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EMPIRICAL ANALYSIS OF TRAFFIC FLOW FEATURES OF A
FREEWAY BOTTLENECK SURROUNDING A LANE DROP
by
MONICA TERESA LEAL SANCHEZ
A research project report submitted in partial fulfillment
of the requirements for the degree of
MASTER OF SCIENCE
in
CIVIL ENGINEERING
Portland State University
2002
PROJECT APPROVAL
The research project report of Mónica Teresa Leal Sánchez for the Master of Science in
Civil Engineering submitted on November 17, 2002, is accepted by the faculty advisor
and the department.
ADVISOR APPROVAL ____________________________________________
Robert L. Bertini, Advisor
DEPARTMENT APPROVAL ____________________________________________
Scott A. Wells, Chair
Department of Civil and Environmental Engineering
iii
ACKNOWLEDGEMENTS
I wish to thank Professor Robert Bertini for supervising this project, and for his
energy, motivation, invaluable support, understanding, and guidance. I feel honored to
have the opportunity to work with him.
I would like to thank Dr. Lall for his advice, and for making my learning
experience more enjoyable and productive. I am also grateful to the transportation
research group with whom I have shared this important time in my life. Dr. Robert
Bertini, Sutti Tantiyanugulchai, Shazia Malik, Roger Lindgren, Ahmed El-Geneidy, and
Edward Anderson not only share their opinions, knowledge, helpful comments, but also
their invaluable friendship. I also thank Jenny Kincaid, Marianne Stupfel-Wallace, and
Marianne Cartwright for their help during the last two years. Special thanks to Dr. Bertini
and Roger Lindgren for their help to finish this report.
This project would not have been possible without the encouragement and support
of my lovely family who gave me advice, energy, and motivation. I would especially like
to thank my parents, Jose Antonio Leal and Cecilia Sánchez de Leal who support me in
the achievement of my dreams. I also thank my sisters, brothers and their families, Clara,
Liliana, Jose, Sandra, Diego, Maria Ale and Felipe who always encourage me and give
me energy to continue with my studies. I would sincerely thank Ray Agosta, who has
always been there for me in the good and bad times to give me support and energy and
my friends for the fun times that we have shared.
I would like to acknowledge Mr. Stuart Beale, Telematics Group, Highways
Agency, Department of the Environment, Transport and the Regions, United Kingdom
and Mr. Tim Rees, Project Manager, Transport Research Laboratory, United Kingdom,
for generously supplying the data used herein. In addition, I would also like to thank the
Department of Civil Engineering and Environmental Engineering, at Portland State
University that funded a portion of this project.
iv
ABSTRACT
An abstract of the research project report of Mónica Teresa Leal Sánchez for the Master
of Science in Civil Engineering submitted on November 17, 2002.
Title: Empirical analysis of traffic flow features of a freeway bottleneck surrounding a
lane drop.
Traffic was studied upstream and downstream of a lane drop bottleneck on a
motorway near London, United Kingdom. Both the bottleneck location and the time of
activation were reproducible from day to day. The average bottleneck discharge flows
were also reproducible and were found to be approximately 9.7% lower than the
prevailing flow observed prior to the queue formation. Flow reductions occurring
sequentially in time and space showed the passage of a backward-moving shock
accompanied by marked reductions in speed. Some features of the shock marking the
transition from uncongested to congested conditions such as travel time and speed were
also evaluated. The mean shock velocities ranged between 3 and 4 mph as they traveled
upstream from the bottleneck. There were only slight differences observed between the
shock speeds from one section to another. Oscillations arising in queues were also
identified. Oscillations propagated upstream at a nearly constant speed of 11 to 12 mph,
upstream of the bottleneck’s location. Locations downstream of the head of the queue
were not affected by these oscillations.
The analysis tools used for this study were curves of cumulative vehicle arrival
number versus time and curves of cumulative time mean speed versus time. These curves
were constructed using data from neighboring loop detector stations along the motorway.
The curves were transformed in order to facilitate the observation of traffic conditions.
v
TABLE OF CONTENTS
1. Introduction 1
1.1. Background……………………………………………………………....2
2. Data 4
3. Methodology 6
3.1. Curves of cumulative count versus time…………………………….…..6
3.2. Curves of cumulative time mean speed versus time……………….……9
4. Observations 11
4.1. First Day……………………………………………………………..….11
4.1.1. First Day: Identification of the bottleneck…………………....12
4.1.2. First Day: Bottleneck discharge features…………………......16
4.1.3. First Day: Shock features…………………………………......18
4.1.4. First Day: Traffic oscillations………..……………….………21
4.2. All Days……………………………………………………….………..23
4.2.1. All Days: Bottleneck’s identification and features…………...23
4.2.2. All Days: Shock characteristics……………………….……...24
4.2.3. All Days: Speed observations………………………………...25
5. Conclusions 26
References 28
Appendix A – Weather conditions 29
Appendix B - Summary of shock characteristics 30
vi
LIST OF TABLES
Table Title Page
1 First Day: Shock Characteristics…………………………………..18
2 All Days: Summary of Traffic Features…………………………..23
3 All Days: Shock Characteristics…………………………………...24
4 Weather Conditions………………………………………………..29
5 First Day: Shock Characteristics……………………………...…...30
6 Second Day: Shock Characteristics………………………………..30
7 Third Day: Shock Characteristics………………………………….31
8 Fourth Day: Shock Characteristics……………………………...…31
9 Fifth Day: Shock Characteristics……………………………...…...31
vii
LIST OF FIGURES
Figure Title Page
1 Site Map…………………………………………………………….4
2 Curve of cumulative counts versus time………………………….…6
3 N-curves at two consecutive loop detector stations…………………7
4 Diagram to evaluate queue features…………………………………8
5 Oblique coordinate system…………………………………………..9
6 Transformed V-curves……………………………………………...10
7 First Day: Speed contours…………………………………………..11
8 First Day: Transformed N-curves……………………………….….12
9 First Day: Transformed V-curves at upstream stations…………..…15
10 First Day: Upstream and downstream transformed N-curves………16
11 First Day: Transformed N-curve and V-curves at station 8……..….17
12 First Day: Speed-Flow diagram for station 5………….……………19
13 First Day: Flow-Density diagram for station 5……………………..20
14 First Day: Flow-Density diagram for station 2..……………………20
15 First Day: N − N 10 curves at each station…………………………..22
16 Fifth Day: Transformed V-curves at station 2 and 9 ………………25
1
1. Introduction
It is shown that a freeway bottleneck became active near a lane drop from three to
two lanes on the M4 motorway near London, United Kingdom. The activation of the
bottleneck was marked by the formation of a queue that propagated several miles
upstream, and resulted in a reduction in average discharge flow. The bottleneck location
was reproducible from day to day, and the average discharge flow measured after
bottleneck activation did not vary substantially on the five days analyzed. The arrival of a
backward-moving queue at a measurement location was accompanied by a reduction in
flow and a simultaneous drop in speed. It is also shown that shock velocities were nearly
constant as the queue propagated upstream from the bottleneck. Oscillations were
identified arising in the direction opposite of the flow at measurement locations upstream
of the active bottleneck. These oscillations were not observed at any locations
downstream of the head of the queue.
In earlier studies, traffic conditions have been examined upstream and
downstream of a freeway bottleneck located near a busy on-ramp (e.g., Cassidy and
Bertini, 1999a; Cassidy and Bertini, 1999b; Bertini, 1999; Bertini and Cassidy, 2002). In
an earlier study, oscillations were also found to propagate in queues at a nearly constant
speed in the direction contrary to the flow (Mauch and Cassidy, 2002). These oscillations
did not affect flows measured downstream of the location where the queue formed. To
promote the visual identification of time-dependent features of the traffic stream, these
previous studies used curves of cumulative vehicle count and curves of cumulative
occupancy constructed from data measured at neighboring freeway loop detectors
(Cassidy and Windover, 1995). These cumulative curves provided the measurement
resolution necessary to observe the transitions from freely flowing to queued conditions
and to identify a number of notable, time-dependent traffic features in and around the
bottlenecks.
Cumulative curves were also used in this study, which adds to previous findings
by reporting on observations taken during five-morning peak periods both upstream and
2
downstream of a freeway lane drop. These observations were made possible via the use
of cumulative curves of vehicle arrival versus time and cumulative curves of speed versus
time, using vehicle actuation times and time mean speeds obtained from inductive loop
detectors located in each travel lane. Through the use of these curves, it has been possible
to verify that the bottleneck became active, guaranteeing that vehicles discharged from an
upstream queue and were unimpeded by traffic conditions from further downstream
(Daganzo, 1997). It was also possible to observe that certain bottleneck features were
reproducible from day to day.
First, a brief background discussion will be provided, followed by a description of
the study site and the loop detector data used for this analysis. Next, a detailed
description of the bottleneck’s location and discharge features will be presented for one
study day, followed by a summary of features found to be reproducible on four additional
days. Finally, some concluding comments will be provided.
1.1. Background
Understanding traffic behavior at a freeway bottleneck provides a foundation for
understanding how a freeway system operates. A bottleneck is any point on the network
upstream of which one finds a queue and downstream of which one finds freely flowing
traffic. Bottlenecks can be static (e.g., a tunnel entrance) or dynamic (e.g., an incident or
a slow moving vehicle). A bottleneck is considered active when it meets the conditions
described above and is deactivated when there is a decrease in demand or when there is a
spillover from a downstream bottleneck (Daganzo, 1997). Bottlenecks are important
components of freeway systems, since the queues that develop upon bottleneck activation
may propagate for several miles, causing delay and potentially blocking off ramps and
access to other facilities.
While discussing the state of traffic flow theory in 1965, Gordon Newell
(unpublished lecture) stated that “the main object” of traffic research should be “to study
time-dependent flows, to determine velocities of propagations of disturbances,” and to
3
determine “how…traffic adjusts to some time- or space-dependent influences such as
traffic lights, bottlenecks, etc.” In a later review of the evolution of traffic flow theory,
Newell (1995) explained that in the early 1960s, it was expected that “new experimental
observations would soon resolve some of the deficiencies of existing theories.” With the
implementation of new surveillance systems, it is appropriate to study and understand
freeway bottlenecks of all kinds—including merges (e.g., Cassidy and Bertini, 1999a;
Cassidy and Bertini, 1999b; Bertini, 1999; Bertini and Cassidy, 2002; Cassidy and
Mauch, 2001), diverges (e.g., Windover, 1998; Muñoz and Daganzo, 2002), lane drops
and other configurations. This study will contribute to a greater understanding of
bottlenecks arising in the vicinity of a freeway lane drop.
4
2. Data
The observations that follow were taken during the morning peak hours of five
study days from a segment of the eastbound M4 motorway near London, United
Kingdom, as shown in Figure 1. This is part of the main highway that connects London
and Heathrow Airport. Inductive loop detectors recorded individual vehicles’ arrival
times (hr:mm:ss) and their time mean speeds in each lane. If no vehicles were observed in
a given second, no values were recorded for vehicle count and speed. Thus the loop
detector data were available in their most raw form and were not aggregated over any
arbitrary periods. Occupancy data were not recorded. The loop detectors are labeled 1
through 9 as shown in Figure 1. The lane drop is located at mile 10.6 (kilometer 17.1).
When these data were collected the motorway speed limit was 70 mph upstream of
station 9 and 50 mph downstream of station 9. Appendix A contains weather conditions
data for London, UK on the days that were analyzed.
Figure 1: Site Map
5
It is noted that the M4 motorway now includes a bus lane at this site, installed in
1999. The bus lane was installed on the fast (right hand) lane of the motorway (Rees,
White and Quick, 2000). However, the observations contained in this study were taken
during the period prior to these modifications to the motorway lane markings.
6
2. Methodology
Curves of cumulative vehicle count and curves of cumulative time mean speed
versus time were used to identify the motion of changing traffic stages. These curves
were constructed from data measured by loop detectors in each lane at the study site
(Cassidy and Windover, 1995). The curves were transformed in order to facilitate the
observations of traffic conditions. The procedure used to create and transform these
curves is described in this section.
2.1. Curves of cumulative count versus time
An example of a curve of N(x,t), where N(x,t) is the cumulative count of vehicles
to pass location x by time t, is shown in Figure 2. For this hypothetical example, as shown
in the vertical axis, a total of four vehicles arrived at location x by t = 6:22:06 AM.
Constructing differentiable interpolations to this stepwise function, it is possible to create
a smooth approximation to N(x,t). The time derivative of this approximation is the flow at
(x,t). The flow is measured as the slope of the curve.
8
7
6
F lo w
5
N(x,t)
4 N (x ,t)
3
2
1
0
6:22:00 AM
6:22:01 AM
6:22:02 AM
6:22:03 AM
6:22:04 AM
6:22:05 AM
6:22:06 AM
6:22:07 AM
6:22:08 AM
6:22:09 AM
6:22:10 AM
6:22:11 AM
6:22:12 AM
T im e @ S t a t io n x
Figure 2: Curve of cumulative counts versus time
7
Curves of cumulative vehicle count constructed at consecutive loop detector
stations can be used to trace the motion of disturbances in time and space. Figure 3 shows
two hypothetical cumulative curves recorded at two consecutive locations along a
freeway. The two curves are constructed using the same collection of vehicles. To
illustrate this situation, suppose that an observer records the arrival time of vehicles as
they pass location x1 while another observer records the time at which the same vehicles
pass location x2. N(x1,t) represents the cumulative number of vehicles that pass location
x1 by time t while N(x2,t) illustrates the cumulative number of vehicles that pass the
downstream location x2 at time t. The vertical distance between the curves at some time t1
is the total vehicle accumulation between x1 and x2 at that time. The horizontal distance at
some N=j between the curves is the actual jth vehicle’s trip time between x1 and x2.
18
16
N(x 1 ,t)
x2
Traffic Direction
14
12
N(x 2 ,t)
10
x1
N(xi,t)
8
Trip Time
j
6 Vehicle
Accumulation
4
2
0
t1
Time , t
Figure 3: N-curves at two consecutive loop detector stations
8
Figure 4 shows a queuing diagram constructed by shifting the upstream N-curve
to the right by the free flow travel time between station x1 and x2 (Newell, 1982; Newell,
1993). If vehicles are conserved between x1 and x2, the N-curves are superimposed when
the traffic is flowing freely between these stations. The resulting vertical separation
between these curves is now the excess vehicle accumulation and the horizontal
separation is delay between x1 and x2.
18
x2
16
Traffic Direction
N(x 1 ,t) Excess Vehicle
14 Accumulation
Delay
12
x1
N(xi,t)
10
N(x 2 ,t)
8
6
4 Free Flow Trip Time
2
0
Time , t
Figure 4: Diagram to evaluate queue features
In order to magnify the curves’ features, an oblique coordinate system was used
where N(x, t) was reduced by q0 (t-to), where q0 is a background flow and to is the curve’s
starting time as shown in Figure 5. The value of q0 was chosen by iteration so that the
range of N-q0 (t-to) was small as compared with the N itself, and obtain the best
visualization that magnified traffic features of interest such as changes in slope, drops and
increases in flow. The same value of q0 was used for all curves and therefore does not
affect the vertical separations (Cassidy and Windover, 1995). The use of the oblique
9
coordinate system is described in several references (Cassidy and Windover, 1995;
Cassidy and Bertini, 1999a).
18
16 N(x i ,t)
14
12 qo
N(x i ,t)- q o (t-t 0 )
10
N(xi,t)
8
6
4
2
0
Time , t
Figure 5: Oblique coordinate system
3.2. Curves of cumulative time mean speed versus time
Curves of cumulative time mean speed, V(x,t), were used in this study to verify
traffic features observed from the N-curves. V(x,t) represents the cumulative time mean
speed measured at station x by time t. As with the N-curves, piecewise linear
approximations to V(x,t) were constructed. While somewhat unconventional, the slope of
the V-curve is a “speed rate” measured at location x at time t. An oblique coordinate was
also used following the same methodology described above for the N-curves. V(x,t) was
reduced by Vo(t-t0), where Vo is a background rate for transforming the V-curve at
location x while t0 represents the curve’s starting time. The V-curves were transformed to
improve visualization of speed features at location x. The estimated speed changes are
marked on the transformed V-curve by linear approximations made by eye (dashed lines).
10
Thus, the transformed V-curves help to visually identify periods of nearly constant
average speed, changes in average speed and other speed features contained in the raw
data, as shown in the Figure 6.
22000
V(x,t) -Vo (t - 6:33:18 AM), Vo = 110,000 mph per hour
d
ee
sp
nt
ta
Speed drops
ns
co
rly
15000
ea
N
8000
6:33:18 AM 6:46:38 AM 6:59:58 AM
Time
Figure 6: Transformed V-curve
11
3. Observations
4.1. First Day
Traffic features surrounding a lane drop were analyzed on the M4 motorway on
Monday, November 16, 1998. The weather conditions show that this was a cloudy day. In
order to gain a macroscopic view of traffic conditions over the morning peak period,
Figure 7 shows a speed contour diagram for an extended section of the eastbound M4. In
this figure the horizontal axis is time (hours—5:30 AM until 11:00 AM) and the vertical
axis is distance. Note that on this day detector data were not available for station 9. The
variations in color represent changes in speed, from blue for lower speed to green for
higher speed. As noted above, the speed limit for the segment shown was 70 mph. The
speed contours indicated that a queue arose in the vicinity of station 6 and propagated
upstream for several miles beginning at around 6:45 AM. This led to lower vehicle
speeds and resulting delays. It is also apparent that the queue began to dissipate
beginning near station 1 at about 8:45 AM.
100
Speed (mph)
0
Figure 7: First Day: Speed Contours
12
4.1.1. First Day: Identification of the bottleneck
Figure 8 shows rescaled curves of cumulative arrival number of vehicles versus
time, N(x, t), constructed from counts measured across all lanes at detectors 2-8 and
collected during a 30-minute period surrounding activation of the bottleneck between
detectors 6 and 7. The curves were constructed by taking linear interpolations through the
individual vehicle arrival times, so that a curve’s slope at time t would be the flow past
location x at that time. The counts for each curve were started (N=0) relative to the
passage of a hypothetical reference vehicle so all curves describe the same collection of
vehicles. Each curve was shifted horizontally to the right by the average free flow trip
time from its respective x to station 8, the downstream most detector. Any resulting
vertical displacement is the excess vehicle accumulation between stations because of
vehicle delays.
440
22000 Station 2 - 7:01:46 AM
400
V(6,t) -Vo t', Vo = 104,000 mph per hour
360 Station 3 - 6:59:25 AM
6:47:28 AM
320 15000
Station 4 - 6:55:27 AM
N(x,t)- qo (t-to), qo = 3600 vph
280
240
8000
6:33:18 AM 6:46:38 AM 6:59:58 AM
Tim e
200
Station 5 - 6:50:08 AM
160
Station 6 - 6:47:28 AM
120
Station 7 - 6:45:26 AM
80
40
0
Station 8 - 6:45:50 AM
-40
6:41:00 AM
6:42:20 AM
6:43:40 AM
6:45:00 AM
6:46:20 AM
6:47:40 AM
6:49:00 AM
6:50:20 AM
6:51:40 AM
6:53:00 AM
6:54:20 AM
6:55:40 AM
6:57:00 AM
6:58:20 AM
6:59:40 AM
7:01:00 AM
7:02:20 AM
7:03:40 AM
7:05:00 AM
7:06:20 AM
7:07:40 AM
7:09:00 AM
Time @ Station 8
Figure 8: First Day: Transformed Cumulative Curves
13
In order to magnify the curves’ features, an oblique coordinate system was used
where N(x, t) was reduced by q0(t-to) where q0 is the background flow and to is the
curve’s starting time. The same value of q0 was used for all curves and therefore does not
affect the vertical separations (Cassidy and Windover, 1995).
As shown in Figure 8, curves for all stations are initially superimposed indicating
freely flowing traffic throughout the entire motorway section. The curves for stations 8
and 7 remain nearly superimposed for the entire period, indicating that traffic continued
to flow freely between these stations. Substantial vehicle accumulations are seen between
stations 6 and 7 subsequent to flow reductions which were observed at stations 7 and 8 at
around 6:45:26 AM and 6:45:50 AM respectively.
The divergence of the curve at station 6 from the one at station 7 (at 6:47:28 AM)
marks the arrival of a backward-moving queue at station 6. There was a pronounced flow
reduction at station 6 that accompanied this divergence. The inset in Figure 6 contains a
transformed curve of cumulative vehicle speed, V(x,t), versus time, measured at station 6.
A sharp reduction in speed is seen at around 6:47:28 AM, verifying the arrival of the
queue. The presence of freely flowing traffic between stations 7 and 8 (as evidenced by
the superimposed station 7 and station 8 curves on Figure 6) accompanied by excess
accumulation of vehicles upstream of station 7 reveals that the bottleneck was located
somewhere between stations 6 and 7 where the transition from three lanes to two lanes
takes place.
Figure 8 also traces the propagation of the queue beyond station 6. As shown in
the figure, a reduction in flow at station 5 was observed at 6:50:08 AM, where the curve
at station 5 deviates from the upstream curves. This indicates excess vehicle
accumulation upstream of station 5. Further deviations are shown, indicating that the
queue arrived at station 2 at 7:01:46 AM. This figure has made it possible to diagnose the
bottleneck’s location (between stations 6 and 7), as well as the time it became active
(around 6:47:28 AM).
14
To verify the arrival of the backward-moving queue at every station, cumulative
time mean speed curves (V(x,t)) were constructed for every detector site. Figure 9 shows
the V(x,t) from stations 6 to 2 of the mean speed for all lanes and the V(x,t) for every lane
at the corresponding location. Note that the times where the average speed dropped
correspond with the times marked by reductions in flow. The figure shows that the
backward-moving queue arrived at slightly different times in each lane.
In order to determine the length of time that this bottleneck remained active,
Figure 10 shows rescaled N(x, t) for stations 2 and 8 for a longer period. As indicated by
the continued vertical displacement between the two curves, the queue between stations 2
and 8 persisted until around 9:07:56 AM when the N-curves became superimposed. This
shows that the vehicles were traveling at their free flow speeds between these stations
after this time. The insets in Figure 10 contain transformed curves of cumulative vehicle
speed at stations 2 and 8. As shown in the lower inset, a speed increase was observed at
station 8 around the time that the queue dissipated. The complete queue dissipation
occurred several minutes after a decrease in flow at station 2 (around 8:48:40 AM)
signaled the end of queuing at that station. The upper inset verifies the timing of the end
of queuing at station 2 by showing that an increase in speed also occurred around 8:48:40
AM.
Figures 8 and 10 have verified the bottleneck’s location, the time it became
active, and the time that it was deactivated. Figure 8 helps to map the time when the
backward-moving queue reached each station. Now it is possible to examine the active
bottleneck’s queue discharge features in detail.
15
22000 14000
V(6,t)-V o(t-6:33:18 AM),Vo= 104,000 mph per hour
Lane 1
Lane 2
Lane 3 6:47:28 AM
V(6,t) -Vo (t-6:21:36 AM)
6:47: 28 AM 6:49:01 AM
15000 7000 6:46:22 AM
8000 0
6:33:18 AM 6:46:38 AM 6:59:58 AM 6:21:36 AM 6:39:36 AM 6:57:36 AM
Time Tim e
STATION 6
9000
Lane 1
30000 Lane 2
V(5,t)-Vo(t-6:38:53 AM),Vo =104,000 mph per hour
Lane 3 6:50:01 AM
V(5,t) -Vo (t-6:34:34 AM)
6:50:08 AM
6:49:06 AM
4500
24500
6:50:08 AM
0
19000 6:34:34 AM 6:46:48 AM 6:59:02 AM
6:38:53 AM 6:49:41 AM 7:00:29 AM Time
Time
STATION 5
28000
Lane 1 7:00:01 AM
39000 Lane 2
V(3,t) -V o(t-6:28:48 AM),Vo= 110,000 mph per hour
Lane 3
V(3,t) -Vo (t-6:07:12 AM)
6:59:25 AM 6:59:22 AM
14000
22000
7:00:51 AM
0
6:07:12 AM 6:48:58 AM 7:30:43 AM
5000
Time
6:28:48 AM 6:57:36 AM 7:26:24 AM
Time
STATION 3
40000 30000
7:01:59 AM
V(2,t)-Vo(t-6:14:24 AM),Vo= 113,000 mph per hour
Lane 1
Lane 2
Lane 3
V(2,t) -Vo (t-6:28:48 AM)
7:01:46 AM
20000 15000
7:01:43 AM
7:02:30 AM
0
0 6:28:48 AM 6:59:02 AM 7:29:17 AM
6:14:24 AM 6:53:17 AM 7:32:10 AM
Time
Time
STATION 2
Figure 9: First Day: Transformed V-curves at upstream stations
16
1700
-12000
V(2,t) - Vo t , Vo = 100,000 mph per hour
1500 Station 2
Flow Reduction
End of Queue
8:48:40 AM
1300
9:07:56 AM
1380
-16000
1100 8:48:40 AM
900
N(x,t)- qo (t-to), qo = 2806 vph
880
700 -20000
8:40:48 AM 8:51:28 AM 9:02:08 AM
500 Time
300 380
100 Station 2
-12100
Station 8
-100
V(8,t)-Vot,Vo =96,000mph per hour
-120
Station 8
-300
-14300
-500
9:07:56 AM -620
-700
-16500
-900 8:49:58 AM 9:00:48 AM 9:11:38 AM
Time
-1100 -1120
5:45:36 AM
6:00:00 AM
6:14:24 AM
6:28:48 AM
6:43:12 AM
6:57:36 AM
7:12:00 AM
7:26:24 AM
7:40:48 AM
7:55:12 AM
8:09:36 AM
8:24:00 AM
8:38:24 AM
8:52:48 AM
9:07:12 AM
9:21:36 AM
9:36:00 AM
9:50:24 AM
10:04:48 AM
10:19:12 AM
Time, t @ Station 8
Figure 10: First Day: Upstream and downstream N-curves
4.1.2. First Day: Bottleneck’s Discharge Features
Cumulative curves from station 8 (downstream of the bottleneck) will be used to
examine the bottleneck’s discharge features. Figure 11 shows a rescaled curve of N(8,t)
along with a rescaled curve of V(8,t) also measured at station 8. In the figure periods of
nearly constant flow and speed are represented with solid lines where the average flows
are in vph and the average speeds are in mph. The average discharge flow is marked with
a dashed line and is given in vph. Figure 11 shows that the formation of an upstream
queue at 6:45:50 AM was marked by a reduction in N accompanied by a reduction in
mean speed. Since the curves in Figure 11 do not display any abrupt reductions in the N
accompanied by reductions in speed between 6:45:50 AM and 9:07:56 AM, it is apparent
that there was no disruption of bottleneck discharge caused by a queue from anywhere
further downstream.
17
200 90000
End of Queue
100 85000
0 80000
N(8,t) - q o (8) t '
Queue Discharge
N(8,t)-qo (t-6:00:00 AM), qo =3200 vph
V(8,t)-vo t', v o =49000 mph per hour
9:07:56 AM
-100 75000
-200 70000
7:35:14 AM
8:38:21 AM
-300 65000
8:07:55 AM
6:45:50 AM
-400 60000
-500 55000
V(8,t) - V o (8) t '
6:28:00 AM
-600 50000
-700 45000
-800 40000
6:00:00 AM
6:08:20 AM
6:16:40 AM
6:25:00 AM
6:33:20 AM
6:41:40 AM
6:50:00 AM
6:58:20 AM
7:06:40 AM
7:15:00 AM
7:23:20 AM
7:31:40 AM
7:40:00 AM
7:48:20 AM
7:56:40 AM
8:05:00 AM
8:13:20 AM
8:21:40 AM
8:30:00 AM
8:38:20 AM
8:46:40 AM
8:55:00 AM
9:03:20 AM
9:11:40 AM
9:20:00 AM
9:28:20 AM
9:36:40 AM
TIME
Figure 11: First Day: Transformed N-curve and V-curve at Station 8
Turning to the bottleneck’s flow features displayed in Figure 11, it is shown that
between 6:28 AM and the beginning of queue discharge (6:45:50 AM) a flow of 3690
vehicles per hour (vph) prevailed in this two-lane section downstream of the bottleneck.
Upon queue discharge, a lower flow of 3470 vph was observed, which prevailed for
about 50 minutes. This was followed by a series of sequences of nearly constant flow
that continued until the queue dissipation. From the perspective of modeling queue
evolution, this sequence of flows does not deviate much from the average discharge flow
of 3300 vph (marked by a dashed line), which was 10.6% lower than the flow that
prevailed prior to bottleneck activation. This average discharge flow prevailed over a
period of 2 hours 22 minutes.
18
4.1.3. First Day: Shock Features
Table 1 shows the shock speeds recorded upon bottleneck activation. As shown,
the shock moved upstream at a speed between 4-8 mph. It is noted that the wave from
station 7 to station 8 is a downstream moving expansion wave of lower flow and higher
speed.
16-Nov-98
Stations Distance Mean Travel Time Mean Speed
miles min:sec mph km / h
7-8 0.31 0:24 + 47 + 75
6-5 0.31 3:40 -5 -8
5-4 0.31 5:19 -4 -6
4-3 0.31 3:58 -5 -8
3-2 0.31 2:21 -8 - 13
Table 1: Shock Characteristics
The upstream shock velocity was verified using basic stream flow diagrams.
Figure 12 shows an example of a speed-flow (v-q) diagram for station 5. Time periods
with nearly constant flows were identified using a transformed N-curve for station 5. The
average speed was calculated for each of the same time periods using the reported speed
data. The resulting speed-flow coordinates were plotted in the v-q plane and the points
were enumerated in accordance with their order in time (1-13) as shown in Figure 12. In
the figure a speed-flow curve can be visualized with uncongested (1-6) and congested (7-
13) conditions. Note that the backward-moving queue reached this station around 6:50:08
AM (See Figure 8). From dimensional analysis, the slope of the line connecting the origin
to a point in the v-q plane is the inverse of the density corresponding to that traffic state.
19
80
70 Uncongested
1 3 5 6:00 AM to
6:50:08 AM
2 4
60 Point before the formation of the queue
6
50
Speed (mph)
40
1/k
30
Point right after the formation of the queue 7
20
12
10
10 11 8
9 Congested
13 6:50:09 to
8:00:00 AM
0
1500 2000 2500 3000 3500 4000 4500
Flow (vph)
Figure 12: Speed-flow diagram for station 5
Figure 13 shows the corresponding flow-density diagram for station 5. Note that
the points were enumerated in accordance with their order in time (1-13). In the figure the
uncongested (1-6) and congested (7-13) conditions can be also identified. Since the slope
of any line on this plane has units of speed, it is possible to show that the shock velocity
is approximately 4.4 mph.
Figure 14 also shows a flow-density diagram for station 2. Note that the queue
reaches this station at 7:01:46 AM (See Figure 8) marking the changes from freely
flowing to queued conditions. The shock velocity is determined to be approximately 7.2
mph.
20
5000
Uncongested - 6:00 AM to 6:50:08 AM
Congested - 6:50:09 to 8:00:00 AM
4500
6
4000
7
4
Point before the formation of the queue 10
3500 12 8
Flow (vph)
Point right after the formation of the queue
2 9
3000 5
3 13
2500 11
2000 1
1500
0 50 100 150 200 250 300 350 400
Density (veh/mile)
Figure 13: Flow-Density diagram for station 5
6000
Uncongested - 6:00 AM to 7:01:46 AM
Congested - 7:01:47 to 8:00:00 AM
5000
4000
Point before the formation of the queue
Point right after the formation of the queue
Flow (vph)
3000
2000
1000
0
0 50 100 150 200 250 300
Density (veh/m ile)
Figure 14: Flow-Density diagram for station 2
21
4.1.4. First Day: Traffic Oscillations
Traffic oscillations in queued conditions are characterized by sharp increases in
flow followed by sharp reductions in flow. To a motorist driving in the queue oscillations
appear as stop and go or slow and go driving conditions. Figure 15 illustrates oscillations
for about 45 minutes of congested conditions by the use of transformed N-curves for
stations 1 to 8. The vertical distance between the transformed N-curves is proportional to
the distance between the loop detector stations along the motorway. Each N-curve was
transformed by subtracting the moving 10-minute average flow as shown in the following
formula: N (t ) − [N (t + 5 min) + N (t − 5 min)] / 2 which is represented as N − N 10 in the
figure. Therefore, the slope of the transformed N-curves in this figure represents
observed flow deviations from average flows (Mauch and Cassidy, 2002).
The oscillations only occur upstream of the formation of the bottleneck and last
for several minutes (stations 1 to 6). The transformed N-curves for the stations
downstream of the bottleneck remain smooth (stations 7 and 8). Thus, oscillations were
not observed where there was not congestion. The amplitudes of each oscillation were no
more than 70 vehicles across all lanes or about 23 vehicles per lane as one can verify in
the figure. The highest amplitude was observed at station 1 where the amplitude was 68
vehicles across all lanes or about 22 vehicles per lane. Other findings report slightly
lower amplitudes with no more than 50 vehicles (Mauch and Cassidy, 2002).
The peaks of the oscillations are connected by dashed lines to show the upstream
motion of the oscillations. The slope of the dashed lines represents the speed at which the
oscillation propagated (Mauch and Cassidy, 2002). The dashed lines are nearly parallel
representing a constant speed of about 11 to 12 mph independent of the location within
the queue. Mauch and Cassidy (2002) reported very similar speeds of about 14 to 15
mph.
22
800
100
1 0
700
2 600
3
500
4 400
N - N10
_
5 300
6
200
Upstream of the formation of the bottleneck
Downstream of the formation of the bottleneck
7 100
8 0
-100
7:30:00 AM
7:35:00 AM
7:40:00 AM
7:45:00 AM
7:50:00 AM
7:55:00 AM
8:00:00 AM
8:05:00 AM
8:10:00 AM
8:15:00 AM
TIME
Figure 15: First Day: N − N 10 curves at each station
23
4.2. All Days
4.2.1. All Days: Bottleneck’s Identification and Features
The analyses described in the previous sections were repeated using data taken
from four additional days on the M4 motorway. Similar traffic conditions were
reproduced during the 4 days, but with some variations. On all five days, the bottleneck
arose between stations 6 and 7. Table 2 reports the sustained flow immediately prior to
queue formation and the average discharge rate that prevailed subsequent to bottleneck
activation for all days including the first day. The mean, standard deviation, and
coefficient of variation are identified for the flows. The duration of queue discharge is
also displayed. Also, the table shows the percentage difference between the higher flow
prior to queue discharge and the sustained average flow that followed.
Flow Immediately Prior to the Percent
Average Discharge Rate
Queue Difference
Date Day
Rate Duration Rate Duration
%
vph hr:min:sec vph h:min:sec
16-Nov-98 Monday 3690 0:17:50 3300 2:22:06 10.6
18-Nov-98 Wednesday 3690 0:14:45 3300 2:19:25 10.6
30-Nov-98 Monday 3840 0:08:07 3430 2:06:09 10.7
2-Dec-98 Wednesday 3750 0:11:57 3500 1:33:32 6.7
3-Dec-98 Thursday 3510 0:13:12 3150 4:52:22 10.3
Mean 3700 3340 9.7
Standard Deviation 121 135
Coefficient of Variation 3.26 4.04
Table 2: Summary of Traffic features
The flow immediately prior to the queue lasted for relatively short periods,
consistent with other studies (e.g., Cassidy and Bertini, 1999a; Cassidy and Bertini,
1999b; Bertini, 1999; Bertini and Cassidy, 2002). At this site, however, these flows
24
appeared to be relatively consistent, with a mean value of 3700 vph measured in the two-
lane section at station 8. This may be at odds with other findings (e.g., Cassidy and
Bertini, 1999a; Cassidy and Bertini, 1999b; Bertini, 1999; Bertini and Cassidy, 2002) that
revealed possible instabilities in the higher flow reported prior to bottleneck activation.
The average discharge flow was also consistent from day to day, with a mean value of
3340 vph. This flow was sustained for much longer periods, ranging from 1 hour 30
minutes to almost 5 hours. The drop in flow observed upon queue formation was also
consistent from day to day. On four of the five days, this percentage drop is between 10
and 11 percent, while on December 2, 1998 the percentage difference was between 6 and
7 percent.
4.2.2. All Days: Shock Characteristics
The shock speed was analyzed for the additional 4 days. Table 3 shows a
summary of the shock speed for all days including the first day. The mean shock speeds
ranged between 3 and 4 mph as they traveled upstream from the bottleneck. There were
only slight differences observed between the shock speeds from one section to another.
This would appear to confirm the validity of a linear q-k relation for predicting queue
propagation (e.g., Newell, 1993; Windover, 1998), but confirmation of this is part of
ongoing research. Appendix B shows the shock speeds for the four days.
Mean of 5 days
Stations Distance Mean Travel Time Mean Speed
miles min:sec mph km / h
* 9-8 0.31 0:21 + 53 + 86
7-8 0.31 0:28 + 40 + 65
6-5 0.31 7:03 -3 -4
5-4 0.31 5:23 -3 -6
4-3 0.31 4:09 -4 -7
3-2 0.31 5:46 -3 -5
*9-8 was measured on December 3, 1998
Table 3: Shock Characteristics
25
4.2.3. All Days: Speed observations
Figure 11 shows rescaled V(x,t) for stations 2 and 9 from December 3, 1998. It is
clear that during the congested period the speed drops at both stations, but the reduction
at station 2 is greater since the speed limit is 70 mph while at station 9 the speed limit is
only 50 mph. These speed limit changes could be one of the reasons the speed at
downstream stations does not increase as much as one would expect. It appears that the
vehicles do not accelerate very rapidly after passing the bottleneck location because of
the drop in speed limit at station 9. This aspect is the subject of further analysis.
140000 7:06:01 AM
120000
70 mph - Speed Limit @ Station 2
Station 2
V(x,t) - Vo t, Vo = 45000 mph per hour
100000
80000
60000 6:43:46 AM
40000
20000 50 mph - Speed Limit @ Station 9
Station 9
0
-20000
5:31:12 AM
5:45:36 AM
6:00:00 AM
6:14:24 AM
6:28:48 AM
6:43:12 AM
6:57:36 AM
7:12:00 AM
7:26:24 AM
7:40:48 AM
7:55:12 AM
8:09:36 AM
8:24:00 AM
8:38:24 AM
8:52:48 AM
9:07:12 AM
9:21:36 AM
9:36:00 AM
9:50:24 AM
10:04:48 AM
10:19:12 AM
10:33:36 AM
10:48:00 AM
Time @ Station 9
Figure 16: Fifth Day: Transformed curves at stations 2 and 9
26
5. Conclusions
This study analyzed traffic conditions upstream and downstream of a lane drop
bottleneck. Transformed curves of cumulative vehicles and cumulative time mean speed
versus time were used in this study. These transformed curves facilitated the observation
of traffic conditions around the lane drop bottleneck. The study shows that at locations
where only count and speed data are available, curves of cumulative time mean speed can
be used as tools to verify observations made using curves of cumulative arrival of
vehicles.
It has been shown that a bottleneck arose in the vicinity of a freeway lane drop in
a predictable way. The flow increased above some level, a queue formed and propagated
upstream until a reduction in demand led to queue dissipation later in the morning. The
bottleneck’s location was reproducible from day to day. Also, the flow can drop
substantially following the formation of the upstream queue, followed by discharge flow
exhibiting nearly stationary patterns. The drop in flow is accompanied by a drop in speed.
The higher flow prior to queue formation was sustained for relatively short periods and
the discharge flow that followed prevailed for much longer periods. The values of both
of these flows appeared to be reproducible from day to day. The long run queue discharge
flow can be considered to be the bottleneck capacity since discharge flows were nearly
constant and they were reproducible from day to day.
It is also shown that the changes in the speed limit can be one of the reasons the
speed at downstream stations did not increase as much as one would expect. It is possible
that the vehicles did not accelerate rapidly after passing the bottleneck location because
of the drop in speed limit at the most downstream station. This aspect is subject of further
studies.
As mentioned earlier, the shock velocity observed is slower than reported
elsewhere in the literature. To what extent this is related to drivers’ familiarity with the
27
roadway geometry and/or the speed limit change at station 9 is the subject of ongoing
research.
The observed oscillations arose within the queue at the stations upstream of the
bottleneck. Oscillations were not observed at locations downstream of the head of the
queue. It was observed that the oscillations displayed a constant speed in the opposite
direction of the flow.
This research is only an initial step toward understanding bottleneck behavior in
relation to lane drops. Thus, further analyses need to be conducted at this site in London
as well as at other lane drop sites in the United States.
28
References
Bertini, R.L. (1999). Time-Dependent Traffic Flow Features at a Freeway Bottleneck
Downstream of a Merge. Ph.D. Thesis, University of California, Berkeley, U.S.A.
Bertini, R.L. and Cassidy, M.J. (2002). Some Observed Queue Discharge Features at a
Freeway Bottleneck Downstream of a Merge. Transportation Research, Vol. 36A, pp.
683-697.
Cassidy, M.J. and Bertini, R.L. (1999a). Some Traffic Features at Freeway Bottlenecks.
Transportation Research, Vol. 33B, pp. 25-42.
Cassidy, M.J. and Bertini, R.L. (1999b). Observations at a Freeway Bottleneck.
Proceedings of the Fourteenth International Symposium on Transportation and Traffic
Theory, Jerusalem, Israel, pp. 107-124.
Cassidy, M.J. and Mauch, M. (2001). An Observed Feature of Long Freeway Traffic
Queues. Transportation Research Vol. 35A, pp. 149-162.
Daganzo, C.F. (1997). Fundamentals of Transportation and Traffic Operations. Elsevier,
New York.
Mauch, M. and Cassidy, M.J. (2002). Freeway Traffic Oscillations: Observations and
Predictions. Proceedings of the Fifteenth International Symposium on Transportation and
Traffic Theory, Adelaide, Australia, pp. 653-673.
Muñoz, J.C. and Daganzo, C.F. (2002). The Bottleneck Mechanism of a Freeway
Diverge. Transportation Research Part A, Vol. 36(6), pp. 483-505.
Newell, G.F. (1982). Applications of queueing theory. Chapman and Hall, New York.
Newell, G.F. (1995). Theory of Highway Traffic Flow 1945-1965. Course Notes UCB-
ITS-CN-95-1. Institute of Transportation Studies, University of California at Berkeley,
U.S.A.
Newell, G.F. (1993). A simplified theory of kinematic waves in highway traffic; I:
General Theory, II: Queueing at freeway bottlenecks, III: Multi-destination flows.
Transportation Research, 27B (4), 281-313.
Rees, T., White, J. and Quick, J. (2000). Monitoring of the bus lane: The first year.
Highway Agency, TRL Limited, UK.
Windover, J.R. (1998). Empirical Studies of the Dynamic Features of Freeway Traffic.
Ph.D. Thesis, University of California, Berkeley, U.S.A
29
Appendix A - Weather conditions
Weather conditions are described in Table 4. Temperatures and a description of
the weather are shown for every day of study. This data comes from a meteorological
center for Heathrow that is very close to the site of study. The data was provided by
George Hood, Enquiry Officer at a Meteorological Office in London, UK.
Temperature
Date Day Description
Low High
°C
16-Nov-98 Monday 1.3 5.3 Little cloudy
18-Nov-98 Wednesday 1.0 6.8 Haze
30-Nov-98 Monday 4.1 5.1 Haze
2-Dec-98 Wednesday 3.5 4.2 Cloudy and haze
3-Dec-98 Thursday 1.7 2.5 Showers and drizzle
Table 4: Weather conditions
30
Appendix B - Summary of Shock Characteristics
The following tables show shock characteristics such as travel time and speed
between the stations for every day studied. It was observed that the average speed
between station 6 and 2 ranged between 2 and 5 mph. In addition, it was clear that a
downstream moving expansion wave of lower flow and higher speed reached stations 7
to 9 with speeds between 29 to 59 mph.
16-Nov-98
Travel Time Distance Speed
Hour Minutes Seconds Time (hours) miles mph km/hr
7-8 0 0 24 0.01 0.31 +47 +75
6-5 0 3 40 0.06 0.31 -5 -8
5-4 0 5 19 0.09 0.31 -4 -6
4-3 0 3 58 0.07 0.31 -5 -8
3-2 0 2 21 0.04 0.31 -8 - 13
6-2 0 14 18 0.24 1.24 -5 -8
Table 5: First Day: Shock Characteristics
November 18, 1998
Travel Time Distance Speed
Hour Minutes Seconds Time (hours) miles mph km/hr
7-8 0 0 32 0.01 0.31 + 35 + 56
6-5 0 4 48 0.08 0.31 -4 -6
5-4 0 5 23 0.09 0.31 -3 -6
4-3 0 3 33 0.06 0.31 -5 -8
3-2 0 10 21 0.17 0.31 -2 -3
6-2 0 24 5 0.40 1.24 -3 -5
Table 6: Second Day: Shock Characteristics
31
November 30, 1998
Travel Time Distance Speed
Hour Minutes Seconds Time (hours) miles mph km/hr
7-8 0 0 39 0.01 0.31 + 29 + 46
6-5 0 9 16 0.15 0.31 -2 -3
5-4 0 4 41 0.08 0.31 -4 -6
4-3 0 1 33 0.03 0.31 - 12 - 19
3-2 0 8 1 0.13 0.31 -2 -4
6-2 0 23 31 0.39 1.24 -3 -5
Table 7: Third Day: Shock Characteristics
December 2, 1998
Travel Time Distance Speed
Hour Minutes Seconds Time (hours) miles mph km/hr
7-8 0 0 19 0.01 0.31 + 59 + 95
6-5 0 12 19 0.21 0.31 -2 -2
5-4 0 6 0 0.10 0.31 -3 -5
4-3 0 6 39 0.11 0.31 -3 -5
3-2 0 5 9 0.09 0.31 -4 -6
6-2 0 30 7 0.50 1.24 -2 -4
Table 8: Fourth Day: Shock Characteristics
December 3, 1998
Travel Time Distance Speed
Hour Minutes Seconds Time (hours) miles mph km/hr
8-9 0 0 21 0.01 0.31 + 53 + 86
7-8 0 0 25 0.01 0.31 + 45 + 72
6-5 0 5 14 0.09 0.31 -4 -6
5-4 0 5 32 0.09 0.31 -3 -5
4-3 0 5 2 0.08 0.31 -4 -6
3-2 0 2 57 0.05 0.31 -6 - 10
6-2 0 18 45 0.31 1.24 -4 -6
Table 9: Fifth Day: Shock Characteristics
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