J. Cent. South Univ. (2012) 19: 2697-2704 

DOI: 10.1007/s11771-012-1329-2

Mixed traffic flow modeling near Chinese bus stops and its applications

YANG Xiao-bao(杨小宝), SI Bing-feng(四兵锋), HUAN Mei(环梅)

Key Laboratory for Urban Transportation Complex Systems Theory and Technology of Ministry of Education

(Beijing Jiaotong University), Beijing 100044, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2012

Abstract: To determine how bus stop design influences mixed traffic operation near Chinese bus stops, a new theoretical method was developed by using additive-conflict-flows procedure. The procedure was extended from homogeneous traffic flow to mixed traffic flow. Based on the procedure and queuing theory, car capacity and speed models were proposed for three types of bus stops including curbside, bus bay and bicycle detour. The effects of various combinations of bus stop type, traffic volume, bus dwell time, and berth number on traffic operations were investigated. The results indicate that traffic volume, bus dwell time and berth number have negative effects on traffic operations for any type of bus stops. For different types of bus stops, at car volumes above approximately 200 vehicles per hour, the bus bay and bicycle detour designs provide more benefits than the curbside design. As traffic volume increases, the benefit firstly increases in uncongested conditions and then decreases in congested conditions. It reaches the maximum at car volumes nearly 1 100 vehicles per hour. The results can be used to aid in the selection of a preferred bus stop design for a given traffic volume in developing countries.

Key words: mixed traffic flow; bus stop; road capacity; speed; bicycles

1 Introduction

A number of alternatives are available when choosing the type of facility for a particular bus stop. These alternatives include curbside, bus bay, bicycle detour designs and berth number. None of these alternatives is completely advantageous under all conditions; however, one design may offer a better balance of benefits over another design under certain conditions. In recent decades, some studies were conducted on the location [1], design [2-3] and operations [4-5] of bus stops and obtained many useful achievements. In addition, researchers mainly focused on the effects of bus stops on traffic flow. For example, FERNANDEZ [6] presented a microscopic model to calculate operational impacts on bus stops such as capacity, delays, queues and waiting time. ARASAN and VEDAGIRI [7] used microscopic simulation to study the effect of provision of reserved bus lanes on the flow of heterogeneous traffic. ZHAO et al [8-9] proposed a two-lane cellular automata model to investigate the capacity drop caused by the combined effects of signalized intersections and bus stops. NAGAI et al [10] developed an extended optimal velocity model to investigate the effect of the bus on car stream. TANG  et al [11] used macro dynamic model to study the effects of bus stop on traffic flow. On the bus stop, unfortunately, little information has been published concerning bus stops with non-motor vehicles.

Mixed traffic between motor vehicles and non-motor vehicles is an important traffic type in China [12]. In many Chinese cities, the roadway is divided to serve motor (cars and buses) and non-motor (bicycles) traffic streams. Typically, there are three types of bus stops in Chinese urban areas: the curbside stops, bus bays, and bicycle detours (see Fig. 1). When a bus dwells in the bicycle lane, bicycles may move to the motor lane and go around the stopped bus. The temporary car-bicycle conflict would reduce road capacity and delay vehicle speed. In addition, the car-bus conflict takes place when a bus departs from the stop to the motor lane.

Due to the special features of mixed traffic, the existing research has seldom investigated the effects of bus stop design on mixed traffic flow. Although YANG et al [13] established a capacity model near the curbside stop with bicycles based on gap acceptance theory, they did not analyze other types of bus stops. KOSHY and ARASAN [14] used simulation technique to study the impact of bus stop type on the speeds of other vehicles under heterogeneous conditions. However, neither the bicycle detour design nor the difference of capacities among various bus stops was considered. Therefore, the first aim of this work is to develop an appropriate methodology to calculate car capacities and speeds for various bus stop types. The second aim is to investigate the effects of bus stop design on mixed traffic operations around bus stops.

Fig. 1 Three types of bus stops in Chinese urban areas

Owing to the complex interactions among buses, cars and non-motor vehicles, the traditional gap acceptance theory loses its applicability. Thus, there is a need for new techniques. In this work, we used additive-conflict-flow (ACF) procedure to study the effects of bus stop design on traffic operations under mixed traffic conditions. ACF procedure was firstly developed for signalized intersection analysis, and then modified by BRILON and WU [15], WU [16] and LI   et al [17] for application to unsignalized intersections. This procedure can simply handle the interaction between the different streams and easily take into account non-motor vehicles. It is the reason why ACF procedure is chosen to analyze the effects of bus stop design on mixed traffic operations. Based on ACF procedure, car capacities and speeds were proposed near various stops including curbside, bus bay and bicycle detour. The factors influencing bus stop design such as bus dwell time, berth number and traffic volume were investigated. The findings of this work can help to determine how bus stop design influences mixed traffic operations for a given traffic volume. It is hoped that the results can help to improve the planning and designing of bus stops in developing countries.

2 Methods

The mathematical background of ACF technique is the graph theory. In the case of a first-in-first-out (FIFO) intersection of two one-way streets, this corresponds to the rule of zipping. The capacities of the streams are not distributed proportionally to their traffic flow rates. In an overloaded departure sequence, the capacities of all streams are equal. In a non-overloaded departure sequence, the capacity of a stream is the traffic flow rates that can departure within the time which cannot be consumed by the other stream [16].

Traffic characteristics in China are different from those in developed countries. The lengths and distance headways of different types of vehicles vary widely in Chinese cities. Therefore, ACF technique under mixed traffic flow is dissimilar to that under homogeneous traffic flow. The car-bicycle conflict and car-bus conflict take place near bus stops with mixed traffic flow. Because no traffic streams near bus stops possess absolute priority of driving, the conflicting areas near bus stops can be considered in such a way that the FIFO discipline is applied [18]. The special cases of so-called limited priority will be discussed.

2.1 ACF procedure in first-in-first-out discipline

Let Q denote traffic volume, vehicles per hour (veh/h), and C denote vehicle capacity, veh/h. For a curbside stop with mixed traffic flow, when buses dwell in the bicycle lane near a bus stop, the car-bicycle conflict takes place because bicycles may move to the motor lane and go around the stopped bus. The vehicles can pass the car-bicycle conflict point in the way of approximate three bicycles after one car based on the observation in Beijing. This is consistent with the result in Ref. [12] that the left-turn bicycle conversion factor at intersections is 0.328. Accordingly, bicycle capacity in the motor lane should be three times of car capacity by permanent queuing if non-motor vehicle volume, Q_n, exceeds its capacity, C_n. Otherwise, non-motor vehicle volume is lower than the admitted capacity. The remaining capacity must be distributed by car stream. Thus, the capacity for car stream is

              (1)

where “c” denotes car stream and “n” denotes non-motor vehicle stream. t_c and t_n are the service times for car stream and non-motor vehicle stream passing the conflict point, respectively.

When buses depart from the stop to the motor lane, the car-bus conflict takes palace. Generally, the vehicles of these two streams can enter the conflict area in the way of two cars after one bus. Accordingly, the capacity for car stream is

             (2)

where “b” stands for bus stream.

For a bus stop, car stream may be affected by both the car-bicycle conflict and the car-bus conflict. According to the situation whether bicycle stream and bus stream are saturated, traffic flow can be classified as four conditions. In this case, car capacity near a curbside stop can be given as

(3)

2.2 ACF procedure in limited-priority discipline

In the above analysis, it is assumed that different streams obey the FIFO traffic discipline. However, Chinese traffic laws regulate that car drivers hold greater responsibility than cyclists in the car-bicycle collision. Thus, some drivers may decelerate to give way to cyclists when they have conflicts, particularly at high flows and low speeds. In this case, bicycle stream has a so-called limited priority. If bicycle stream is overloaded, the cyclists with the priority would lead to the decrease of car capacity. The service time for bicycle stream must be subtracted from the total time. Otherwise, bicycle stream is non-overloaded, and car capacity is not affected by bicycle stream.

It is assumed that cars give priority to a fraction γ of cyclists among the bicycle stream. Considering the limited priority of bicycle stream under congested flow, car capacity near the stop with mixed traffic flow can be given by

  (4)

3 Models              

3.1 Queuing model of bus stream

Consider a road link near a bus stop. A sophisticated queuing theory model can be developed by the assumption that the simple bus stream system can be represented by an M/M/k queue. The service counter is the bus stop. Let λ_b denote the mean arrival rate of bus stream (vehicles per second). The buses approaching from upstream are assumed to arrive at random, i.e. negative exponentially distributed arrival headways with mean 1/λ_b (s). The dwell time at the stop is the service time, which is also assumed to be independent and negative exponential distributed random variable with mean s_b (s). The “k” in M/M/k stands for the number of existing berths at a bus stop.

Let p_r denote the equilibrium probability that there are r buses in the system, ρ_b denote the service rate of bus system and N_b denote the number of buses in the system that is in equilibrium. For the M/M/k system, on the basis of MEDHI’s monograph [19], P(N_b=r) can be given by the following equation:

     (5)

The probability p₀ follows the normalization, yielding

      (6)

Note that, here, ρ_b=λ_bs_b, and for the existence of a steady-state solution, λ_bb, that is, the mean arrival rate of bus stream must be less than the mean maximum potential service rate of the system. In the bus queuing system, P(N_b=r) stands for the probability that there are r buses at the stop.

In addition, the expected number in the system (i.e. the expected number of buses both in service and in queue at the stop) at steady state is

                 (7)

For a curbside stop or a bicycle detour, all the stopped buses dwell in the non-motor lane. Thus, the probability of one or more buses in the bicycle lane may be expressed by

                (8)

where the subscripts 1, 2 and 3 denote the curbside stop, bus bay and bicycle detour designs, respectively. Correspondingly, the expected number of stopped buses in the bicycle lane at steady state is

        (9)

For a bus bay, when the number of arriving buses is larger than the number of existing berths, residual buses only dwell in the bicycle lane. Therefore, the probability of one or more stopped buses in the bicycle lane is

   (10)

For a bus bay, the expected number of stopped buses in the bicycle lane at steady state is

        (11)

3.2 Car capacity near bus stops under mixed traffic flow

According to the definition for the capacity of a facility in the Highway Capacity Manual [4], car capacity near a bus stop under mixed traffic flow conditions can be defined as the maximum hourly rate at which cars can pass the stop during a given time with other flow rates (including the arrival rates of both bicycle stream and bus stream, and the dwell time of bus stream) remaining unchanged.

3.2.1 Car capacity model near curbside stop

Firstly, for a curbside stop, traffic conditions can be classified into two types: presence and absence of stopped bus at the stop. Under the former condition, the conflict among cars, bicycles and buses leads to a negative effect on car capacity. Under the latter condition, the car stream and other streams have no conflict and car capacity is not affected by other streams. The probabilities of no bus and presence of stopped bus can be obtained by using Eqs. (6) and (8). Then, the total capacity for car stream near the curbside stop can be expressed by

C_T1=P(N_b=0)·C(N_b=0)+P(N_b≥1)·C(N_b≥1)=

p₀·C_max+(1- p₀)·C_c-b-n                               (12)

where C_T1 denotes the total capacity for car stream near the curbside stop and C(N_b=0) denotes the car capacity that is not affected by other streams, that is, the car capacity on an uninterrupted roadway section, which is given by the following formula:

                    (13)

where t_cf denotes the follow-up headway for car stream in the motor lane (i.e., the minimum saturation headway for car stream). In addition, C(N_b≥1) denotes car capacity when there are one or more buses at the stop, which can be obtained by Eq. (3). C_T1, C(N_b=0) and C(N_b≥1) are all expressed in vehicles per hour.

3.2.2 Car capacity model near bus bay

For a bus bay, it is assumed that there are k berths at the bus stop. Traffic conditions can be divided into three cases: 1) No bus dwells in the stop; 2) The number of stopped buses is from 1 to k; 3) The number of stopped buses is more than k. In the first case, there is no conflict between car stream and other streams. In the second case, no stopped bus needs to dwell in the bicycle lane, and only the car-bus conflict takes place when the bus departs from the stop to the motor lane. Finally, if there are more than k buses in the bus queuing system, the conflict among three kinds of streams with cars, buses and bicycles takes place. Thus, the total capacity for car stream near the bus bay with k berths can be expressed by

     (14)

3.2.3 Car capacity model near bicycle detour

For a bicycle detour, traffic conditions can be classified into two types: presence and absence of stopped bus at the stop. Under the former condition, there is no conflict between car stream and other streams. Car capacity is not affected by other streams. Under the latter condition, only the car-bus conflict may take place. Then, the total capacity for car stream near the bicycle detour can be expressed by

C_T3=P(N_b=0)·C(N_b=0)+P(N_b≥1)·C(N_b≥1)=

p₀·C_max+(1-p₀)·C_c-b                      (15)

3.3 Car delay and speed near bus stops under mixed traffic flow

Car delay is defined as the average delay time when car stream passes a bus stop with mixed traffic flow. To calculate the average delay for car stream (d_c), a classical approach can be used. The formula contained in the HCM [4] as Eq. (17-38) may be applied to calculate the average delay for car stream. Accordingly, car delay near bus stops under mixed traffic flow can be given as

         (16)

where Q_c is the flow rate for car stream; C_T is the total capacity for car stream, t_cf is the minimum saturation headway for car stream (s), and T is analysis time (h).

Speed is defined as the distance per unit time (m/s). Car speed near the stop with mixed traffic flow can be expressed by

                               (17)

where L is the distance of observed road segment which contains the bus stop (m); d₀ is the time that a car passes the observed segment with free-flow speed; d_c is the average delay for car stream near the stop.

4 Model applications

The capacity and speed models for the curbside, bus bay and bicycle detour designs in China were analyzed for various combinations of traffic volume, bus dwell time and the number of berths. The results from the curbside versus bus bay and bicycle detour study are analyzed. Firstly, it is assumed that the service time for bus stream, car stream and bicycle stream in the conflicting area are 3.5, 2.4 and 0.9 s, respectively, the follow-up time of car stream is 2.0 s, the average dwell time of bus stream is 15 s, the number of berths is 2, Q_b=0.2Q_c, and Q_n=1.4Q_c. In addition, to study the trends between these factors and varying traffic volume (0 to 1500 veh/h), bus dwell times (10, 15, and 20 s) and the number of berths (1, 2, and 3), several figures are generated.

4.1 Effects of traffic volume on mixed traffic operations for various bus stops

Using Eqs. (12), (14), (15) and (17), the capacities and speeds of car stream were computed for the curbside, bus bay and bicycle bay designs. Then, the effects of traffic volume on car capacity and speed can be obtained (see Figs. 2 and 3). From the two figures, we can conclude the following results:

1) For car volumes below approximately 200 veh/h, there is no significant difference in the car capacities or speeds for various stop designs. Therefore, in this case, curbside stop design is advantageous. Above 200 veh/h, the car capacities and speeds for the bus bay and bicycle detour designs are obviously greater than those for the curbside design. This is because the interactions  between motor vehicles and non-motor vehicles for the curbside design are greater than those for the bus bay and bicycle detour designs.

Fig. 2 Relationship between car capacity and volume for three types of bus stops

Fig. 3 Relationship between car speed and volume for three types of bus stops

2) For car volumes above 600 veh/h, both car capacities and speeds for the bicycle detour design are slightly larger than those for the bus bay design. In this case, traffic operations for bicycle detour design are relatively advantageous over those for bus bay design. However, the land use area for the bicycle detour design is significantly larger than that for the bus bay design. As the bicycle detour design is rare in actual traffic, following is only a discussion of the results from the curbside versus bus bay design.

4.2 Effects of bus dwell time on mixed traffic operations for various bus stops

To illustrate the effects of bus dwell time on car capacities and speeds, Figs. 4 and 5 are plotted. In addition, Figs. 6 and 7 illustrate the benefits of a bus bay design over a curbside design for different dwell times. From these figures, we can obtain the following findings:

1) For any type of bus stops, as the dwell time increases, the capacity and speed for car stream decrease (see Figs. 4 and 5). This is because increasing dwell time leads to the increase of the conflict between motor vehicles and non-motor vehicles around bus stops.

Fig. 4 Relationship between car capacity and volume for different dwell times

Fig. 5 Relationship between car speed and volume for different dwell times

2) For any dwell time, as traffic volume increases, car capacity difference (i.e. car capacity for bus bay design minus car capacity for curbside design) firstly increases in uncongested conditions and then decreases in congested conditions (see Fig. 6). Similarly, car speed difference (i.e. the average speed for bus bay design minus the speed for curbside design) firstly increases and then decreases with increasing traffic volumes (see   Fig. 7). For car volumes nearly 1 100 veh/h, the benefits for bus bay over curbside stop design reach the maximum.

3) For different dwell times, in uncongested conditions (car volumes below approximately 1 000 veh/h), the longer the dwell time is, the greater the benefit for bus bay design over curbside stop design becomes. On the contrary, in congested conditions (car volumes above 1 000 veh/h), the benefits decrease with increasing dwell times (see Figs. 6 and 7).

Fig. 6 Car capacity difference between bus bay and curbside design for different dwell times

Fig. 7 Car speed difference between bus bay and curbside design for different dwell times

4.3 Effects of berth number on mixed traffic operations for various bus stops

To illustrate the effects of berth number on car capacities and speeds, Figs. 8 and 9 are plotted. In addition, Figs. 10 and 11 illustrate the benefits of a bus bay design over a curbside design for different berths. From these figures, we can conclude the following results:

1) For any type of bus stops, car capacity and speed increase with increasing berth number, and the growth rate decreases with increasing berth number (see Figs. 8 and 9). This is because increasing berths reduces bus queue time and then gives rise to the decrease of the conflict between motor vehicles and non-motor vehicles around bus stops. As berth number increases, the positive effect may be weakened.

2) For any bus berth, as traffic volume increases, the benefits for bus bay design over curbside stop design firstly increases in uncongested conditions and then decreases in congested conditions. The benefits for the 3 berths, 2 berths and 1 berth reach the maximum at approximately 1200, 1100 and 900 veh/h, respectively (see Figs. 10 and 11).

3) For different bus berths, the benefit for bus bay design over curbside stop design increases with increasing berth number. However, the growth rate of the benefits decreases with increasing berth number (see Figs. 10 and 11).

Fig. 8 Relationship between car capacity and volume for different berths

Fig. 9 Relationship between car speed and volume for different berths

Fig. 10 Car capacity difference between bus bay and curbside design for different berths

Fig. 11 Car speed difference between bus bay and curbside design for different berths

5 Conclusions

1) In virtue of the additive-conflict-flows procedure, car capacities and speeds near various stops including curbside, bus bay and bicycle detour are proposed, in which the bicyclist limited priority and the interaction between motor vehicles and non-motor vehicles are easily handled.

2) Based on the proposed models, the factors influencing mixed traffic operations such as bus dwell time, berth number and traffic volume for various bus stop designs are analyzed. For any type of bus stops, car capacities and speeds decrease significantly with increasing traffic volumes, bus dwell times and berth number.

3) The effects of bus stop design on mixed traffic operations are investigated. For different type of bus stops, at car volumes above 200 veh/h, the bus bay and bicycle detour designs provide significantly greater benefit than the curbside design. As traffic volume increases, the benefit firstly increases in uncongested conditions and then decreases in congested conditions. Meanwhile, at car volumes above 600 veh/h, the bicycle detour design provides slightly greater benefit than the bus bay design. The results can be used to aid in the selection of a preferred bus stop design for a given traffic volume in China.

4) Although this work has given valuable insights into the effects of bus stop design on mixed traffic flow, possible further research work is suggested. Firstly, other queuing models can be developed to describe the queuing behavior of bus system on the basis of actual traffic survey. In addition, considering the effect of signalized intersection, the total capacity for car stream should be multiplied by green ratio.

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(Edited by YANG Bing)

Foundation item: Project(2012CB725400) supported by the National Basic Research Program of China, Projects(70901005, 71071016, 71131001) supported by the National Natural Science Foundation of China; Project(2011JBM055) supported by the Fundamental Research Funds for the Central Universities of China

Received date: 2011-07-21; Accepted date: 2012-04-24

Corresponding author: YANG Xiao-bao, Associate Professor, PhD; Tel: +86-10-51687070; E-mail: yangxb@bjtu.edu.cn