A Figure-eight Hysteresis Pattern in MacroscopicFundamental Diagrams for an Urban Freeway Network

in Beijing, China

Zhengbing He, PhDMOE Key Laboratory for Urban Transportation Complex System Theory and Technology,

Beijing Jiaotong University, Beijing, Chinahe.zb@hotmail.com

Shuyan He, PhD candidateMOE Key Laboratory for Urban Transportation Complex System Theory and Technology,

Beijing Jiaotong University, Beijing, Chinaheshuyan@gmail.com

Wei Guan, PhD (corresponding author)MOE Key Laboratory for Urban Transportation Complex System Theory and Technology,

Beijing Jiaotong University, Beijing, Chinaweig@bjtu.edu.cn

Total 6948: 4198 words + 10 figures + 1 tables

November 15, 2012

ABSTRACT1

This paper presents Macroscopic Fundamental Diagrams (MFDs) for an urban freeway network2

in Beijing, China. In the diagrams, a figure-eight hysteresis pattern is observed. To understand3

the causes, analyses are made ranging from spatialtemporal heterogeneity of vehicles to the flow-4

occupancy relation for individual locations. Eventually, at individual locations we observe that5

free-flow traffic with the same occupancy exhibits different flows in the onset and offset of a rush6

hour; it is attributed to the counter-clockwise loop in the figure-eight hysteresis pattern at the7

macroscopic level. Different lane-changing rates in the onset and offset of a rush hour are discussed8

as the deeper causes of the multi-branch flow-occupancy diagram at individual locations; it is9

closely related to the denseness of ramps on the urban freeway network in Beijing. The paper10

enriches the knowledge about MFDs and provides some empirical support for the existing theory.11

2

INTRODUCTION12

Nowadays, most of approaches of traffic management and control still highly rely on traffic da-13

ta that are difficult to be obtained sometimes. Combined with complexity of traveler behavior14

and network topology, the practical effects are compromised. A recently proposed macroscopic15

fundamental diagram (MFD) for a large urban area provides a new thought on aggregate traffic16

management and control that are less affected by details.17

Understanding the shape and characteristics of an MFD for a network is basic and signif-18

icant to take advantage of the diagram in practice. In the MFDs for an urban freeway network19

in Beijing, China, we observe a figure-eight hysteresis pattern combining clockwise and counter-20

clockwise loops, which is only theoretically mentioned in Gayah and Daganzo (1). Accompanying21

with the counter-clockwise loop, lower occupancy variance is associated with lower mean flow;22

it is inconsistent with the observation in Geroliminis and Sun (2). The paper is dedicated to re-23

porting the MFDs with the figure-eight hysteresis pattern, and to investigating the causes of the24

counter-clockwise loop and the association between lower occupancy and lower mean flow.25

The remainder of the paper is organized as follows: literature review, the urban freeway26

network in Beijing and the data used are presented in the next two sections; these are followed27

by a presentation of the figure-eight hysteresis pattern and an investigation of the formation of the28

counter-clockwise loop; discussion and conclusions are made at last.29

LITERATURE REVIEW30

Investigations regarding relationships between macroscopic variables in an urban area could be31

traced back to Godfrey (3). In the literature the relationship between average speed and vehicle32

density was explored in a macroscopic view. The two-fluid model based on the fraction of moving33

and standing traffic was later addressed in Herman and Prigogine (4) and Herman and Ardekani (5).34

Some literature, e.g. Ardekani and Herman, Mahmassani et al., Mahmassani and Peeta, Olszewski35

and Fan (6, 7, 8, 9), also investigated the aggregate traffic relationships.36

More recently, Daganzo (10) proposed an MFD that reflected invariant macroscopic rela-37

tionships among space-mean flow, density and speed in a large urban area. Daganzo and Geroliminis38

(11) theoretically proved the existence of the MFD using variational formulation of the kinematic39

wave theory (see Daganzo, Daganzo (12, 13)) , and conjectured four regularity conditions ensuring40

a well-defined (low scatter) MFD. Meanwhile, Helbing (14) also derived analytical solutions for41

the MFD by using a utilization-based approach (see Daganzo (15)). Empirical evidence was pro-42

vided by Geroliminis and Daganzo (16), in which data collected from an urban area in Yokohama,43

Japan, was used and a well-defined MFD was first observed.44

A number of investigations regarding the MFD were conducted theoretically and practically45

since the seminal papers were released. In empirical study, Buisson and Ladier (17) first reported46

hysteresis phenomena with clockwise loops existing in an MFD for the Toulouse road network in47

France, and showed that heterogeneity in types and topology of road networks as well as locations48

of detectors had strong impacts on the shape of the MFD. Geroliminis and Sun (2) explicitly49

investigated causes of the clockwise hysteresis loops by utilizing data collected from the Twin50

Cities metropolitan area freeway network in Minnesota, USA. Two reasons of the clockwise loops51

were unveiled: different spatialtemporal distributions of congestion in the onset and offset, and52

synchronized occurrence of capacity drop at individual locations. An association between higher53

occupancy variance and lower mean flow was also observed. Geroliminis and Sun (18) compared54

the MFDs for the urban areas in Yokohama and Twin Cities, and analyzed characteristics of the55

3

road network presenting a well-defined MFD. A sufficient existence condition for a well-defined56

MFD was addressed. It was also indicated that surface networks more likely exhibited MFDs with57

low scatter due to the characteristics of network redundancy, traffic signals, etc. To the authors’58

knowledge, only the MFDs for the three cities have been reported. More empirical observations59

either supporting or contradicting existing findings are still expected to enrich the understanding60

of the MFD.61

In analytical study, Daganzo (19) modeled traffic dynamics on a ring freeway with on- and62

off-ramps by using the kinematic wave theory. The model illustrated how the distribution of flow63

and density became uneven in the offset of a rush hour even when the ring was symmetric and the64

demand was uniform. Clockwise hysteresis loops arose with the unevenness. Daganzo and Gayah65

(20) modeled a square grid by using a two-ring idealization and further simplified into a two-bin66

model. The results showed that random turning at intersections aggravated congestion and thereby67

led to uneven congestion and hysteresis phenomena in the MFD. Gayah and Daganzo (1) incorpo-68

rated trip ends into the two-bin model and came to a conclusion that traffic usually exhibited more69

instability in the offset of a rush hour than in the onset; it also implied that hysteresis phenomena70

could also arise due to occurrence of unexpected disturbance even in a symmetric network with71

uniform demand. Meanwhile, the literature illustrated a figure-eight hysteresis pattern, and stated72

that the pattern occurred if the loading demand was very unbalanced and the maximum density73

was quite high; the conditions were rare and no empirical observation has been reported yet.74

To provide more state-of-the-art information, we keep reviewing the simulation and ap-75

plication study, although this paper doesn’t belong to the types of study. In simulation study,76

Mazloumian et al. (21) proposed a traffic flow simulation model based on the section-based traf-77

fic model (see Helbing (22)). A variety of simulation scenarios were conducted, and the spatial78

distribution of vehicles measured by variability of vehicle densities was considered as a key vari-79

able of traffic performance and the scatter in the MFD. Knoop and Hoogendoorn (23) developed80

a road network simulation model based on the cell transition model, and further investigated the81

influence of the variability. A two-variable macroscopic fundamental diagram incorporating a di-82

mension of the variability was suggested. Ji et al. (24) modeled the A10 west in Amsterdam, the83

Netherlands on VISSIM. The influence of various factors on the MFD was demonstrated, such as84

ramp-metering, the onset and offset of congestion, rapidly changing demands, etc.85

In application study, Daganzo (10) proposed an accumulation-based (AB) rule for optimiz-86

ing arrival rates of vehicles based on a given MFD. Gonzales et al. (25) demonstrated the appli-87

cations of the AB rule via a simulation model of an urban area in San Francisco, USA. Perimeter88

control approaches could be used to implement the AB rule, such as modifying signal control,89

rationing license plate, etc. Interaction of multiple modes in the MFD scheme was also analyzed.90

Daganzo et al. (26) discussed similar issues in an example to show the benefits of parsimonious91

models . Zheng et al. (27) developed an MFD-controlled cordon pricing scheme. In the scheme,92

a toll was determined based on the MFD of the target network, and the objective was to maintain93

mean flow of the network at the maximum value of the MFD. Knoop et al. (28) attempted to apply94

the MFD in routing. A few of routing strategies were compared in a network simulation model.95

The results showed improved traffic flow, importance of properly partitioning network, etc. Had-96

dad and Geroliminis (29), Remezani et al. (30) and Haddad et al. (31) partitioned an urban network97

into two regions each with an independent MFD: a city center and its periphery. Stability of the98

two-region system was analyzed and a few of optimal traffic control problems were explored.99

4

URBAN FREEWAY NETWORK IN BEIJING AND THE DATA100

Urban ring freeways in Beijing101

Beijing is one of the largest cities in the world. At present, the urban area of Beijing is enclosed102

by four two-way urban ring freeways, i.e., the 2nd-5th rings. Among these, the 3rd ring with three103

lanes in each direction is 48.3 km and the speed limits are 80 km/h for straight sections and 60104

km/h for curves. 74 Remote Transportation Microwave Sensors (RTMS) covering two-way traffic105

have been installed on the ring (see Figure 1). Traffic flow data (i.e., occupancy, flow and speed)106

used in the paper are collected on the ring from 6 am to 12 pm on four weekdays, i.e., June 3-6107

(Mon-Thu), 2002. The data are aggregated every two minutes.108

FIGURE 1 The urban freeway network in Beijing and locations of RTMS on the 3rd urbanring freeway.

Data processing109

The data are processed as the following three steps and Table 1 presents the results:110

step 1: eliminating ineffective RTMS by checking if a data file contains data;111

step 2: observing the flow-occupancy diagram drawn by using data from each individual lane, and112

discarding entire data pertaining to a lane whose diagram looks obviously incorrect. In the step, the113

flow-occupancy relation and similarity of the diagrams for adjacent lanes are mainly considered.114

The step is manual and relies on basic traffic flow knowledge;115

step 3: removing missing and out-range data. If one item in a data unit of occupancy, flow and116

speed at a time slice is out of given intervals, the unit will be removed. The intervals of occupancy,117

flow and speed are chosen as [0,100]% , [0,2500] veh/h and [0,100] km/h, respectively.118

5

TABLE 1 All data and the number of ineffective data in each stepDate step 1: RTMS step 2: lane data step 3: data unit

total # ineff. # total # ineff. # total # missing # out-range #

June 3 74 9 419 30 280080 5831(2.08%) 11345(4.05%)June 4 74 9 419 26 282960 8631(3.05%) 11012(3.90%)June 5 74 11 407 16 281520 5359(1.90%) 12896(4.58%)June 6 74 9 419 11 293760 14600(4.97%) 12362(4.20%)

Building Macroscopic Fundamental Diagrams119

Since entire data from some lanes are excluded, we take an average of data of the rest of lanes in a120

direction covered by a RTMS ( we regard a direction of each RTMS as a location in the rest of the121

paper) to represent traffic conditions at the location. Then, the space-mean flow and occupancy on122

the ring is derived by using the formula introduced in Geroliminis and Daganzo (16).123

Specifically, let i and N be the index and the total number of locations covered by all124

effective RTMS, and denote by Ni the number of effective lanes at location i. Occupancy is125

directly used without being converted to density as usual. Mean flow and mean occupancy at time126

interval k are obtained as follows:127

Q(k) =1

N

N∑i=1

qi(k), qi(k) =1

Ni

Ni∑j=1

αij(k)qij(k) (1)

128

O(k) =1

N

N∑i=1

oi(k), oi(k) =1

Ni

Ni∑j=1

βij(k)oij(k) (2)

where qij(k) and oij(k) are flow and occupancy collected on lane j at location i every two minutes;129

αij(k), βij(k) ∈ {0, 1} are dummy coefficients that are equal to 1 if the data unit is effective; 0,130

otherwise.131

MACROSCOPIC FUNDAMENTAL DIAGRAMS FOR THE URBAN FREEWAY NETWORK132

IN BEIJING133

Existence of a figure-eight hysteresis pattern134

MFDs for the four weekdays are built in Figure 2. Meanwhile, the global variance of occupancy135

among all locations in a time interval (denoted by V (k)) is calculated to represent spatial hetero-136

geneity, and a relation between occupancy variance and mean occupancy is also plotted in the same137

figure.138

In the figure, two distinguishing features could be observed: (i) figure-eight hysteresis139

combining clockwise and counter-clockwise loops; (ii) an association between lower occupancy140

variance and lower mean flow accompanying with the counter-clockwise loops; it is inconsistent141

with the observation in Geroliminis and Sun (2). The causes of the clockwise loops have been142

explicitly investigated theoretically and empirically in Daganzo (19), Buisson and Ladier (17) and143

Geroliminis and Sun (2); refer to the review of the papers. Therefore, the remainder of the paper144

concentrates on the formation of the counter-clockwise loop and the accompanied association.145

6

(a) June 3, 2002 (b) June 4, 2002

(c) June 5, 2002 (d) June 6, 2002

FIGURE 2 Mean flow vs. mean occupancy (upper plot) and occupancy variance vs. meanoccupancy (lower plot) for the 3rd ring on June 3-6, 2002 (the gradually changing colors fromred to blue demonstrate the time growth from 6:00 am to 12:00 pm)

Formation of the counter-clockwise loop in the figure-eight hysteresis pattern146

We select the counter-clockwise direction of the 3rd ring as an example and deeply look at its spa-147

tialtemporal heterogeneity of vehicles (note that there are two directions on a ring road, which are148

usually called the clockwise and counter-clockwise directions). Figure 3 first shows the relations149

between mean flow and mean occupancy and between occupancy variance and mean occupancy;150

the aforementioned features are also observed. Figure 4 combines time, locations and occupancy,151

and provides a clear look at the spatialtemporal heterogeneity. Heavy congestions occur at the152

locations around 3040 and 3070. The congestion around location 3070 vanishes at the end of the153

rush hour, while the congestion around location 3040 lasts to the end; it implies higher occupancy154

variance in the offset of the rush hour, and provides an empirical evidence that unevenness of vehi-155

cle distribution will arise in the offset of a rush hour on a ring road, which was theoretically stated156

in Daganzo (19).157

7

0 3 6 9 12 15 18 21 24 270

300

600

900

1200

1500

Mea

n flo

w (

veh/

h)

Mean occupancy (%)

7:03

11:45

0 3 6 9 12 15 18 21 24 270

100

200

300

400

500

Occ

upan

cy v

aria

nce

(%)

7:03

11:45

FIGURE 3 Mean flow vs. mean occupancy (upper plot) and occupancy variance vs. meanoccupancy (lower plot) for the counter-clockwise direction in the 3rd ring on June 4, 2002(the gradually changing colors from red to blue demonstrate the time growth from 6:00 amto 12:00 pm)

FIGURE 4 Spatialtemporal heterogeneity of vehicles in the counter-clockwise direction ofthe 3rd ring from 6:00 am to 12:00 pm on June 4, 2002

To see more details, we select a pair of two-minute intervals on the counter-clock loop,158

i.e., k1 and k2, which start from 7:03 and from 11:45, respectively. In the paired time intervals the159

mean occupancy is approximate, while lower occupancy variance is associated with lower mean160

8

flow, i.e., V (k1) < V (k2) and Q(k1) < Q(k2), when O(k1) ≈ O(k2). We plot the occupancy, flow161

and speed in each time interval at all locations in Figure 5. From the figure, two distinguishing162

traffic conditions can be observed, those are, (a) the congested condition existing at location 3042163

and 3043 (denoted by M the set of the two locations), where higher occupancy is associated with164

lower speed comparing with other locations; (b) the free-flow condition at the other locations165

(denoted by M ), where all occupancy, speed and flow are close. We now check the association:166

FIGURE 5 Traffic conditions in the two-minute intervals respectively starting from 7:03 and11:45 in the counter-clockwise direction of the 3rd ring on June 4, 2002 (the blue: occupancy,the green: flow, and the red: speed)

(i) V (k1) < V (k2). It can be seen from all occupancy in the paired time intervals that all167

locations are in the free-flow condition in k1. In contrast, congestion still exists at M in k2. The168

variance of occupancy in k2 is thus higher than that in k1. It is just as what existing findings stated.169

(ii) Q(k1) < Q(k2). The locations of M have170 ∑i∈M

qi(k1) ≈∑i∈M

qi(k2),∑i∈M

oi(k1) <∑i∈M

oi(k2) (3)

9

To achieve Q(k1) < Q(k2) and simultaneously O(k1) ≈ O(k2), other locations M should have171 ∑i∈M

qi(k1) <∑i∈M

qi(k2),∑i∈M

oi(k1) >∑i∈M

oi(k2) (4)

However, we can not see the relations in the figure, and they are rare to the free-flow condition, in172

which flow and occupancy at an individual location is usually positively correlated as well as the173

sum of flow and occupancy from different locations based on the fundamental traffic flow theory.174

To understand the relations, flow-occupancy relations at different locations in the paired175

time intervals k1 and k2 are plotted in Figure 6. It can be seen in general that the flow in k1 is176

smaller than that in k2, and the occupancy in k1 is greater than that in k2, which lead to inequality177

(4). It provides insight into the cause of Q(k1) < Q(k2) and simultaneously O(k1) ≈ O(k2).178

The observation, however, is interesting: in the free-flow condition, the same occupancy in k1179

is associated with lower flow than that in k2. To show no coincidence, we further present the180

flow-occupancy relations on other days; see Figure 7.181

0 10 20 30 40 50 600

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

−3.602x2+114.3x+212 (R2=0.6331)

−2.757x2+109.4x+297.2 (R2=0.824)

7:0311:45

M

M

FIGURE 6 Flow vs. occupancy at all locations at 7:03 and 11:45 on June 4, 2002 (The curvesare fitted using data units which occupancy is lower than 20%)

To provide more evidences at the microscopic level, we plot flow-occupancy diagrams for182

individual locations and present some of them in Figure 8 and 9. Two free-flow branches can be183

seen in the diagrams, i.e., a lower free-flow branch in the onset of the rush hour and a higher branch184

in the offset; it is interesting and we will discuss the causes in the next section. Importantly note185

that speed limits during the selected weekdays were invariant; only a part of locations exhibit the186

feature, and the locations in Figure 8 and 9 are some of those with the quite obvious feature.187

10

0 10 20 30 40 50 600

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

−2.371x2+92.24x+239.4 (R2=0.6402)

−3.003x2+102.1x+370.1 (R2=0.6187)

6:4511:31

(a) 20020603(clockwise)

0 10 20 30 40 50 600

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

0.08171x2+51.51x+382.8 (R2=0.7152)−2.097x2+97.31x+366.5 (R2=0.6838)

6:4511:31

(b) 20020603(counter-clockwise)

0 10 20 30 40 50 600

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

−1.715x2+86.73x+290.6 (R2=0.5966)

0.09916x2+55.62x+547.7 (R2=0.7453)

7:0710:11

(c) 20020605(clockwise)

0 10 20 30 40 50 600

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

−1.523x2+76.48x+374.8 (R2=0.7053)

−2.098x2+94.69x+411.4 (R2=0.6725)

7:0910:27

(d) 20020605(counter-clockwise)

0 10 20 30 40 50 600

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

−2.044x2+85.31x+322.6 (R2=0.5659)

−0.5844x2+68.35x+481.2 (R2=0.6449)

7:0310:45

(e) 20020606(clockwise)

0 10 20 30 40 50 600

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

−1.84x2+82.88x+352.8 (R2=0.672)−4.359x2+133.8x+290.7 (R2=0.6473)

7:0510:43

(f) 20020606(counter-clockwise)

FIGURE 7 Flow vs. occupancy at all locations at pairs of time slices with the approximatemean occupancy on June 3, 5 and 6, 2002 (The curves are fitted using data units whichoccupancy is lower than 20%)

11

off-rampon-ramp off-rampon-ramp

3002

underpath

clockwise direction

860 m

(a) The location of RTMS 3002 in front of National Agriculture Exhibition Center of China

0 20 40 60 80 1000

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

(b) Median lane, clockwise, 3002

0 20 40 60 80 1000

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

(c) Center lane, clockwise, 3002

0 20 40 60 80 1000

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

(d) Shoulder lane, clockwise, 3002

FIGURE 8 Flow-occupancy diagrams for location 3002 on the 3rd ring on June 4, 2002 (thegradually changing colors from red to blue demonstrate the time growth from 6:00 am to12:00 pm)

overpath

3072

off-ramp counter-clockwise direction on-ramp

688 m

(a) The location of RTMS 3072 on Sanyuan West Bridge

0 20 40 60 80 1000

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

(b) Median lane, counter-clockwise,3072

0 20 40 60 80 1000

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

(c) Center lane, counter-clockwise,3072

0 20 40 60 80 1000

400

800

1200

1600

2000

Occupancy (%)

Flo

w (

veh/

h)

(d) Shoulder lane, counter-clockwise, 3072

FIGURE 9 Flow-occupancy diagrams for location 3072 on the 3rd ring on June 4, 2002 (thegradually changing colors from red to blue demonstrate the time growth from 6:00 am to12:00 pm)

12

DISCUSSION OF THE MULTI-BRANCH FLOW-OCCUPANCY DIAGRAM188

The urban freeway network in Beijing has unique characteristics, such as a large number of aux-189

iliary roads surrounding and connecting with the urban freeways, dense ramps (a ramp per 0.5190

km on average, approximately; refer to Figure 8(a) and 9(a) for examples), short ramp length,191

many interchanges, etc. All of these have significant impacts on the traffic and driver behavior, in192

particular the dense ramps, which result in frequent lane-changing maneuvers.193

In recent work on the traffic on the urban freeways, we proposed an inhomogeneous macro-194

scopic traffic flow model with a multi-branch fundamental diagram (see He et al., He and Guan195

(32, 33)). Such two free-flow branches could be explained as a result of different lane-changing196

rates in the onset and offset of a rush hour.197

We briefly introduce the macroscopic driver perception (MDP) model proposed in He and198

Guan (33). The model extends the LWR model (34, 35) by considering a speed-density relation199

u = U(ρ, µ), where u and ρ are speed and density, respectively, and the diver perception factor µ200

changes with surrounding traffic situations. The equation of the MDP model reads:201

∂(ρu)

∂t+∂(ρuµ)

∂x= ψµe(ψ) +

ρu(µe(ψ)− µ)

ι(5)

where ψ representing lane-changing frequency is a known function on space time point (x, t), ι202

is a relaxation factor in units of distance, and µe(ψ) is a desired perception factor dependent on203

ψ. The flow ρuµ is named as “perception flow" as µ is a driver perception factor. The first part204

ψµe(ψ) in the source term of Equation 5 indicates the increased perception flow per distance unit205

caused by lane-changing vehicles with perception factor µe(ψ). The second part ρu(µe(ψ)− µ)/ι206

means the increased perception flow per distance unit caused by vehicles in the target lane with207

perception factor µ and relaxes to µe(ψ) as lane-changing vehicles enter.208

The perception-dependent speed-density relation U(ρ, µe(ψ)) is constructed in the paper,209

and the multi-branch fundamental diagram is calibrated using the empirical data collected on a210

road section with dense ramps on an urban ring freeway in Beijing; see Figure 10. The solution to211

the corresponding Riemann problem is further provided, and numerical simulations show that the212

model is able to reproduce the patterns observed at on-ramp inhomogeneity.213

Based on the multi-branch flow model, the lower free-flow branch in the onset of congestion214

is caused by higher lane-changing rates. In an urban freeway network with dense ramps, like the215

urban freeways in Beijing, locating detectors closely to ramps is difficult to be avoided, and more216

lane-changing maneuvers are thus included in the detected traffic flow data. Indeed, lane-changing217

maneuvers on the urban freeways are more frequent due to the denseness of ramps, and the kind218

of data describes the reality. It is obvious that the lane-changing maneuvers are closely related to219

inflow and outflow via on- and off-ramps as well as the OD matrices. Due to lack of the lane-220

changing data or the data on ramps, we can not directly show the change of the lane-changing rates221

at the study locations of the paper. However, it is not difficult to imagine that higher demands for222

the urban freeway in the onset of the rush hour result in higher inflow and higher lane-changing223

rates in the vicinity of on-ramps, and consequently lead to a lower free-flow branch in the flow-224

occupancy relations.225

13

FIGURE 10 Empirical multi-branch fundamental diagrams: (a) flow-density curves, (b)speed-density curves.

CONCLUSIONS226

A figure-eight hysteresis pattern is observed in the MFDs for the 3rd urban ring freeway in Beijing,227

China. To understand the causes, analyses are made ranging from spatialtemporal heterogeneity228

of vehicles to the flow-occupancy relation for individual locations. At individual locations, it229

is observed that the free-flow traffic with the same occupancy exhibits lower flow in the onset230

of the rush hour and higher flow in the offset. The multi-branch flow-occupancy relation at the231

microscopic level, consequently, results in the counter-clockwise loop in the figure-eight hysteresis232

pattern and the association between lower occupancy variance and lower mean occupancy at the233

macroscopic level.234

It is discussed that the multi-branch flow-occupancy relation is caused by different lane-235

changing rates in the onset and offset of a rush hour. Frequent lane-changing maneuvers due to236

dense ramps are an important characteristic of the traffic on the urban freeway network in Beijing.237

Although more work is still needed to shed light on the phenomena, the results still indicate that238

both lane-changing rates (or detector locations) and the shape of the fundamental diagram for239

individual locations have significant impacts on the shape of the MFD.240

Moreover, this paper presents the MFDs for an urban ring freeway. The hysteresis phe-241

nomena are also observed in the MFDs for the urban freeway network with more ramps (i.e., more242

route choices for drivers than regular freeways). Meanwhile, the results also provide empirical243

support that unevenness of the vehicle distribution will arise in the offset of a rush hour on a ring244

road.245

ACKNOWLEDGMENT246

The authors are grateful to the anonymous reviewers for their constructive comments and Mis-247

s Wang L. for checking the phrasing patiently. This research has been funded by 973 program248

(2012CB725403), National Natural Science Foundation of China (71131001), Fundamental Re-249

search Funds for the Central Universities (2012JBM064) and Innovation Foundation for Ph.D Stu-250

dent (2011YJS245).251

14

REFERENCES252

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