $U (a 1, b 1)$ is a uniform distribution function. Two parameters $a 1$ and $b 1$ denote the upper and lower boundaries, respectively. A uniform distribution is assumed for two reasons. On the one hand, the deceleration of vehicles is always negative, and on the other hand there are always the upper and lower boundaries of the deceleration of vehicles approaching an intersection.

Meanwhile, there will be a comparison between this minimum stopping distance and the distance from the vehicle to the stop line. If the minimum stopping distance is less than or equal to the distance to the stop line, the vehicle will be decelerated smoothly and stopped in front of the stop line. Otherwise, it will proceed continuously. If there is a leading vehicle on the approach, the vehicle will move forward according to formula (1). In addition, if there is already a stopping vehicle in front of the stop line, the minimum stopping distance will be compared with the distance from the vehicle to the stopping vehicle. If the minimum stopping distance is less than or equal to the distance to the stopping vehicle, the vehicle will be decelerated smoothly and stopped behind the stopping vehicle.

Similarly, for a bus, when approaching the bus stop, the minimum stopping distance is compared with the distance from the bus to the bus stop. If the minimum stopping distance is less than or equal to the distance to the bus stop, the bus will be decelerated smoothly and stopped at the bus stop. Otherwise, it proceeds continuously according to formula (1). If there is already a bus dwelling at the bus stop, the minimum stopping distance will be compared with the distance to the dwelling bus. When the minimum stopping distance is less than or equal to the distance to the dwelling bus, the bus will be decelerated smoothly and stopped behind the dwelling bus.

Furthermore, according to previous studies,^11,19 in this study the dwelling time $T_{d}$ of the bus is also assumed to follow a uniform distribution, $T_{n} ~ U (a 2, b 2)$ . Two parameters $a 2$ and $b 2$ denote the upper and lower boundaries, respectively.

Lane-changing rules

Generally, if there is a more ideal lane, the lane-changing behavior will be happed when a vehicle is hindered by the leading vehicle. But the lane-changing behavior must be based on two essential criteria. One is the incentive criterion which means that the driving condition of the target lane is better than the original one. For example, the vehicle could move forward at a desired speed. The other is the security criterion which means that the vehicle could not collide with the neighboring vehicles on the target lane or the original lane during the lane-changing behavior.

In Figure 1, $Δ x_{n, 1}$ denotes the distance between vehicle n and its leading vehicle n + 1 on Lane1. $Δ x_{n, 2}$ and $Δ x_{n, 3}$ denote the distances between vehicle n and its neighboring vehicles on Lane2. $Δ v_{n, 1}$ is the velocity difference between vehicle n and its leading vehicle n + 1 on Lane1 which corresponds to $(v_{n + 1} (t) - v_{n} (t))$ . $Δ v_{n, 2}$ denotes the velocity difference between vehicle n and its front vehicle on Lane2 which equals $(v_{n + 2} (t) - v_{n} (t))$ . $D_{s}$ denotes the distance between the bus stop and the signalized intersection. The lane-changing incentive criterion for vehicle n is related to the attributes $Δ x_{n, 1}$ , $Δ x_{n, 2}$ , $Δ v_{n, 1}$ , and $Δ v_{n, 2}$ . The security criterion is related to $Δ x_{n, 3}$ . Consequently, for all vehicles except the buses which are approaching the bus stop, the comprehensive lane-changing rules can be written as

Δ x_{n, 2} > Δ x_{n, 1}, Δ x_{n, 3} > d_{safe} and rand () < p

(4)

or as

\begin{matrix} Δ x_{n, 2} ⩽ Δ x_{n, 1}, Δ v_{n, 2} > Δ v_{n, 1}, Δ x_{n, 2} > v_{n} (t) \times T, \\ Δ x_{n, 3} > d_{safe} and rand () < p \end{matrix}

(5)

where $d_{safe}$ denotes the safety distance which means the critical distance between the front and rear vehicles. Here, we assume that the lane-changing probability of a vehicle is p.

Figure 1.

Configuration of a road link with an intersection and a bus stop.

For buses, when approaching or entering the bus stop, there is a different lane-changing rule to follow. Generally, in order to facilitate dwelling at the bus stop, a bus is inclined to move forward on Lane2. If a bus is proceeding on Lane2, it will not change to Lane1. If a bus is on Lane1 currently, there will be a lane-changing behavior as long as the condition of Lane2 is not worse than Lane1.

As shown in Figure 1, $D_{cl}$ is the distance from a bus on Lane1 to the end of the bus stop that is trying to change to Lane2 for the convenience of dwelling at the bus stop. $D_{bstop}$ is the distance from vehicle n to the end of the bus stop which corresponds to $(x_{bstop} - x_{n})$ . Here $x_{bstop}$ is the position of the end of the bus stop and $x_{n}$ is the position of vehicle n. As a consequence, when $D_{bstop} ⩾ 0$ and $D_{bstop} ⩽ D_{cl}$ , the comprehensive rules for a bus to change from Lane1 to Lane2 are written as

\begin{matrix} Δ x_{n, 2} > min (Δ x_{n, 1}, (v_{n} (t) - d v_{n}) \times T) Δ x_{n, 3} \\ > min (d_{safe}, v_{nback} (t) \times T) and rand () < p_{b} \end{matrix}

(6)

where $v_{nback} (t)$ is the velocity of the vehicle on Lane2 behind the bus and $p_{b}$ denotes the lane-changing probability of each bus. Since the closer the bus is to the bus stop, the more the bus tends to change from Lane1 to Lane2, and the probability $p_{b}$ increases as the distance $D_{cl}$ increases. Furthermore, there is no doubt that a bus on Lane1 which has passed the starting point of the bus stop would change to Lane2 if the lane-changing conditions are satisfied. And then the schematic of probability $p_{b}$ is shown in Figure 2, and the equation of probability $p_{b}$ can be written as

p_{b} = p + (1 - p) \times ξ (\frac{D_{cl} - D_{bstop}}{D_{cl} - L_{bstop}})

(7)

where $L_{bstop}$ denotes the length of the bus stop and $ξ (x)$ is a piecewise function, where $ξ (x) = 0$ for $x < 0$ , $ξ (x) = x$ for $0 ⩽ x < 1$ , and $ξ (x) = 1$ for $x ⩾ 1$ .

Figure 2.

The schematic of probability $p_{b}$ .

The condition $Δ x_{n, 2} ⩾ Δ x_{n, 1}$ or ( $Δ x_{n, 2} < Δ x_{n, 1}$ and $Δ x_{n, 2} > max (0, (v_{n} (t) - d v_{n} (t)) \times T)$ ) means that the driving condition of Lane1 is not much better than that of Lane2. The condition $Δ x_{n, 3} > min (d_{safe}, v_{nback} (t) \times T)$ ensures that the lane-changing behavior is safe. If a bus cannot change from Lane1 to Lane2 until it is near the end of the bus stop, it has to stop on Lane1 and wait for the chance to change lane. In addition, if the bus stop is completely occupied by other buses, the approaching bus can only be stopped nearby to wait for entering the bus stop.

Simulation and results

The simulation of this two-lane dynamic traffic state with bus stops close to a signalized intersection is carried out under the open boundary condition. At each time step, the position of the last vehicle on each lane is checked. If the position of the last vehicle is greater than $d_{safe}$ , a vehicle with maximal expected velocity gets into the corresponding lane with the entering probability $p_{e}$ . Each time step corresponds to 0.1 s. The first 10 min is discarded to prevent the unstable behavior, and the program is executed for 15 min.

There are two types of vehicles: buses and cars. The mix probability $p_{gb}$ corresponds to 0.05. In this study, the length and maximal expected velocity of buses correspond to 10 m and 40 km/h, respectively, and the length and maximal expected velocity of cars are equal to 5 m and 60 km/h, respectively. Therefore, the OV functions of two types of vehicles are defined using formulae (8) and (9).

For buses

V_{b} (Δ x_{n, 1} (t)) = 5.52 + 6.02 \tanh [0.13 (Δ h_{n} (t) - 10) - 1.57]

(8)

For cars

V_{c} (Δ x_{n, 1} (t)) = 8.0 + 8.67 \tanh [0.13 (Δ h_{n} (t) - 5) - 1.57]

(9)

The length of the road $(L_{road})$ corresponds to 5000 m. The intersection is set at the position 4000 m. Other parameters in this study are set as follows: $L_{bstop} = 29 m$ , $D_{cl} = 60 m$ , $d_{safe} = 5 m$ , and $p = 0.5$ . The length of the signal cycle is fixed and corresponds to 90 s. It is assumed that the green time is equal to the red time. According to a previous study,¹¹ the dwelling time $T_{d}$ of each bus is always uniformly distributed within the range [35 s, 55 s]. In addition, considering the comfort degree of passengers, the distribution of each vehicle’s deceleration is set as shown in formula (10).

d v_{n} (t) ~ U (- 2, - 2.6)

(10)

Impact of the bus stop on traffic volume

In order to analyze the impact of a bus stop close to the signalized intersection on the traffic volume, the traffic volume changes under different $p_{e}$ and $D_{s}$ are described as shown in Figure 3. Here traffic volume is defined as the average number of vehicles passing through the intersection. The simulation program is run 10 times and the average traffic volume is obtained.

Figure 3.

The relationship between the traffic volume and the entering probability $p_{e}$ with different $D_{s}$ : (a) for the case with a bus stop locating upstream of the intersection and (b) for the case with a bus stop locating downstream of the intersection.

As shown in Figure 3, regardless of whether the bus stop resides upstream or downstream of the intersection, with the increase of the entering probability $p_{e}$ , the volume increases until it reaches a saturated value. This saturated volume is the capacity of the intersection and the corresponding entering probability is called the critical entering probability value. When $p_{e}$ is below this critical value, the traffic on the road is free-flow, and it is easy for a vehicle to find a sufficient gap to change lane, and then a few of the passing vehicles are hampered by the stopping buses in the green period. As a result, the impact of the bus stop close to the intersection on the traffic volume is likely to be minimal. However, when $p_{e}$ is above this critical value, the traffic becomes congested-flow. The stopping buses as well as the buses waiting to stop always block other vehicles from passing through the intersection at the green light. Consequently, the impact is significant and great.

Furthermore, as shown in Figure 3, the capacity of the intersection increases as the distance $D_{s}$ increases. But, with the increase of the distance $D_{s}$ , the increment gradually drops until it is close to zero. This means that, when the distance $D_{s}$ is large enough, the impact of the bus stop on the traffic volume gradually disappears.

It can also be seen from Figure 3 that, in the case with an upstream bus stop, the increment of capacity becomes slow when the distance $D_{s}$ exceeded 140 m and, in the case with a downstream bus stop, it almost does not change when the distance $D_{s}$ is more than 200 m. In order to further analyze the different impacts on the traffic volume between the two cases, the traffic volume variation versus the distance $D_{s}$ with different entering probabilities $p_{e}$ is shown in Figure 4. Here the traffic volume variation is defined as the difference between the volume with an upstream bus stop and the volume with a downstream bus stop at the same distance $D_{s}$ and the same entering probability $p_{e}$ .

Figure 4.

The traffic volume variation versus the distance $D_{s}$ with different $p_{e}$ in the two cases.

As shown in Figure 4, the location of the bus stop influences the traffic volume at the signal intersection significantly. When $D_{s}$ is less than 70 m, the volume difference is great and negative, which means that the traffic volume in the case with an upstream bus stop is lower than the one in the case with a downstream bus stop. This implies that the downstream bus stop is greatly superior to the upstream one when $D_{s}$ is less than 70 m. On the contrary, when the distance $D_{s}$ is between 70 and 200 m, the volume difference is almost positive expect the case with p_e = 0.008, which means that the traffic volume in the case with an upstream bus stop is slightly higher than the one in the case with a downstream bus stop. This implies that the downstream bus stop becomes slightly inferior to the upstream one when $D_{s}$ ranges from 70 to 200 m. Moreover, when $D_{s}$ is more than 200 m, the volume difference becomes very little. This means that the traffic volume is roughly the same regardless of whether the bus stop is placed upstream or downstream when $D_{s}$ is more than 200 m.

Impact of the bus stop on delay

Delay is always viewed as an important measure to analyze the operating characteristics and service level of a signal intersection. Numerous models have been established to evaluate the delay (the details shown in the literature^20–25). In general, delay often refers to the difference between the ideal time and the actual time when traversing a road section,²⁵ such as an intersection. Here, the ideal time corresponds to the travel time under ideal conditions. Similarly, the actual time means the travel time under real conditions. In order to investigate the impact of the bus stop on the delay, in this study, the ideal time is calculated as

T_{ideal} = \frac{L_{rs}}{V_{ideal}}

(11)

where $T_{ideal}$ corresponds to the ideal travel time, $L_{rs}$ means the length of a road section, and $V_{ideal}$ represents the ideal velocity when traversing this road section.

In order to determine the length of the road section, the time–space diagrams of the trajectories for the simulated vehicles are depicted in Figures 5 and 6. Here the entering probability $p_{e}$ during each time step corresponds to 0.026, which comes closer to the critical value (see Figure 3). Figure 5 presents the time–space diagrams for the case with a bus stop locating upstream of the signal intersection. The time–space diagrams for the case with a bus stop locating downstream of the signal intersection is illustrated in Figure 6. As shown in Figures 5 and 6, due to the red light as well as the stopping buses or the buses waiting to stop, traffic congestions inevitably come up in the upstream sections of the intersection as well as the bus stop, which will be bound to cause a travel delay of each vehicle. Therefore, the road section used to compare the travel time should contain all the congestion sections.

Figure 5.

Time–space diagram for the case with a bus stop locating upstream of the intersection. The black lines are trajectories of cars and the red lines are trajectories of buses. The left row denotes the time–space diagram of Lane1, whereas the right one denotes that of Lane2.

Figure 6.

Time–space diagram for the case with a bus stop locating downstream of the intersection. The black lines are trajectories of cars and the red lines are trajectories of buses. The left row denotes the time–space diagram of Lane1, whereas the right one denotes that of Lane2.

Furthermore, in order to determine the ideal velocity, it is necessary to investigate the velocity trajectory for any simulated vehicle. The velocity for the 200th vehicle with different $p_{e}$ and $D_{s}$ is depicted in Figures 7 and 8. It can be seen that, regardless of whether the bus stop is upstream or downstream of the intersection, the velocity of the 200th vehicle is relatively stable in the range of 2000–3000 m. This indicates that the velocity of the 200th vehicle in this range is not affected by the intersection or the bus stop, so the average velocity of this range can be used to calculate the ideal velocity $V_{ideal}$ . Moreover, from Figures 7 and 8, it is also found that the velocity of the 200th vehicle tends to fluctuate widely at the nearby section of the intersection as well as the bus stop, especially in the range of 3000–4500 m. This means that the intersection as well as the bus stop has an observable impact on the velocity of the 200th vehicle in this range. In view of the above analysis, the starting point of the delay comparison road section is set at 3000 m and the end point is set at 4500 m. The total length is 1500 m, that is, $L_{rs} = 1500 m$ .

Figure 7.

The trajectory of velocity for the 200th vehicle for the case with a bus stop locating upstream of the intersection.

Figure 8.

The trajectory of velocity for the 200th vehicle for the case with a bus stop locating downstream of the intersection.

When $L_{rs}$ and $V_{ideal}$ are identified, the ideal time traversing this road section can be calculated according to formula (11). Meanwhile, the actual travel time for each vehicle can also be observed during the simulation. Then, the delay caused by the intersection as well as the nearby bus stop will be deduced. Here, the vehicle delay is caused by two aspects: on the one hand the problem of the red light at the intersection, and on the other hand the problem of the stopping buses. By contrasting the total delay time for the case with the upstream bus stop and the case with the downstream bus stop under the same values of $D_{s}$ and $p_{e}$ , the superiority and inferiority of these two cases can be investigated.

In order to further analyze the impact of these two parameters $D_{s}$ and $p_{e}$ on the delay, the delay time versus the distance $D_{s}$ with different entering probabilities $p_{e}$ is shown in Figure 9. Obviously, the vehicle delay is closely related to the distance $D_{s}$ and the entering probability $p_{e}$ . With the increasing value of $D_{s}$ , the delay time tends to decrease gradually. Because, when the bus stop is close to the intersection, vehicles are affected not only by a red light but also by the stopping buses at the green light on. With the increase of $D_{s}$ , vehicles only have to stop at a red light when passing through the intersection, and the delay time decreases correspondingly. Furthermore, by comparing Figure 9(a)–(f), it can be seen that, with the increasing value of $p_{e}$ , the total delay increases dramatically. Because with the increase of the entering probability, the number of vehicles on the road will increase and the gaps will decrease correspondingly. As a result, it is difficult for vehicles or buses to change lane freely. The vehicle that is hampered by the stopping buses will wait for a longer time and then the delay time will inevitably increase.

Figure 9.

The relationship between the delay and the distance $D_{s}$ with different $p_{e}$ : (a) p_e = 0.008, (b) p_e = 0.016, (c) p_e = 0.026,(d) p_e = 0.036, (e) p_e = 0.042, and (f) p_e = 0.046.

In addition, from Figure 9, the upstream bus stop is advantageous over the downstream one in terms of vehicle delay. When a bus stop is located upstream of the intersection, the stopping buses or the buses waiting to stop will hinder not only the vehicles that get through the intersection without stopping, but also the vehicles that stop and wait at a red light. On the other hand, when the bus stop is downstream of the intersection, the stopping buses will influence the traffic flow leaving the intersection, leading to small scope congestion. Hence, vehicles take more time to traverse the road section, and the total delay time is relatively longer. Furthermore, it is also shown that the superiority of the upstream bus stop is obvious, while the distance $D_{s}$ is around 50–200 m. If the distance $D_{s}$ is small, whether the bus stop resides upstream or downstream, it will make a strong impact on the traffic flow at the intersection. Otherwise, when the distance $D_{s}$ increases to more than 200 m, the impact of the bus stop will be reduced and the difference between the two cases becomes smaller. Moreover, by comparing Figure 9(a)–(f), it can be found that the greater the value of $p_{e}$ , the more obvious the superiority of the upstream bus stop. Overall, with regard to the vehicle delay, the upstream bus stop is better than the downstream one, especially when the entering probability $p_{e}$ is larger and the distance $D_{s}$ is in the range of 50–200 m.

Conclusion

Public transport has been regarded as the main means to solve urban traffic problems, such as pollution and traffic congestion. In order to facilitate passengers to travel, many bus stops will be located near the road intersection in urban areas. However, the location of the bus stops near the intersection will directly affect the traffic efficiency of the road. This study establishes an extended OV model to investigate the impacts of different bus stop locations from both the traffic volume and vehicle delay aspects.

The simulation results are summarized as follows:

The bus stop close to the intersection has a greater impact on the traffic volume only when the entering probability $p_{e}$ is above the critical value. Furthermore, as the distance $D_{s}$ between the bus stop and the intersection increases, the impact gradually disappears.

In terms of the impact on traffic volume, the downstream bus stop is greatly superior to the upstream one when $D_{s}$ is less than 70 m. Nevertheless, when $D_{s}$ is between 70 and 200 m, this superiority disappears completely and the downstream bus stop becomes slightly inferior to the upstream one.

With respect to the impact on vehicle delay, the upstream bus stop outperforms the downstream one, especially when the entering probability $p_{e}$ is larger and the distance $D_{s}$ is in the range of 50–200 m.

These results indicate that, if the entering probability $p_{e}$ is small or the distance $D_{s}$ is far enough, the bus stop can be placed upstream or downstream of the intersection. Furthermore, in terms of traffic volume and delay, the upstream bus stop is better than the downstream one when $D_{s}$ is in the range of 70–200 m. At this point, in order to decrease the transfer distance of passengers, it is favorable to place the bus stop upstream of the intersection. We hope that the results of this study may provide valuable suggestion for decision makers when determining the optimum location of a bus stop close to the intersection.

There are also two limitations in this study. One is the identical parameters in the OV model for buses and cars, such as the sensitive parameter $κ$ and the constant parameter $λ$ . Actually, the moving processes of buses are always different from cars. The further study is to calibrate the OV model for buses with the field data. The other is the fixed cycle time and green time in the simulation. According to the results of Zhao et al.,¹³ the traffic cycle time made a significant impact on the relationship between the maximum traffic volume and $D_{s}$ . Therefore, investigating the impact of the bus stop on the traffic flow with different cycle time is critical. This will also be the study emphasis in our further works. In addition, we will continue to evaluate the model and optimization method through some specific intersection cases to improve the model in the future.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the Higher Educational Scientific Research Projects of Inner Mongolia Autonomous Region (Project No. NJZY17012).

ORCID iD

Zhenyu Liu

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