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The capacity drop caused by the combined effect of the intersection and the bus stop in a CA model
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Physica A 385 (2007) 645–658 www.elsevier.com/locate/physa
The capacity drop caused by the combined effect of the intersection and the bus stop in a CA model Xiao-mei Zhao, Zi-you Gao, Bin Jia School of Traffic and Transportation,Beijing Jiaotong University,Beijing 100044,China Received 28 September 2006; received in revised form 10 May 2007 Available online 28 July 2007
Abstract The aim of this work is to investigate the combined effect of the signalized intersection and its near-by bus stop, by using a two-lane CA model. Four cases that the stop locates upstream or downstream the intersection, and ones with the special stop lane or not are considered. The effect of the distance LD between the stop and the intersection on the capacity is studied, with respect to the traffic light cycle T and the bus dwell time T s . It is found that acting as a bottleneck, the bus stop near the intersection causes the drop of the capacity. The negative effect only appears below a critical point LDc , which is related to the T and the T s in no stop lane cases. The larger T and T s have the tendency to create the higher loss of the capacity. While for stop lane cases, the critical value LDc changes little. Comparisons among four cases suggest that the special stop lane can effectively enhance the capacity, and the downstream stops perform better than the upstream ones at small LD or small T or large T s . The results imply that the capacity can be maximized by adjusting both the position of the bus stop and the cycle time, or adding a special stop lane. These findings may be useful to offer scientific guidance for the management and the design of traffic networks. r 2007 Elsevier B.V. All rights reserved. PACS: 89.40.a; 64.60.Cn; 05.10.a; 05.70.Fh Keywords: Intersection; Bus stop; The combined effect
1. Introduction Recently, the dynamics of traffic systems have been broadly investigated by using various traffic models [1,2], such as cellular automaton (CA) model, car-following models, gas kinetic models and hydrodynamic models. Especially, the CA models have been extensively studied to understand traffic behaviors, due to its simplicity and flexibility in modelling different aspects of real traffic. Intersections are key ingredients in the management of the traffic systems. The study of intersections is of significance on both offering scientific advices for real traffic management and the development of traffic theory. Therefore, it is worthy of devoting to understand the traffic behaviors controlled by traffic lights. The traffic dynamics controlled by traffic lights have been revealed by different ways over the past several years, Corresponding author.
E-mail addresses: [email protected] (X.-m. Zhao), [email protected] (Z.-y. Gao), [email protected] (B. Jia). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.07.040
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and some useful achievements are obtained. Brockfeld et al. have studied optimizing traffic lights for city traffic and found that the problem of the optimal cycle time in the network equals to that of a single street [3]. Sasaki and Nagatani studied the traffic flow controlled by traffic light, based on an optimal velocity car-following model [4]. Huang and Huang adopted the NS(NaSch) model to study the traffic flow controlled by traffic light [5]. The effects of traffic flow states on different roads on capacity of T-shaped intersection system are analyzed by Wu et al. [6]. Jiang and Wu developed a stopped time dependent randomization CA model to explain the dependence of the saturated current on the cycle time [7]. However, little information has been published concerning the impacts of the bus stop close to the signalized intersection. Nowadays, researches on the traffic behaviors controlled by traffic lights mainly focus on the effects of different traffic policies on the traffic capacity, without considering the interference with other traffic conditions, like close-by bus stops. That bus stops are set near intersections to facilitate passengers passing through intersections is often observed. If the bus stop is close to the stop line of an approach to a signalized intersection, buses waiting to stop accumulate outside the bus stop. The stopping buses seriously affect the performance of the intersection. It is really true in Beijing that stopping buses often hinder other vehicles to move forward in the green-light time. Only Wong et al. [8] investigated the delay at signal-controlled intersection with bus stop upstream. The traffic lights can operate as a bottleneck. And it is argued for us that the bus stop might have a bottleneck effect on its close-by intersection and would strongly influence the traffic behavior. That is to say, there may be interactions between the bus stop and the signalized intersection. As mentioned above, traffic lights are significant entities to manage urban traffic system. Obviously, bus stops are crucial for urban public transport. Thus, the study on the combined effects between the intersection and its neighboring bus stop is very important. In this paper, the combined effects of the upstream (downstream) bus stop and the signalcontrolled intersection will be discussed, using a new two-lane CA model. The following problems will be focused on. How the bus stop interferes with the traffic flow controlled by the traffic light? How the transition from free flows to congestions occurs? Are upstream cases and downstream cases different distinctly? Could the capacity be enhanced by adding special stop lane or changing the position of the near-by bus stop? This paper is organized as follows: the models used in the simulations are introduced in Section 2. In Section 3, the simulation results are discussed. Finally, our conclusions are summarized. 2. Models Intersections and bus stops exist widely in city traffic system. Sometimes the bus stop locates downstream the intersection, and sometimes upstream the intersection. The bus stops mainly have two forms, stops with special stop lane or not. Therefore, four cases will be considered. Case 1(2) denotes that the bus stop without(with) the special stop lane lies upstream the intersection. Case 3(4) denotes that the bus stop without(with) the special stop lane lies downstream the intersection. These cases are illustrated in Fig. 1. The road with two lanes, the right and the left, consists of six sections, including sections A, B, C, C1, D and E. Section A is the start of the road. Section B is the upstream part near the bus stop and special lane-changing rules will be used in this section. Section C is the bus stop. Section C1 is the downstream part of the bus stop. Section D is the part between the bus stop and the intersection. Section E is the end of the road. There are two types of vehicles, cars with one-cell lengths and buses with two-cell lengths. Buses mix in the traffic flow with the mixture probability Pm , which is the percentage of buses. And some of the buses must halt at the bus stop, called as stoping buses, and the others need not, called as non-stop buses. The stop probability Ps stands for the percentage of the stoping buses. T s denotes the dwell time of stoping buses at the stop. A two-lane CA model will be used in this system. The vehicle movement includes forward motions and lanechanging motions. 2.1. Forward motions The NS model is used to control the vehicles moving forward. The road is divided into L cells. Each cell may be either empty or occupied by one vehicle with an integer velocity v between 0 and the speed limit vmax . At each discrete time step t ! t þ 1, the system is updated by the following rules: (1) acceleration, vn ! minfvn þ 1; vmax g; (2) deceleration, vn ! minfvn ; d n g; (3) randomization, vn ! maxfvn 1; 0g with
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Traffic Light
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Fig. 1. Configuration of the road in the simulations.
probability p; (4) motion, xn ! xn þ vn . Here, xn and vn are the position and velocity of vehicle n, d n ¼ xn1 xn l n1 is the gap between vehicle n and its preceding vehicle n 1, l n1 is the length of vehicle n 1. p is the randomization probability. When the traffic light is red, the gap of the nearest vehicle behind the intersection should be calculated as d n ¼ xlight xn , where xlight is the position of the intersection. If the nearest vehicle behind the bus stop is a stopping bus, the gap between it and its front vehicle is computed as d n ¼ xbstop xn , where xbstop is the end position of the bus stop. In cases with the stop lane, if the leader on the stop lane has stopped, its front gap is given as d n ¼ xC1 xn , where xC1 is the position of the end of the bus stop lane. Assume the cycle time of the traffic light is denoted as T, and the green-light time equals to the red-light time. Since the bus has the length of two cells, it may stay at the intersection when the traffic light changes from green to red. To avoid the situation, when the leader vehicle n upstream the intersection is a bus in the last time step of the green light, the following rules should be used after the velocity and position update process, if xhead ¼ xlight þ 1 and vn ovmax and xlight þ 2 is empty then vn ¼ vn þ 1; xhead ¼ xlight þ 2, otherwise n n vn ¼ vn 1; xhead ¼ x , light n where xhead is the position of the head of the bus n. The rule means that the bus should accelerate one unit to n pass through the intersection, if its head lies in the next cell of the intersection and its velocity is less than the
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maximal velocity, and its front cell is empty; otherwise, the bus should decelerate one unit to wait behind the intersection. For the stopping bus n staying at the bus stop, if its dwelling time step TS n is less than T s , then it cannot move forward, and the dwelling time is updated as TS n ¼ TS n þ 1. Otherwise, the bus becomes a non-stop bus. 2.2. Lane-changing motions For simplicity, all cars and non-stop buses on the road and the stopping buses which do not lie in Sections B and C will change lanes symmetrically, if the following condition is fulfilled: d n ominfvn þ 1; vmax g and d n;other 4d n and d n;back 4d safe ,
(1)
d n;other ðd n;back Þ represents the gap between the nth vehicle and its preceding (back) vehicle on the destination lane at time step t, d safe is a safety distance to avoid crash. In Sections B and C, special lane-changing rules will be employed for the stopping buses. For simplicity, the lane-changing rules for off-ramp traffic systems in Ref. [9] are used in this paper, because the lane-changing behaviors of vehicles on the main road, which leave the main road and enter the off-ramp, are similar to those of buses entering the bus stop. The drivers of these buses are willing to run on the right side lane to stop conveniently when they are close to the bus stop. These buses will change from the left lane to the right lane as long as conditions on the right lane are not worse than those on the left lane. Namely, if the following condition is satisfied: ½d n ¼ d n;other ¼ 0 or ðd n;other a0 and d n d n;other p2Þ and d n;back 4vob ,
(2)
the stopping bus will change from the left lane to the right lane. vob represents the velocity of its back vehicle on the destination lane. Condition d n ¼ d n;other ¼ 0 means that there is no gap to move forward on both lanes in the next time step; Condition d n;other a0 and d n d n;other p2 means that the road situation on the present lane is not much better than that on its right neighbor. d n;back 4vob means that there are enough space to guarantee the safety. The same changing rules of stopping buses in Eq. (2) are used for both lane-changes from the left lane to the right lane in Sections C and C1 and from the right lane to the stop lane in Sections B and C. If a stopping bus cannot change to the right destination lane, until it approaches the position xbstop of the bus stop, it would stop to wait for the change chance (e.g. it will change the lane as soon as the corresponding position on its right-side lane is empty). In Sections B and C, the stopping buses are prohibited changing from the present lane to its left-side lane. Cars and non-stop buses are forbidden running on sections B and C for no stop lane cases, while forbidden running on the stop lane for the cases with the special stop lane. The stopping buses will become non-stop buses, and can change to its left neighbor by using the rules in Eq. (2). If a bus on the stop lane cannot change to the left-side destination lane, until it reaches the position xC1 , it would stop to wait for the change chance mentioned above. 2.3. Boundary conditions The simulations are carried out under open boundary condition. In each time step, when the update of vehicles on the road is finished, the positions xlast of the last vehicles on each lane is checked. If xlast 4vmax , a vehicle with velocity vmax is injected with the entering probability Pe at the cell minfvmax ; xlast vmax g. The leading vehicle on each lane is removed, if its position xfirst 4L and its following vehicle becomes the new leader. 3. Simulation results and discussions In the simulations, the road is divided into L ¼ 500 cells. Section C contains LC ¼ 6 cells, section E contains LE ¼ 100 cells, section D contains LD cells. When the bus stop locates downstream the intersection, section B consists of LB ¼ minf8; LD g cells, section C1 consists of LC1 ¼ 8 cells, and section A contains LA ¼ L LC LD LE . For the upstream cases, LB ¼ 8 cells, LC1 ¼ minf8; LDg cells, LA ¼ L LC LD LE . Each cell
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corresponds to 7.5 m and each time step corresponds to 1 s. The model parameters are set as follows: vcar max ¼ 3, car bus vbus ¼ 2, d ¼ 3, d ¼ 2, p ¼ 0:3, T ¼ 100 s, where the superscript car(bus) denotes the parameter of the max safe safe car (bus). According to the Transit Co-operative Research Program (TCRP) Report 19 (1996) [10], the average peak-period dwell time exceeds 30 s per bus, so T s is set as 30 s. The mix probability Pm is 0.15, and the stop probability Ps is 0.3. The first 50,000 time steps are discarded to avoid the transient behavior. The flow is averaged by 100,000 time steps. 3.1. Cases the bus stop locating upstream the intersection In this section, the cases that the bus stop lies upstream the intersection (case 1 and 2) are considered. Firstly, the effect of the distance LD between the bus stop and the intersection on the traffic flow is investigated. The capacity versus the entering probability for different LD is shown in Fig. 2 ((a) case 1
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Fig. 3. The capacity versus the LD . Circle denotes case 1 and square denotes case 2.
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Fig. 4. The time–space diagram for case 1 (a1–b2) and case 2 (c1–d2) from 149,500 to 150,000. Black points denote cars, red and green points denote non-stop and stopping buses. The left denotes that of left lane, the right denotes that of right lane. (a,c) LD ¼ 3, (b,d) LD ¼ 15.
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(without the stop lane), (b) case 2 (with the stop lane)). It can be seen that the flow increases with the entering probability until the flow reaches a saturated value. There is a critical point of the entering probability Pec , above which the traffic flow on the road transforms from free flow to congested flow. The saturated flow is the capacity of the intersection, denoted as Qc . In comparison with no bus stop case, the capacity in both case 1
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and 2 drops when the LD is small. The drop of case 1 is much larger than that of case 2. While for enough large LD , the capacities in both case 1 and case 2 are the same with no stop case. This means that on both cases 1 and 2, the upstream bus stop has a bottleneck effect on the intersection at small LD , and the negative effect of case 1 is much stronger than that of case 2. Fig. 3 depicts the relationship between the Qc and the LD . It is clear that the Qc increases with the LD , and approaches a maximum Qc;max ¼ 0:407 at a critical value LDc;Q . The LDc;Q1 (LDc;Q2 ) is 36 (9) for case 1 (2). The Qc in case 1 is less than that in case 2 below LDc;Q1 . To understand these phenomena, the time–space diagram and the velocity distribution are shown in Figs. 4 and 5, respectively. From Fig. 4(a1,a2) and (c1,c2), it is obvious that due to the stop near the intersection, the waiting queue upstream the intersection grows stretching to the bus stop. The queue upstream the intersection mixes with the coming stopping buses. In case 1, during green time cycle, lots of blanks downstream the intersection appears. That means that the stopping buses at the bus stop and buses waiting to stop obstruct other vehicles to pass through the intersection in the green-light period (see Fig. 4(a2)). Whereas in case 2, the special stop lane can provide more spaces near the bus stop for waiting-stop buses, and seldom vehicle passing through is hindered by stopping buses in the green light time step (see Fig. 4(c2)). These also can be verified by the velocity distribution in Fig. 5. It can be seen that vehicles with larger speeds in case 2 are much more than those in case 1 on both the lanes, when LD is small. Thus, the capacity in case 2 is greater than that in case 1 at small LD . When the LD is larger, less vehicles blend with stopping buses behind the bus stop. Therefore, the
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capacity raises with the increase of the LD . If LD is enough large, the bus stop would not bother vehicles behind the intersection. The capacity can approach its maximum. These results suggest that the upstream bus stop near the intersection has a strong bottleneck effect on the intersection. The negative effect decreases with the increase of the LD . Also, the special stop lane can create more benefit. And these two policies provide no additional effects when the LD 4LDc;Q . In Fig. 6, the phase diagram in (Pe ,LD ) space is plotted. The space can be divided into three regions. The solid-dot-dot line represents case 1, and the solid line represents case 2. The traffic flow on the road is free in region I, while it is congested in both regions II and III. The capacity in region II is less than the maximal capacity in region III, because the shorter the LD is, the stronger the bottleneck effect becomes. Comparisons as indicated in Fig. 6 reveal that the critical point of the entering probability Pec has little difference between the two cases. Due to the benefit of the special lane, the region III of case 2 is wider than that of the case 1. Next, the influence of LD on the capacity Qc under conditions of different T is studied. Fig. 7 displays the relationship between the capacity and the LD for various values of T. It is found that there exists a critical value LDc;T , below which the bus stop creates a negative effect on the capacity. The critical values LDc;T1 and
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Fig. 8. The relationship between the capacity and the LD for different T s . (a) T s ¼ 5, (b) T s ¼ 10, (c) T s ¼ 20, (d) T s ¼ 30, (e) T s ¼ 40, (f) T s ¼ 50.
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LDc;T2 are 24 and 9 (T ¼ 50), 36 and 12 (T ¼ 100), 39 and 12 (T ¼ 150) and 42 and 15 (T ¼ 200) for cases 1 and 2, respectively. The critical point LDc;T1 of case 1 greatly depends on the T and becomes larger and larger with the increase of the T. Whereas the critical point LDc;T2 of case 2 changes little with the T. Because in case 1 the longer the cycle time is, the longer the queue behind the traffic light, and the stronger the combined effect of the upstream bus stop and the intersection will become. This means that the cycle time of the signalcontrolled intersection cannot be too long, when a upstream bus stop is close to it, especially in case 1. It is also observed that there exists the capacity difference between the two cases below a critical value LDc;T0 , and the capacity difference in the two cases vanishes at LDc;T0 . Obviously, the LDc;T0 is equal to LDc;T1 . That means that the capacity can be enhanced by either increasing the LD or adding a stop lane if LD oLDc;T0 . Finally, the influence of LD on the capacity Qc is studied in the case of different T s . The relationship between the capacity Qc and the length LD is displayed in Fig. 8. Similarly, at small LD the capacity Qc increases with the LD and approximately becomes a constant when LD is greater than a critical value LDc;Ts (LDc;Ts1 for case 1 and LDc;Ts2 for case 2). This means that the capacity can be enhanced by increasing the LD and the further increase of the LD will have no effect if LD 4LDc;Ts , because the waiting queue upstream the intersection has less interference with waiting-stop buses, as the large LD . And at the critical value LDc;Ts , the interference disappears. Also, the impact of the LD in case 1 is much more obvious than that in case 2. LDc;Ts1 is dependent on the T s , whereas LDc;Ts2 hardly varies with the T s . The capacity difference between the two cases vanishes at a critical value LDc;Ts0 , which equals to LDc;Ts1 . Different from the situation in Fig. 7, the maximal capacities Qcmax;Ts for case 1 (Qcmax;Ts1 ) decline greatly with the T s , and the maximal capacities for case 2 (Qcmax;Ts2 ) change little. As the dwell time becomes large, stopping buses waiting outside the bus stop accumulate greatly and obstruct all lanes for case 1, leading to the drop of the maximal capacity. While for case 2 these stopping buses have more chances running on the right lane and the stop lane, as a result the capacity drop is slight. Therefore, the magnitude of the capacity drop in case 2 is much smaller than that in case 1. 3.2. Cases the bus stop locating downstream the intersection In this section, the cases that the bus stop locates downstream the intersection (Cases 3 and 4) are considered. Similarly, the effect of the length LD between the downstream bus stop and the intersection on the 0.42
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Fig. 9. The capacity versus the LD for cases 1–4. The inlet is the amplification of the range [3, 30].
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Fig. 10. The time–space diagram for case 3 (a1–b2) and case 4 (c1–d2) from 149,500 to 150,000. Black points denote cars, red and green points denote non-stop and stopping buses. The left denotes that of left lane, the right denotes that of right lane. (a,c) LD ¼ 3, (b,d) LD ¼ 15.
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Fig. 11. The relationship between the capacity and the LD for different T.
traffic flow is investigated in respect of the cycle time and the dwell time, as shown in Figs. 9–12. The simulations show that the situations in the downstream cases qualitatively resemble those in the upstream cases. When the downstream bus stop is close to the signal-controlled intersection, it negatively affects the capacity. The increase of the LD can lessen the bottleneck effects when LD oLDc , and adding the special stop lane can create more benefit on the capacity. The critical values LDc;T and LDc;Ts for case 3 increase with the T and T s , while those for case 4 almost not depends on the T and T s . There is a little difference between the upstream cases and the downstream cases at small LD . The situations of the downstream cases have less capacity drop than those of the upstream cases at small T or large T s . From the time–space diagram shown in Fig. 9, it can be observed that in the course of red light there is no any hindrance downstream the bus stop for the waiting-stop buses to proceed. Some or all of the buses finish their stoping procedure at the stop and leave out of the bus stop to move forward further. Thus, the queue constituted by the stopping buses will dissolve before the upcoming green period. While for the upstream cases, the queue behind the intersection still hinders stopping buses waiting upstream the stop in the red-light time step. More and more vehicles continue to accumulate upstream the bus stop, and the situation gets worse. Thus, it can be concluded that the bus stop should be set downstream the intersection, especially for long dwelling time and short time cycle of traffic light.
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Fig. 12. The relationship between the capacity and the LD for different T s .
4. Conclusions The combined effect of the signal-controlled intersection and the near-by bus stop is studied by using a twolane CA model. Four cases are considered, according to two conditions, that the stop is upstream or downstream the intersection, and the special stop lane exists or not. The simulations results indicate that the bus stop near the intersection serves as a bottleneck. The traffic flow transits from free flow to the congestion at a critical entering probability, where the flow is saturated and corresponds to the capacity. The capacity grows gradually with the increase of the distance LD between the stop and the intersection, and reaches a maximum at a critical point LDc , which is mainly affected by two significant factors, the traffic cycle time T and the dwell time T s in no stop lane cases (cases 1 and 3). For no stop lane cases, the larger the T and the T s are, the larger the critical value of the LD are, and the stronger the damage are. This means that the T should not be set too long when a bus stop near the traffic light. Additionally, comparisons among four cases reveal that the capacity using the special stop lane is greater than that without the lane, and the benefit disappears
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when LD is large enough; there is no qualitative difference between the upstream and the downstream cases, and the capacity of the upstream cases is appreciably less than that of the downstream ones, and the difference decreases with the LD . These findings imply the following:
A special stop lane is necessary and worthy to diminish the damage of the negative effect and guarantee higher capacity of the traffic flow if a bus stop is much close to intersections. While its benefit can be negligible when a bus stop much far away from intersections. It is favorable to replace the upstream bus stop with the downstream one.
These implications may provide suggestions on positioning the bus stop near-by a intersection and the design of the traffic light cycle time. It should be mentioned that in this work the simplest situation that identical models of forward motions for buses and cars are used. In reality, there are distinctions between buses and cars. The future works are to consider differences in the acceleration(deceleration) between buses and cars and to validate in practice. Acknowledgments This paper is financially supported by 973 Program (2006CB705500), Project (NSFC nos. 70631001, 70471088 and 70501004), and Program for Changjiang Scholars and Innovative Research Team in University (IRT0605). The authors thank anonymous reviewers for the critical remarks and good suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
D. Chowdhury, L. Santen, A. Schadschneider, Phys. Rep. 329 (2000) 199. D. Helbing, Rev. Mod. Phys. 73 (2001) 1067. E. Brockfeld, R. Barlovic, A. Schadschneider, M. Schreckenberg, Phys. Rev. E 64 (2001) 056132. M. Sasaki, T. Nagatani, Physica A 325 (2003) 531. D.W. Huang, W.N. Huang, Phys. Rev. E 67 (2003) 056124. Q.S. Wu, X.B. Li, M.B. Hu, R. Jiang, Eur. Phys. J. B 48 (2005) 265. R. Jiang, Q.S. Wu, Physica A 364 (2006) 493. S.C. Wong, H. Yang, W.S. Au yeung, S.L. Cheuk, M.K. Lo, J. Transp. Eng. 124 (1998) 229. B. Jia, R. Jiang, Q.S. Wu, Phys. Rev. E 69 (2004) 056105. R.Z. Koshy, V.T. Arasan, J. Transp. Eng. 131 (2005) 640.
Physica A 385 (2007) 645–658 www.elsevier.com/locate/physa
The capacity drop caused by the combined effect of the intersection and the bus stop in a CA model Xiao-mei Zhao, Zi-you Gao, Bin Jia School of Traffic and Transportation,Beijing Jiaotong University,Beijing 100044,China Received 28 September 2006; received in revised form 10 May 2007 Available online 28 July 2007
Abstract The aim of this work is to investigate the combined effect of the signalized intersection and its near-by bus stop, by using a two-lane CA model. Four cases that the stop locates upstream or downstream the intersection, and ones with the special stop lane or not are considered. The effect of the distance LD between the stop and the intersection on the capacity is studied, with respect to the traffic light cycle T and the bus dwell time T s . It is found that acting as a bottleneck, the bus stop near the intersection causes the drop of the capacity. The negative effect only appears below a critical point LDc , which is related to the T and the T s in no stop lane cases. The larger T and T s have the tendency to create the higher loss of the capacity. While for stop lane cases, the critical value LDc changes little. Comparisons among four cases suggest that the special stop lane can effectively enhance the capacity, and the downstream stops perform better than the upstream ones at small LD or small T or large T s . The results imply that the capacity can be maximized by adjusting both the position of the bus stop and the cycle time, or adding a special stop lane. These findings may be useful to offer scientific guidance for the management and the design of traffic networks. r 2007 Elsevier B.V. All rights reserved. PACS: 89.40.a; 64.60.Cn; 05.10.a; 05.70.Fh Keywords: Intersection; Bus stop; The combined effect
1. Introduction Recently, the dynamics of traffic systems have been broadly investigated by using various traffic models [1,2], such as cellular automaton (CA) model, car-following models, gas kinetic models and hydrodynamic models. Especially, the CA models have been extensively studied to understand traffic behaviors, due to its simplicity and flexibility in modelling different aspects of real traffic. Intersections are key ingredients in the management of the traffic systems. The study of intersections is of significance on both offering scientific advices for real traffic management and the development of traffic theory. Therefore, it is worthy of devoting to understand the traffic behaviors controlled by traffic lights. The traffic dynamics controlled by traffic lights have been revealed by different ways over the past several years, Corresponding author.
E-mail addresses: [email protected] (X.-m. Zhao), [email protected] (Z.-y. Gao), [email protected] (B. Jia). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.07.040
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and some useful achievements are obtained. Brockfeld et al. have studied optimizing traffic lights for city traffic and found that the problem of the optimal cycle time in the network equals to that of a single street [3]. Sasaki and Nagatani studied the traffic flow controlled by traffic light, based on an optimal velocity car-following model [4]. Huang and Huang adopted the NS(NaSch) model to study the traffic flow controlled by traffic light [5]. The effects of traffic flow states on different roads on capacity of T-shaped intersection system are analyzed by Wu et al. [6]. Jiang and Wu developed a stopped time dependent randomization CA model to explain the dependence of the saturated current on the cycle time [7]. However, little information has been published concerning the impacts of the bus stop close to the signalized intersection. Nowadays, researches on the traffic behaviors controlled by traffic lights mainly focus on the effects of different traffic policies on the traffic capacity, without considering the interference with other traffic conditions, like close-by bus stops. That bus stops are set near intersections to facilitate passengers passing through intersections is often observed. If the bus stop is close to the stop line of an approach to a signalized intersection, buses waiting to stop accumulate outside the bus stop. The stopping buses seriously affect the performance of the intersection. It is really true in Beijing that stopping buses often hinder other vehicles to move forward in the green-light time. Only Wong et al. [8] investigated the delay at signal-controlled intersection with bus stop upstream. The traffic lights can operate as a bottleneck. And it is argued for us that the bus stop might have a bottleneck effect on its close-by intersection and would strongly influence the traffic behavior. That is to say, there may be interactions between the bus stop and the signalized intersection. As mentioned above, traffic lights are significant entities to manage urban traffic system. Obviously, bus stops are crucial for urban public transport. Thus, the study on the combined effects between the intersection and its neighboring bus stop is very important. In this paper, the combined effects of the upstream (downstream) bus stop and the signalcontrolled intersection will be discussed, using a new two-lane CA model. The following problems will be focused on. How the bus stop interferes with the traffic flow controlled by the traffic light? How the transition from free flows to congestions occurs? Are upstream cases and downstream cases different distinctly? Could the capacity be enhanced by adding special stop lane or changing the position of the near-by bus stop? This paper is organized as follows: the models used in the simulations are introduced in Section 2. In Section 3, the simulation results are discussed. Finally, our conclusions are summarized. 2. Models Intersections and bus stops exist widely in city traffic system. Sometimes the bus stop locates downstream the intersection, and sometimes upstream the intersection. The bus stops mainly have two forms, stops with special stop lane or not. Therefore, four cases will be considered. Case 1(2) denotes that the bus stop without(with) the special stop lane lies upstream the intersection. Case 3(4) denotes that the bus stop without(with) the special stop lane lies downstream the intersection. These cases are illustrated in Fig. 1. The road with two lanes, the right and the left, consists of six sections, including sections A, B, C, C1, D and E. Section A is the start of the road. Section B is the upstream part near the bus stop and special lane-changing rules will be used in this section. Section C is the bus stop. Section C1 is the downstream part of the bus stop. Section D is the part between the bus stop and the intersection. Section E is the end of the road. There are two types of vehicles, cars with one-cell lengths and buses with two-cell lengths. Buses mix in the traffic flow with the mixture probability Pm , which is the percentage of buses. And some of the buses must halt at the bus stop, called as stoping buses, and the others need not, called as non-stop buses. The stop probability Ps stands for the percentage of the stoping buses. T s denotes the dwell time of stoping buses at the stop. A two-lane CA model will be used in this system. The vehicle movement includes forward motions and lanechanging motions. 2.1. Forward motions The NS model is used to control the vehicles moving forward. The road is divided into L cells. Each cell may be either empty or occupied by one vehicle with an integer velocity v between 0 and the speed limit vmax . At each discrete time step t ! t þ 1, the system is updated by the following rules: (1) acceleration, vn ! minfvn þ 1; vmax g; (2) deceleration, vn ! minfvn ; d n g; (3) randomization, vn ! maxfvn 1; 0g with
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Traffic Light
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probability p; (4) motion, xn ! xn þ vn . Here, xn and vn are the position and velocity of vehicle n, d n ¼ xn1 xn l n1 is the gap between vehicle n and its preceding vehicle n 1, l n1 is the length of vehicle n 1. p is the randomization probability. When the traffic light is red, the gap of the nearest vehicle behind the intersection should be calculated as d n ¼ xlight xn , where xlight is the position of the intersection. If the nearest vehicle behind the bus stop is a stopping bus, the gap between it and its front vehicle is computed as d n ¼ xbstop xn , where xbstop is the end position of the bus stop. In cases with the stop lane, if the leader on the stop lane has stopped, its front gap is given as d n ¼ xC1 xn , where xC1 is the position of the end of the bus stop lane. Assume the cycle time of the traffic light is denoted as T, and the green-light time equals to the red-light time. Since the bus has the length of two cells, it may stay at the intersection when the traffic light changes from green to red. To avoid the situation, when the leader vehicle n upstream the intersection is a bus in the last time step of the green light, the following rules should be used after the velocity and position update process, if xhead ¼ xlight þ 1 and vn ovmax and xlight þ 2 is empty then vn ¼ vn þ 1; xhead ¼ xlight þ 2, otherwise n n vn ¼ vn 1; xhead ¼ x , light n where xhead is the position of the head of the bus n. The rule means that the bus should accelerate one unit to n pass through the intersection, if its head lies in the next cell of the intersection and its velocity is less than the
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maximal velocity, and its front cell is empty; otherwise, the bus should decelerate one unit to wait behind the intersection. For the stopping bus n staying at the bus stop, if its dwelling time step TS n is less than T s , then it cannot move forward, and the dwelling time is updated as TS n ¼ TS n þ 1. Otherwise, the bus becomes a non-stop bus. 2.2. Lane-changing motions For simplicity, all cars and non-stop buses on the road and the stopping buses which do not lie in Sections B and C will change lanes symmetrically, if the following condition is fulfilled: d n ominfvn þ 1; vmax g and d n;other 4d n and d n;back 4d safe ,
(1)
d n;other ðd n;back Þ represents the gap between the nth vehicle and its preceding (back) vehicle on the destination lane at time step t, d safe is a safety distance to avoid crash. In Sections B and C, special lane-changing rules will be employed for the stopping buses. For simplicity, the lane-changing rules for off-ramp traffic systems in Ref. [9] are used in this paper, because the lane-changing behaviors of vehicles on the main road, which leave the main road and enter the off-ramp, are similar to those of buses entering the bus stop. The drivers of these buses are willing to run on the right side lane to stop conveniently when they are close to the bus stop. These buses will change from the left lane to the right lane as long as conditions on the right lane are not worse than those on the left lane. Namely, if the following condition is satisfied: ½d n ¼ d n;other ¼ 0 or ðd n;other a0 and d n d n;other p2Þ and d n;back 4vob ,
(2)
the stopping bus will change from the left lane to the right lane. vob represents the velocity of its back vehicle on the destination lane. Condition d n ¼ d n;other ¼ 0 means that there is no gap to move forward on both lanes in the next time step; Condition d n;other a0 and d n d n;other p2 means that the road situation on the present lane is not much better than that on its right neighbor. d n;back 4vob means that there are enough space to guarantee the safety. The same changing rules of stopping buses in Eq. (2) are used for both lane-changes from the left lane to the right lane in Sections C and C1 and from the right lane to the stop lane in Sections B and C. If a stopping bus cannot change to the right destination lane, until it approaches the position xbstop of the bus stop, it would stop to wait for the change chance (e.g. it will change the lane as soon as the corresponding position on its right-side lane is empty). In Sections B and C, the stopping buses are prohibited changing from the present lane to its left-side lane. Cars and non-stop buses are forbidden running on sections B and C for no stop lane cases, while forbidden running on the stop lane for the cases with the special stop lane. The stopping buses will become non-stop buses, and can change to its left neighbor by using the rules in Eq. (2). If a bus on the stop lane cannot change to the left-side destination lane, until it reaches the position xC1 , it would stop to wait for the change chance mentioned above. 2.3. Boundary conditions The simulations are carried out under open boundary condition. In each time step, when the update of vehicles on the road is finished, the positions xlast of the last vehicles on each lane is checked. If xlast 4vmax , a vehicle with velocity vmax is injected with the entering probability Pe at the cell minfvmax ; xlast vmax g. The leading vehicle on each lane is removed, if its position xfirst 4L and its following vehicle becomes the new leader. 3. Simulation results and discussions In the simulations, the road is divided into L ¼ 500 cells. Section C contains LC ¼ 6 cells, section E contains LE ¼ 100 cells, section D contains LD cells. When the bus stop locates downstream the intersection, section B consists of LB ¼ minf8; LD g cells, section C1 consists of LC1 ¼ 8 cells, and section A contains LA ¼ L LC LD LE . For the upstream cases, LB ¼ 8 cells, LC1 ¼ minf8; LDg cells, LA ¼ L LC LD LE . Each cell
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corresponds to 7.5 m and each time step corresponds to 1 s. The model parameters are set as follows: vcar max ¼ 3, car bus vbus ¼ 2, d ¼ 3, d ¼ 2, p ¼ 0:3, T ¼ 100 s, where the superscript car(bus) denotes the parameter of the max safe safe car (bus). According to the Transit Co-operative Research Program (TCRP) Report 19 (1996) [10], the average peak-period dwell time exceeds 30 s per bus, so T s is set as 30 s. The mix probability Pm is 0.15, and the stop probability Ps is 0.3. The first 50,000 time steps are discarded to avoid the transient behavior. The flow is averaged by 100,000 time steps. 3.1. Cases the bus stop locating upstream the intersection In this section, the cases that the bus stop lies upstream the intersection (case 1 and 2) are considered. Firstly, the effect of the distance LD between the bus stop and the intersection on the traffic flow is investigated. The capacity versus the entering probability for different LD is shown in Fig. 2 ((a) case 1
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Fig. 4. The time–space diagram for case 1 (a1–b2) and case 2 (c1–d2) from 149,500 to 150,000. Black points denote cars, red and green points denote non-stop and stopping buses. The left denotes that of left lane, the right denotes that of right lane. (a,c) LD ¼ 3, (b,d) LD ¼ 15.
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(without the stop lane), (b) case 2 (with the stop lane)). It can be seen that the flow increases with the entering probability until the flow reaches a saturated value. There is a critical point of the entering probability Pec , above which the traffic flow on the road transforms from free flow to congested flow. The saturated flow is the capacity of the intersection, denoted as Qc . In comparison with no bus stop case, the capacity in both case 1
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and 2 drops when the LD is small. The drop of case 1 is much larger than that of case 2. While for enough large LD , the capacities in both case 1 and case 2 are the same with no stop case. This means that on both cases 1 and 2, the upstream bus stop has a bottleneck effect on the intersection at small LD , and the negative effect of case 1 is much stronger than that of case 2. Fig. 3 depicts the relationship between the Qc and the LD . It is clear that the Qc increases with the LD , and approaches a maximum Qc;max ¼ 0:407 at a critical value LDc;Q . The LDc;Q1 (LDc;Q2 ) is 36 (9) for case 1 (2). The Qc in case 1 is less than that in case 2 below LDc;Q1 . To understand these phenomena, the time–space diagram and the velocity distribution are shown in Figs. 4 and 5, respectively. From Fig. 4(a1,a2) and (c1,c2), it is obvious that due to the stop near the intersection, the waiting queue upstream the intersection grows stretching to the bus stop. The queue upstream the intersection mixes with the coming stopping buses. In case 1, during green time cycle, lots of blanks downstream the intersection appears. That means that the stopping buses at the bus stop and buses waiting to stop obstruct other vehicles to pass through the intersection in the green-light period (see Fig. 4(a2)). Whereas in case 2, the special stop lane can provide more spaces near the bus stop for waiting-stop buses, and seldom vehicle passing through is hindered by stopping buses in the green light time step (see Fig. 4(c2)). These also can be verified by the velocity distribution in Fig. 5. It can be seen that vehicles with larger speeds in case 2 are much more than those in case 1 on both the lanes, when LD is small. Thus, the capacity in case 2 is greater than that in case 1 at small LD . When the LD is larger, less vehicles blend with stopping buses behind the bus stop. Therefore, the
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capacity raises with the increase of the LD . If LD is enough large, the bus stop would not bother vehicles behind the intersection. The capacity can approach its maximum. These results suggest that the upstream bus stop near the intersection has a strong bottleneck effect on the intersection. The negative effect decreases with the increase of the LD . Also, the special stop lane can create more benefit. And these two policies provide no additional effects when the LD 4LDc;Q . In Fig. 6, the phase diagram in (Pe ,LD ) space is plotted. The space can be divided into three regions. The solid-dot-dot line represents case 1, and the solid line represents case 2. The traffic flow on the road is free in region I, while it is congested in both regions II and III. The capacity in region II is less than the maximal capacity in region III, because the shorter the LD is, the stronger the bottleneck effect becomes. Comparisons as indicated in Fig. 6 reveal that the critical point of the entering probability Pec has little difference between the two cases. Due to the benefit of the special lane, the region III of case 2 is wider than that of the case 1. Next, the influence of LD on the capacity Qc under conditions of different T is studied. Fig. 7 displays the relationship between the capacity and the LD for various values of T. It is found that there exists a critical value LDc;T , below which the bus stop creates a negative effect on the capacity. The critical values LDc;T1 and
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Fig. 8. The relationship between the capacity and the LD for different T s . (a) T s ¼ 5, (b) T s ¼ 10, (c) T s ¼ 20, (d) T s ¼ 30, (e) T s ¼ 40, (f) T s ¼ 50.
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LDc;T2 are 24 and 9 (T ¼ 50), 36 and 12 (T ¼ 100), 39 and 12 (T ¼ 150) and 42 and 15 (T ¼ 200) for cases 1 and 2, respectively. The critical point LDc;T1 of case 1 greatly depends on the T and becomes larger and larger with the increase of the T. Whereas the critical point LDc;T2 of case 2 changes little with the T. Because in case 1 the longer the cycle time is, the longer the queue behind the traffic light, and the stronger the combined effect of the upstream bus stop and the intersection will become. This means that the cycle time of the signalcontrolled intersection cannot be too long, when a upstream bus stop is close to it, especially in case 1. It is also observed that there exists the capacity difference between the two cases below a critical value LDc;T0 , and the capacity difference in the two cases vanishes at LDc;T0 . Obviously, the LDc;T0 is equal to LDc;T1 . That means that the capacity can be enhanced by either increasing the LD or adding a stop lane if LD oLDc;T0 . Finally, the influence of LD on the capacity Qc is studied in the case of different T s . The relationship between the capacity Qc and the length LD is displayed in Fig. 8. Similarly, at small LD the capacity Qc increases with the LD and approximately becomes a constant when LD is greater than a critical value LDc;Ts (LDc;Ts1 for case 1 and LDc;Ts2 for case 2). This means that the capacity can be enhanced by increasing the LD and the further increase of the LD will have no effect if LD 4LDc;Ts , because the waiting queue upstream the intersection has less interference with waiting-stop buses, as the large LD . And at the critical value LDc;Ts , the interference disappears. Also, the impact of the LD in case 1 is much more obvious than that in case 2. LDc;Ts1 is dependent on the T s , whereas LDc;Ts2 hardly varies with the T s . The capacity difference between the two cases vanishes at a critical value LDc;Ts0 , which equals to LDc;Ts1 . Different from the situation in Fig. 7, the maximal capacities Qcmax;Ts for case 1 (Qcmax;Ts1 ) decline greatly with the T s , and the maximal capacities for case 2 (Qcmax;Ts2 ) change little. As the dwell time becomes large, stopping buses waiting outside the bus stop accumulate greatly and obstruct all lanes for case 1, leading to the drop of the maximal capacity. While for case 2 these stopping buses have more chances running on the right lane and the stop lane, as a result the capacity drop is slight. Therefore, the magnitude of the capacity drop in case 2 is much smaller than that in case 1. 3.2. Cases the bus stop locating downstream the intersection In this section, the cases that the bus stop locates downstream the intersection (Cases 3 and 4) are considered. Similarly, the effect of the length LD between the downstream bus stop and the intersection on the 0.42
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Fig. 9. The capacity versus the LD for cases 1–4. The inlet is the amplification of the range [3, 30].
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Fig. 10. The time–space diagram for case 3 (a1–b2) and case 4 (c1–d2) from 149,500 to 150,000. Black points denote cars, red and green points denote non-stop and stopping buses. The left denotes that of left lane, the right denotes that of right lane. (a,c) LD ¼ 3, (b,d) LD ¼ 15.
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traffic flow is investigated in respect of the cycle time and the dwell time, as shown in Figs. 9–12. The simulations show that the situations in the downstream cases qualitatively resemble those in the upstream cases. When the downstream bus stop is close to the signal-controlled intersection, it negatively affects the capacity. The increase of the LD can lessen the bottleneck effects when LD oLDc , and adding the special stop lane can create more benefit on the capacity. The critical values LDc;T and LDc;Ts for case 3 increase with the T and T s , while those for case 4 almost not depends on the T and T s . There is a little difference between the upstream cases and the downstream cases at small LD . The situations of the downstream cases have less capacity drop than those of the upstream cases at small T or large T s . From the time–space diagram shown in Fig. 9, it can be observed that in the course of red light there is no any hindrance downstream the bus stop for the waiting-stop buses to proceed. Some or all of the buses finish their stoping procedure at the stop and leave out of the bus stop to move forward further. Thus, the queue constituted by the stopping buses will dissolve before the upcoming green period. While for the upstream cases, the queue behind the intersection still hinders stopping buses waiting upstream the stop in the red-light time step. More and more vehicles continue to accumulate upstream the bus stop, and the situation gets worse. Thus, it can be concluded that the bus stop should be set downstream the intersection, especially for long dwelling time and short time cycle of traffic light.
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4. Conclusions The combined effect of the signal-controlled intersection and the near-by bus stop is studied by using a twolane CA model. Four cases are considered, according to two conditions, that the stop is upstream or downstream the intersection, and the special stop lane exists or not. The simulations results indicate that the bus stop near the intersection serves as a bottleneck. The traffic flow transits from free flow to the congestion at a critical entering probability, where the flow is saturated and corresponds to the capacity. The capacity grows gradually with the increase of the distance LD between the stop and the intersection, and reaches a maximum at a critical point LDc , which is mainly affected by two significant factors, the traffic cycle time T and the dwell time T s in no stop lane cases (cases 1 and 3). For no stop lane cases, the larger the T and the T s are, the larger the critical value of the LD are, and the stronger the damage are. This means that the T should not be set too long when a bus stop near the traffic light. Additionally, comparisons among four cases reveal that the capacity using the special stop lane is greater than that without the lane, and the benefit disappears
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when LD is large enough; there is no qualitative difference between the upstream and the downstream cases, and the capacity of the upstream cases is appreciably less than that of the downstream ones, and the difference decreases with the LD . These findings imply the following:
A special stop lane is necessary and worthy to diminish the damage of the negative effect and guarantee higher capacity of the traffic flow if a bus stop is much close to intersections. While its benefit can be negligible when a bus stop much far away from intersections. It is favorable to replace the upstream bus stop with the downstream one.
These implications may provide suggestions on positioning the bus stop near-by a intersection and the design of the traffic light cycle time. It should be mentioned that in this work the simplest situation that identical models of forward motions for buses and cars are used. In reality, there are distinctions between buses and cars. The future works are to consider differences in the acceleration(deceleration) between buses and cars and to validate in practice. Acknowledgments This paper is financially supported by 973 Program (2006CB705500), Project (NSFC nos. 70631001, 70471088 and 70501004), and Program for Changjiang Scholars and Innovative Research Team in University (IRT0605). The authors thank anonymous reviewers for the critical remarks and good suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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