A Queuing Analysis of Freeway Bottleneck Formation and Shockwave Propagation

Submitted by

SHANTANU DAS

IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

IN CIVIL ENGINEERING.

i

TABLE OF CONTENTS ACKNOWLEDGMENT ii LIST OF FIGURES iii LIST OF TABLES iv LIST OF SYMBOLS v CHAPTER 1

• 1.1 INTRODUCTION 1 CHAPTER 2

• 2.1 LITERATURE REVIEW 4 CHAPTER 3

• DATA 9 o 3.1 Data consistency 13 o 3.2 Data correction 14

3.21 Finding and calibrating ‘Truth Couplet’ 14 3.22 Section flow adjustment 16

CHAPTER 4 • THEORY AND METHODOLOGY 22

o 4.1 Queuing theory and it is application 25 o 4.2 Density Calculation (I) 28 o 4.3 Active Bottleneck Identification 30 o 4.4 Tracking bottleneck formation and propagation 38 o 4.5 Density Calculation (II) 40

CHAPTER 5

• RESULTS 45 o 5.1 Bottleneck formation and propagation 45 o 5.2 Shockwave tracking 47 o 5.3 Density calculations 51 o 5.4 Active bottleneck identification 53

CHAPTER 6

• CONCLUSIONS AND SUMMARY 63 REFERENCE 65

ii

APPENDIX 67 ACKNOWLEDGEMENT

This thesis is dedicated to my parents who taught me integrity, sincerity, and

responsibility and to my brothers who shared my deepest secrets, many punishments and

my best moments. I owe a lot to all my friends at the University of Minnesota who have

shared my joys and sorrows alike and helped me throughout. They include Kaustubh,

Sudarshan, Satya, Vamsi, Prasoon, Lei, Akash, Rama, Bhanu, Sujay, Adarsh, Ravi, Jiji

and Zou.

This study could not have been possible without the timely advice and tips from

Amaresh who has had a profound effect on what I am. All my undergrad friends played

key roles by encouraging me and showing me the right way all the time and they deserve

mention here. I thank all of my high-school teachers and friends who showed me the

values of courage and discipline early on.

Words cannot sum up the enormous support provided by my advisor Dr. David

Levinson during the course of my graduate study. I cannot imagine how hard my life

would have been without his constant encouragement and patient advice. He molded my

likes and dislikes in a big way. I offer my sincerest gratitude to him.

iii

LIST OF FIGURES 1.1 Freeway Inductive Loop Detector at work 2 3.1 Study locations 10 3.2 Typical flow – density relationship 18 3.3 Typical regression result of upstream counts on downstream counts 12 3.4 A typical freeway section consisting of upstream and downstream detector stations 14 3.5 Consecutive freeway sections 16 3.6 Flowchart depicting implementation of computer program for correcting flows. 18 4.1 General representation of phases of traffic flow 23 4.2 Variation of flow over density for detector 1906 (Near Excelsior Boulevard) 24 4.3 Queuing Diagram 27 4.4 Active bottleneck identification typology 32 4.5 Traffic phase diagram: relationship with cases 36 4.6 Queuing diagram – relationships with phases and cases. 37 4.7 Density calculation for individual detectors on freeway sections 41 5.1 Cumulative flow differences across time on TH 169 (November 7th, 2000) 46 5.2 Afternoon peak period, cumulative flow differences on TH 169 47 5.3 Transformed cumulative curves for successive detector stations on TH 169 48 5.4 Afternoon queuing on TH 169, transformed cumulative flow curves 49 5.5(a) Downstream queuing taking place on TH 169 50 5.5(b) Upstream queuing taking place on TH 169 50 5.6 Sample normalized density curve on I 94 East Bound 52 5.7 Flow-density relationship for detector 52 (Lowry Avenue), on I 94 EB on November 3rd, 2000 55 5.8 Flow and speed varying by density on detector 83 on November 2nd 57 2000 5.9 Flow density relationship for detector 688 (49th/53rd Avenue on I 94) on 61 November 3rd, 2000

iv

LIST OF TABLES 3.1 Regression result of flow over upstream detector1915 on the flow over downstream detector 1922 12 3.2 A sample correction of detector counts 20 3.3 Error removal example 21 5.1 Regression results (speed over density) for detector 52 on I 94 54 5.2 Regression results (flow over density, for speeds below free-flow speed) for detector 83, near 42nd Avenue on I 94 on November 2nd, 2000 56 5.3 Cases for detector 600 (Near Dupont Avenue) on I 94 for November 1st, 2000 58 5.4 Cases for detector 83 (Dowling Avenue) on I 94 on November 1st, 2000 59 5.5 Cases for detector 666 (Shingle Creek parkway) on I 94 on November 2, 2000 60 5.6 Cases for individual sections on I 94 for the period from November 1st to November 6th, 2000 (Summary Table) 62

v

LIST OF SYMBOLS

�

Ca = Correction needed for detector ‘a’

�

CA = Correction needed for the entire detector station, ‘A’

�

Cat = Correction needed in detector ‘a’ count at time ‘t’

�

Cfin = Correction required for input freeway count

�

Cfout= Correction required for output freeway count

�

Cron

= Correction required for on-ramp count

�

Croff= Correction required for off-ramp count

(Ka)t = Density over a detector ‘a’ at time ‘t’

�

K = Density in cars per lane mile Ka = Density over a detector ‘a’ Kn = Normalized density L = Length of a freeway section L1 = Length of a section. ‘L1’ L2 = Length of a section. ‘L2’

�

Leff = Effective Length of vehicles N = Number of cars stored in a freeway section after time ‘t’ Nst = Number of cars stored in any section after a time ‘t’

�

O = Occupancy as a percentage of time a detector is occupied

�

QA = Cumulative flow over a detector stationn ‘A’ which contains the detector ‘a’

�

Qa = Cumulative flow over any detector ‘a’

�

Qin = Flow entering a freeway section

�

Qfin = Input flow on a freeway section

�

Qout = Flow exiting a freeway section

�

Qfout= Output flow from a freeway section

QL1 = Flow over the length L1 after time ‘t’

�

Qroff=Flow from an off-ramp ‘j’ out of the freeway section

�

Qron

=Flow from an on-ramp ‘i’ onto the freeway section

�

Qat =Flow over detector ‘a’ at time ‘t’ station ‘A’

ra = Ratio of cumulative flow over detector ‘a’ over cumulative flows on an entire Va = Speed over a detector ‘a’

1

CHAPTER 1

INTRODUCTION

A modified approach to explaining the nature of traffic flow on freeways is

described in this research. It has its beginnings in the observations of actual traffic data,

which are not easily and convincingly explained by the conventional traffic flow theories.

For more than half a century, people have studied flow, density and speed. Yet it appears

no theory explains everything we see on freeways (Science News Online, July, 1999).

This is expected because of the stochastic nature of traffic flows. The nature of traffic has

responded to years of changes in vehicle parameters, highway design, land use patterns

and socio-economic conditions of people accessing the freeways on a regular basis. In

part, this explains why many theories that were developed earlier seem to be at odds with

the traffic data that is collected now.

The present analysis stems from our observation of the data collected by the

Traffic Management Center (TMC) of the Minnesota Department of Transportation over

major highways crossing the Twin Cities area.

This research applies queuing theory to traffic data to determine the location of

bottlenecks and generation and propagation of shockwaves. Using analytical and

graphical techniques, it proceeds to explain how these potential bottleneck locations and

shockwaves identified.

The work in this paper has its roots in the Ramp Meter shut-off experiment that

was conducted in Twin Cities during the fall of 2000 (Minnesota Department of

Transportation, 2000). Along with determining the effectiveness of ramp metering in

controlling the freeway traffic and maintaining it at an optimal level, this experiment

2

provided an opportunity to study traffic flow characteristics under both metered and un-

metered conditions. This was made possible by the considerable amount of data collected

both during the period preceding the experiment and during the experiment. The data

primarily consisted of freeway flows and occupancies during successive 5 minute

periods.

Most conventional approaches to studying the relationships between traffic flow

characteristics use loop detector data. The loop detectors are typically 6 by 6 ft inductive

loops of electric wires. When a vehicle passes or stops over such a detector, loop

inductance decreases and that induces a higher oscillation frequency, which then invokes

a pulse indicating the presence of a vehicle (California PATH, 2000). A typical loop

detector at work is shown in Figure 1.1:

Figure 1.1: Freeway Inductive Loop Detector (ILDs) on work (California PATH)

3

The Inductive Loop Detectors (ILD)s typically measure flow (the number of

vehicles that pass it in some time period) and occupancy (the percentage of time for

which the ILD is occupied in that time period). But the data provided by such ILDs are

often fraught with errors. ILDs under-count or over-count under different freeway traffic

conditions. Also the accuracy and consistency of detector data depend strongly on their

installation and calibration procedures. A loop detector with a percentage accuracy within

5% is considered a ‘good’ one (Minnesota Department of Transportation, May 2002). For

a freeway with a daily output in the range of tens of thousands of vehicles, the 5% error

can amount to hundreds or thousands of vehicles per day. Also an underlying assumption

is that effective vehicle lengths are uniform. This research aims to present a simple

queuing analysis of freeway traffic that does not rely on vehicle occupancy data or

effective vehicle lengths.

The next chapter gives a brief background of the studies that have been conducted

and which were found to be relevant to this research. Chapter 3 describes the data sets

and data formatting that were needed to apply to this research. The description of the

underlying theory and the methodology involved come next in Chapter 4. Chapter 5

describes the results and Chapter 6 summarizes the findings of the study and provides

conclusions.

4

CHAPTER 2

LITERATURE REVIEW

Since the first attempts to apply probability theory to describe traffic flow

relationships pioneered by Greenshields (1935), transportation researchers have put

forward various theories that explain the relationship between Flow, Density and Speed.

Over the last few decades, traffic flow theory has continually progressed with newer

studies engendering insights into the relationship between traffic flow parameters. With

increasing highway congestion, renewed focus has been shed on the role of traffic flow

theory.

Among the earliest investigators, Greenshields (1935) proposed a parabolic

relationship between traffic flow and density and a linear relationship between traffic

speed and density. Other investigators like Greenberg (1959), Drew (1968) and

Underwood (1961) proposed various models, which took into account the logarithms and

exponentials of the flow parameters. Edie (1961) combined logarithmic and exponential

models to explain a discontinuity in data near the critical densities. Recently however, the

focus has started to shift toward explaining the traffic flow parameters taking into

account the detection and tracking the bottlenecks and shockwaves on the freeway.

Daganzo et al. (1999) analyzed the traffic flow patterns of individual freeway lanes

accounting for the driver characteristics and the ‘reverse-lambda’ pattern seen in the

flow-density scatter plots. They investigated the behavior of car platoons during and after

queuing.

Lovell and Levine (2001) focused on ramp meter connected freeways and tried to

explain the basic structure of the freeway access control problem. Taking the stochastic

5

nature of travel demand patterns, they examined the applicability of ‘fluid’

approximations to explain the dynamic nature of entrance ramp waiting times and then

optimizing those waiting times. They analyzed the freeway access problem from the

points of view of various ramp control strategies available.

Zhang et al. (2001) examined the continuum model of traffic flow theory basing

their work on the Lighthill-Whitham-Richards (LWR) fluid model of traffic flow. They

also considered the effect of different vehicle classes and lane types in adapting the LWR

model in order to model traffic flows. In this study, they conclude that the highly non-

linear behavior of traffic flow cannot be explained by a pure “fluid approximation”,

because LWR models can be unstable and this instability leads to what they call “violent

phase transitions”.

The work of Jia, Chen, Coifman and Varaiya (2001) was significant in the light of

strategies used to get speeds from loop detector data, (loop detectors do not measure

speeds directly, they only give an estimate of it using g-factors which relate flow,

occupancy and speed for each loop). They present an alternative method to estimate

vehicle speeds using adaptive PeMS algorithms, which calculate the g-factors in real

time.

Cassidy et al. (2000) discussed formation of a bottleneck purely due to spillover

on the freeway segment’s off-ramp which blocked the exit lane. In this paper, Cassidy

demonstrates an intuitive method of tracking bottleneck formation and propagation in

which instead of plotting cumulative number of vehicles that went over a detector,

subtracted a successively incrementing ‘base flow’ for each time interval from the

cumulative flow count. In addition, they map out the cumulative occupancy on the

6

detectors across time as the total time that the particular detector was occupied until the

time of measurement. Superimposing the transformed cumulative flow curve with the

cumulative occupancy curve indicates the arrival and sustenance of congestion over a

certain stretch of the freeway. Their use of cumulative flow curves while following the

propagation of bottlenecks upstream finds enormous use in this thesis as will be

described at a later stage. They considered a 1.4-mile stretch of freeway between the

entrance ramp and the downstream exit ramp to observe the formation of a bottleneck in

the intermediate freeway stretch. Keeping detailed track of the length of the queues

formed on the exit ramp over time, they observed the formation of the freeway bottleneck

which they then validated using videotapes of the same region.

Cassidy and Bertini (2000), applied similar techniques while studying the

upstream and downstream bottlenecks on a Toronto freeway. While they used the

‘transformed cumulative curves’, they further shifted them toward the right by the

average free flow time between successive detector stations, so that the vertical

separation between the curves gave the number of vehicles stored between successive

detector stations. Also, a widening gap between such ‘transformed cumulative curves’

(input and output) indicated bottleneck formation and their closing-in indicated

bottleneck dissipation. An important finding of this paper was that the analysis produced

results that indicated that the bottlenecks formed at the same location even on different

days. Similar traffic flow patterns were exhibited over different days. However, this

paper could not successfully identify precise causes of flow reductions as a result of

queuing.

7

Zhang and Kim (2000) present a new car-following model that attempts to

describe multiphase vehicular traffic flow. They conclude that capacity drop and traffic

hysteresis are defining features of multi-phase vehicular traffic flow. Their model

proposes that drivers adopt a speed that depends on spacing from the leading car and their

reaction times. Considering different functional forms of response times, their car

following theory claims to model traffic hysteresis and capacity drop, in separate cases,

while phase transitions occur. They validate their claim by using simulations.

Daganzo et al. (1999) made some observations on long queues that form on a two

lane bi-directional highway. The focus of this paper was on the stability of flow

oscillations (stop-and-go traffic) and they concluded that traffic is more stable in terms of

flow oscillations, while vehicles are closer to the bottleneck and the variability increases

as one moves farther from the bottleneck. Apart from the pure analysis using transformed

cumulative curves, this paper also details the possible sources of error and the correction

strategies. The involvement of a traffic signal on the study stretch however diminishes the

importance of this in the context of describing traffic flow characteristics on freeways.

Lovell, Daganzo and Lawson (1997) showed a simple modification of an input-

output queuing diagram in order to show the time and distance that the vehicles piled

upstream of the bottleneck have to spend before exiting the bottleneck. They examine the

question under two situations, constant departure rate and departure rate that changes

once. The authors distinguish between ‘delay’ and the ‘time spent in the queue’.

Barbour and Fricker (1990), discuss strategies to balance link counts on nodes on

a small network. They propose solutions that include node-balancing algorithms and

mathematical programming approaches to balance the incoming and outgoing flows at

8

each node in a network. The methods include strategies like trial and error, placing link

weights on different paths and a minimax variation where flows are changed on links that

have been relatively unchanged in the balancing process and avoiding links that have had

relatively large changes. Linear programming approaches are detailed. In this research a

unique flow balancing algorithm is deployed that is described in a later chapter.

9

CHAPTER 3

DATA

The data used consisted primarily of 5-minute flows and occupancies collected by

the loop detectors on the freeways. This data was available for both the periods leading to

and during the ramp meter shut down experiment. A ‘Python’ script (written by my

research colleague, Lei Zhang) was used to extract the 5-minute flows and occupancies

for any specified date. The data consists of flows and occupancies on the following

freeway sections:

• Trunk Highway 169 – from Valley View Road to Trunk Highway 55

• Interstate 35W – from Lyndale Avenue to 38th Street

• Interstate 94 – from County Road 152 to Plymouth Avenue and

• Interstate 94 – from Riverside Avenue to Dale Avenue going into St. Paul

All the data used for analysis in this paper was collected between November 1st

and November 8th, 2000, part of the period during which the ramp meters over the Twin

Cities freeway network were shut-off. The study locations are shown in Figure 3.1.

Another important data that was used were the lengths of the various freeway

sections. In the absence of a recorded database of freeway lengths on TH 169, most of

these lengths were measured from the detector maps and using online resources.

10

Figure 3.1: Study locations (shown by dark bands)

The first step of analysis consisted of pure visual inspection of the flows and

densities (derived from the occupancies) from their scatter plots. The scatter plots show

169

35W

I 94

I 94

11

the noisy and unpredictable relationships of flow and density after the critical density is

reached. This approach suggests, to some extent, the regions of the curves on which we

should concentrate more on. This is illustrated in the following Figure:

Figure 3.2: Typical relationship between flows and density (detector 1915 near

36th Street on TH 169 on November 8th, 2000).

The next step regresses the upstream detector flows on the downstream detector

flows. This was done to see the extent to which upstream detector flows could have been

influenced by the flow occurring immediately downstream. A sample regression result is

shown next for illustration.

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120

Density in cars per lane mile

5 minute flows

(vehicles/5mins)

12

Figure 3.3: A typical regression result (upstream detector 1915 count, on

downstream detector 1922, count)

Dependent Variable: Upstream_flow Variable Hypothesis Coefficient P > |Z| Constant 15.44 0

Downstream_flow +S 0.862 0 Significance based on 90% confidence interval Number of observations = 200 R-squared = 0.796

Table 3.1: Regression result of flow over upstream detector 1915 on the flow over

downstream detector 1922 (both are in the same lane)

1922

62 182

69

174

Downstream detector (1922) counts

Upstream detector 1915 counts (5 min flows)

65 80 95 110 125 140 155 170 185

175 150 125 100 75

13

From the high R2 value it is easy to conclude that there exists a definite

relationship between the flows occurring over the successive detectors. But it is not

enough to be able to just say that. We need to investigate further, which is where the data

that are being used here assumes a significant role. This necessity becomes particularly

pressing in the light of the noisy relationship between traffic flow and density when

vehicles are allowed to move without restrictions imposed by ramp metering.

3.1 Data Consistency:

We need to ensure that we are dealing with good, reliable data. The methodology

involved in ensuring balanced flows consists of comparing the flows that entered and

exited the freeway during some long time period (here 24 hours). The first step consists

of getting the data for the days concerned. The data is obtained for a carefully selected

freeway section, which affords working detectors and which is long enough to be able to

offer a possibility of observing bottleneck formation and their propagation. Then we

detail the detector stations that form each section’s starting and ending points. Once we

have the detector stations, we add the flow data. Comparison between the cumulative

counts of all vehicles that went past the starting and ending points of the freeway section

over a long period of time, which is 24 hours in the analyses carried out here, gives the

extent of detector error. To minimize the occurrence of existing vehicles on the roads, we

start out at 3:30 am when the freeway sections would be closest to being empty. While

controlling for the on-and off-ramps over a freeway section, we check if flow is

conserved.

14

Over a long period of time,

�

Qfin+! Qr

on

i

I

! = Qfout+! Qroff

j

J

!

where the Left Hand Side (LHS) is the sum of the flows occurring over all the ‘I’ on-

ramps and the upstream freeway section and the Right hand Side (RHS) is the sum of the

flows occurring over all the ‘J’ off-ramps and the downstream freeway section. This is

illustrated by Figure 3.4:

Figure 3.4: A typical freeway section consisting of upstream and downstream detector

stations

Unfortunately, the measured flows between the various detector stations are not in

balance, probably due to detector errors.

3.2 Data Correction

3.2.1 Finding and calibrating the ‘Truth Couplet’

Our next step is to clean the data such that the flows across various sections of

the freeway balance over a long period of time (24 hours). The logic that was arrived at

15

was: There must be some section on the freeway where the differences between inflows

and outflows were at a minimum. This region would have around it a pair of detector

stations that were defective to the least extent or not defective at all. Such a pair would be

called a ‘Truth Couplet’. (Note that the detector station on the freeway is mostly a

combination of two or more individual detectors.)

What follows next is a calibration of the truth couplet it itself wherein minor

errors are removed by allocating the differences into individual detector stations involved

in the truth couplet.

Mathematically, if the input flows (freeway + on-ramp) differs from the output flow

(freeway + off-ramp) then, correction for the input freeway count (

�

Cfin) is given by,

�

Cfin=

Qfin

Qfin+ Qron

+ Qfout+ Qroff

* diff

where

diff = (Qfin+ Qron

) ! (Qfout+ Qroff

){ }

Similarly, Corrections for the on-ramp count (

�

Cron

), output freeway count (

�

Cfout) and

off-ramp count (

�

Croff) would be respectively given by:

�

Cron=

Qron

Qfin+ Qron

+ Qfout+ Qroff

* diff

C fout=

Qfout

Qfin+ Qron

+ Qfout+ Qroff

* diff

Croff=

Qroff

Qfin+ Qron

+ Qfout+ Qroff

* diff

16

Now all the detector stations involved in the truth couplet would be regarded

‘correct’ and so their final counts would be assumed to be as close to accurate as possible

3.2.2 Section flow adjustment

Once the truth couplet is identified, the adjustments in the total daily flows are

done over successive detector stations both upstream and downstream of the truth

couplet. The adjoining detector stations were calibrated based on the value of the

common detector station count (from the truth couplet) involved in the next section. For

example, in Figure 3.5, the ovals show the truth couplet and the letters show the detector

stations. A, B, C and D are each true and so when we go downstream, ‘G+F-E’ should

equal D. This equality does not hold if there are errors in the final total counts on G, F

and E. These errors are eliminated by allocating some portion of the error to each of the

detector stations G, F and E weighted by their ratios. If for example the ratio of the total

counts on G, F and E are 5:2:1 then G will share 5/8 of the error, F will share 2/8 of the

error and E will get the rest 1/8 of the error. (Appendix A.1 shows an example.)

Figure 3.5: Consecutive freeway sections

17

Mathematically, the corrections for each of E, F and G can be represented as:

�

CG =QG

QG + QE + QF

* diff

CE =QE

QG + QE + QF

* diff

CF =QF

QG + QE + QF

* diff

where,

diff = QD ! (QG + QE !QF )

We either add or subtract the correction factor depending on the need to increase

or decrease the counts.

This process is continued farther downstream until we reach the end of the entire

freeway section considered. The same procedure is also carried out in the upstream

direction. The procedure described above is shown in the flowchart in Figure 3.6.

A computer program was coded in C++ to automate the above analysis.

18

Figure 3.6: Flowchart depicting the computer program

Yes

No

Correct the flows over the freeway and ramps based on truth couplet counts by

allocating the errors to individual detector stations.

Check for any more section up or downstream Give the output in

terms of corrected freeway and ramp

counts.

End program

Start

Enter all the freeway, on-ramp and off-ramp counts.

Find the differences between incoming flow and outgoing flow for each freeway section.

Designate the section with the least difference between incoming and

outgoing flows as the ‘’truth couplet’ and calibrate it to remove the smallest error.

19

Allocation to each lane:

After the corrected flows are obtained from the program, the differences between

the original counts and the new corrected counts are allocated to the individual detectors

again according to the original flow proportions. This is because when we get the new

freeway counts, it is the aggregated count of all the detectors involved in that section of

the freeway and when we have to distribute the ‘adjustments’ we have to do it such that

the detector (in the station) on the maximum flow lane gets the maximum amount of

adjustment content. Similarly the detector that was in the least flow lane needs to be

changed by the least amount.

Mathematically,

�

Ca

=Qa

QA

*CA

where

Ca = Correction needed to apply on detector, ‘a’

Qa= Cumulative (24 hour) flow over detector ‘a’

QA= Cumulative (24 hour) flow over detector station, ‘A’, which includes all ‘a’

detectors

CA= Correction needed for detector station, ‘A’

A question arises whether the original flow ratios are correct in the first place

given that the original counts may be fraught with errors that we are trying to remove.

However when considered the whole day or a longer time period, the errors themselves

20

are observed to be only a small percentage of the total flow and the probability that the

ratios may have been erroneous in the first place is small. An example of this process of

allocating the corrections is shown in Table 3.2.

Table 3.2: A sample correction of detector counts

Allocation over time:

Moreover, after adjusting for the total 24-hour flow for each detector, we have to

also adjust for every 5 minute count. This is again done based on the flow. The

corrections for each detector were based on individual detector shares in the original

detector station count. Now we need to ensure that that correction is applied most when

the errors are maximum in the day and vice versa. In other words, the correction is

allocated on a percentage basis. If the percentage correction needed for the 24 hour

period (for an individual detector) is say 1%, then for each 5 minute interval, we change

(add or subtract) the count by 1% of the original count in that 5 minute period.

Mathematically,

�

Cat

=Qat

Qa

*Ca

where,

Cat = Correction needed in detector ‘a’ count at time ‘t’

21

Ca= Total correction required on detector ‘a’ over a 24 hour period.

Qa= Cumulative flow over detector ‘a’ over a 24 hour period.

Qat= Flow over detector ‘a’ at time ‘t’.

All this is done to minimize all possible sources of errors and end up with the

most correct data for the main analysis.

Quantification of error reduction:

Applying the above mentioned techniques we reduce errors in detector counts.

This is illustrated by Table 3.3 where six freeway sections are laid out and the differences

in daily and mean hourly flows are shown.

Table 3.3: Correction of flows on individual freeway sections

We observe that the cumulative daily differences between flows coming into any freeway

section and flows going out of the same freeway section almost vanishes. There are also

large percentage reductions in mean hourly flow differences between input and output

flows for each freeway section.

Interstate 94

Section between Before After Before After

CR 152 and Xerxes Avenue 369 1 21 2.12 89.90

Xerxes Avenue and Shingle Creek Parkway 319 1 19.63 3.30 83.16

Shingle Creek Parkway and Dupont Avenue 35 2 5.75 4.05 29.50

I 694 and 57th Avenue 38 1 9.25 6.71 27.47

57th Avenue and 49th Avenue 52 1 4.75 1.41 70.25

49th Avenue and 42nd Avenue 30 1 6 4.08 31.97

Note: Daily differences do not reduce to zero due to rounding error

Daily Differences Mean hourly differences Percentage

reduction

22

CHAPTER 4

THEORY AND METHODOLOGY

The three important characteristics for highway traffic operational analyses are

flow, speed and density. It is hypothesized that if freeway traffic is allowed to behave

without any restrictions imposed by ramp meters, it displays four phases in the flow-

density curve.

• Phase 1 is the un-congested phase when there is no influence of the increasing

density on the speeds of the vehicles. The level of service remains high enough to

ensure that the speed does not drop with the introduction of newer vehicles onto

the freeway.

• Phase 2 starts at the point of critical density. In phase 2, the freeway cannot

sustain the speed with injection of newer vehicles into the traffic stream. The

density increases while speed falls, maintaining the flow.

• Phase 3 is when we observe decreased speed and decreased flows. Very low

speeds cause the flow to drop at an active bottleneck or a downstream bottleneck

may be constraining the flow.

• Phase 4 is the recovery phase. With fewer cars entering the traffic stream and the

output flow increasing at the same time, we are led to this phase of recovery.

During this phase, the density of traffic starts decreasing and speed starts

23

increasing at a faster pace leading to increasing flow. This is the period when the

traffic flow is recovering and trying to reach the initial speeds. We start to see

‘Traffic Hysteresis’ taking place. The hysteresis occurs because the rate at which

the flow drops during the phase 3 is not the same as the rate at which the flow

recovers during the phase 4, as apparently acceleration rate is lower than the

deceleration rate. A general representation of the freeway traffic flow is shown in

Figure 4.. Variations may occur from station to station:

Figure 4.1: A general representation of the phases of traffic flow

The circled region, ‘A’ is typically where we start observing ‘freeway breakdown’. In

other words, this region occurs when flow exceeds some critical capacity and there is a

drop in speed. This speed drop occurs with vehicles taking more and more time to cover

the same distance as they enter the traffic stream. In the circled region, ‘B’, flow drop

occurs due to very low speeds. The phenomenon is particularly evidenced by the

formation of queues upstream of where the breakdown occurs and a low discharge rate of

24

vehicles due to sustained low speeds, as the vehicles only start to gradually accelerate

soon after.

An example of such a pattern is shown for detector 1906 (near Excelsior Boulevard), on

trunk highway 169, for November 7th, 2000 in Figure 4.2, with lines for each phase fitted

onto the graph:

Figure 4.2: Variation of flow over density for detector 1906 (Near Excelsior Boulevard)

It is hypothesized that upstream detector flows are governed to some extent by

what flows occur downstream. This is particularly true during peak hours. In other words,

we may be able to see queuing and back propagation taking place, which influence the

Flow versus density

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120 Density in cars per lane mile

5 minute flows

25

following drivers to behave accordingly. And that’s where the application of Queuing

Theory becomes significant.

4.1 Queuing Theory and its application

One of the most visible applications of queuing theory has been in traffic flow.

Queuing theory analyzes the lines that form while the servers serve the waiting

customers. Queuing analysis highly depends on the queue characteristics. This includes

the ways in which the cars arrive in order to form a queue and the way in which they are

cleared from the queue and allowed to move forward. Queues can be classified based on

some system characteristics:

A/B/C:DISC

Where, A Arrival pattern characteristics which include the average rate of arrivals

and the statistical distribution of the inter-arrival times.

B Service time facility characteristics that include the average rates of

service and the distribution of the inter-service times.

C The number of servers

DISC The queue discipline characteristics i.e., the way in which the next

customer is selected. For example: FIFO (first in first out), LIFO (Last

in first out), random or the most profitable customer first.

A and B are further letter coded as:

M : exponentially distributed

D: deterministic or constant arrivals or departures.

G: general distribution of service times.

26

Various equations (for flow, waiting time, number of cars in the queue, average

waiting time for an arrival that wait is etc.) can be arrived at if we know the queue

characteristics.

One of the problems with applying queuing theory to freeway traffic flow is that

there is no ‘service time’ on the freeway unless there is formation of an active bottleneck

that lets the queue dissipate in some distinguishable and regular fashion. It is often

assumed that the arrival patterns are normally Poisson distributed in cases where

bottlenecks form with traffic arriving randomly at the tail of the queue and service rates

holding constant.

To conduct the queuing analysis, we need a way of predicting the occurrence and

location of bottlenecks. This requires consistent observation of repeated bottlenecks and

shockwaves forming over space and time on the freeway.

Conventional traffic flow analyses have used flow and occupancy data that are

provided by freeway detectors. From measured occupancies, applying the following

empirical formula gives densities:

�

K =O

Leff

where

K = Density

O = Occupancy

Leff = Effective length of the average vehicle, that is the length of the vehicle from the

front to the end plus the length of the detector.

27

Thus the calculation of densities and their resulting role in traffic flow analyses

requires assumption of vehicle lengths and reliance on occupancy data. There is little

scope to check for errors in the occupancy data. However, flow counts given by

detectors, which can be calibrated using the theory of conservation of flows as has been

shown earlier. Moreover, occupancy is a point measurement and cars move over space. In

such a scenario, queuing theory seems more appropriate and applicable to analyze traffic

flow characteristics. The basic queuing diagram can be represented as shown in Figure

4.3:

Fig 4.3: Queuing Diagram

The upper curve on the queuing diagram represents the arrival curve and the

lower curve (the straight line) represents the departure curve. Just before the onset of

queuing, both the arrival and departure curves coincide. With arrivals exceeding

28

departures, the upper curve starts rising, while departure rate remains constant and shows

up as the straight line. After some time however, arrival rate drops below departure rate

and the two curves close in and finally merge to become one again. This is when the

queue completely dissipates. The area between the two curves represents the total delay

of all the cars that were queued up. At any point of time, ‘j’ in the Figure, there would be

as many cars queued up as shown by the difference in the vertical distance between the

two curves at that point of time. For any ‘i’th car in the Figure, the total waiting time in

the queue is the horizontal distance between corresponding points of the two graphs. And

there may definitely be more than just one queue regime, unlike the one shown here.

4.2 Density Calculation (I)

The principal advantage of applying Queuing Theory to traffic flow analyses is

that we don’t have to rely on occupancy data obtained from individual detectors. Rather,

we can find the densities directly from the corrected flows. This becomes advantageous

as density is a space measurement and cars move over space, whereas occupancy is a

point measurement. As cars pass over freeway sections, we keep a count of all the cars

that passed the start of the section and the end of the section. At any particular point of

time, ‘t’ we find the number of cars that are present in any particular freeway section by

subtracting the cumulative number of cars that have passed the end of the freeway section

from the cumulative number of cars that passed the start of the freeway section. We

divide this number by the length of that section to find the density within that freeway

section at that point of time.

29

Mathematically,

�

N = Qin! " Q

out!

where

N = Number of cars in the freeway section after a time, ‘t’ from the start of counting.

Qin = Number of cars that have entered the section

Qout = Number of cars that have exited the section

And then density in the section is given by:

�

K =N

L

where

L = Length of the section.

Thus, a queuing analysis of freeway traffic flow does not involve any

presumption of effective vehicle lengths or acceptance of detector occupancy

measurements. Moreover, a queuing analysis helps us graph vehicle flows over time, so

that we can actually see the sequence of events that have occurred up to any particular

point in time. Apart from this, we can detect bottleneck formation and their propagation.

One operation that we do repeatedly is to accumulate the vehicle flows over

freeway sections. This leads to the drawback of applying queuing theory in traffic flow

analyses, namely, errors tend to accumulate. However the premise that we started out

with a clean set of data points tends to assure us that such errors are rare, if at all present.

30

While we talk of bottleneck detection and their tracking, one issue that deserves

attention is whether we can detect the active bottleneck. There may be cases where the

bottleneck propagated backwards and caused a queue to form at a location where it would

otherwise not have formed. This is important because identification of the active

bottlenecks will allow us to apply control policies in order to prevent formation of

bottleneck at that particular location. This will also curb the queues from propagating

backward and affecting other freeway locations.

4.3 Active Bottleneck Identification

We apply a systematic procedure in order to analyze this phenomenon. First, we

look for any drop in speed. If speed drops then we watch out for flow drops on the

downstream sections. If during the same time, flows drop on the upstream sections then it

is probably due to backward propagation of shocks. Otherwise, the upstream section is an

active bottleneck. Analyzing the flow – density curves in such cases will indicate whether

the section being analyzed is an active bottleneck or the result of a backward propagating

queue. The cases are briefly discussed in the following section:

Case 1: Neither flow nor speed drops on a freeway section. Vehicles move at free-flow

speeds.

Case 2: Speed drops on a section but flow does not. In this case, flows are maintained

with occurrence of high densities but low speeds.

Case 3a: Flow on the section drops and at the same time speed also drops, while flow

downstream does not. Speeds drop to such an extent that vehicle cannot travel the vehicle

length in 2 seconds. This indicates formation of active bottleneck on the section.

31

Case 3d: Both flow and speed drop on the section but flows downstream drop just

before, thus indicating that the section is subject to active bottleneck downstream.

Case 4: Speeds rise on the section indicating hysteresis.

This framework is illustrated in the Figure 4.4.

32

Speed drops?

Speed rises?

Section flow drops?

Downstream flow drops

just before?

Q

K

Uncongested Case 1

Q

K

Only flow drop Case 2

Q

K

Hysteresis Case 4 Q

K

Section subject to downstream bottleneck

Case 3d Q

K

Active bottleneck on section Case 3a

Figure 4.4: Active bottleneck identification typology

33

This typology of identifying the active bottlenecks is applied using statistical tools as

described in the next section.

Statistical determination of active bottleneck formation

A major effort of this research has gone into identifying the active bottlenecks

successfully. The typology that was developed earlier has been implemented statistically

and is detailed here.

We follow a particular sequence of steps:

Case 1:

No bottleneck formation, neither speeds nor flows drop. We regress speed as a function

of density and test the significance of density. Statistically it is represented as:

�

Va

= !0

+ !1*K

a

where

Va = Speed over detector ‘a’

Ka = Density over detector ‘a’

If α1 is not statistically significant, we conclude that this is case 1.

Case 2:

Speed drops but flow does not.

We encounter a significant α1 and conclude that it is not case 1. For such a detector, we

lay out the time intervals when speed drops by more than one standard deviation and

regress the flows over densities for such time intervals.

34

Statistically,

�

Qa

= !0

+ !1*K

a

where

Qa = Flow over detector ‘a’

Ka = Density over detector, ‘a’

And we test for statistical significance of β1.

If β1 is not statistically significant, we have Case 2.

Case 3a, Case 3d and Case 4:

If we encounter a significant β1 we know that it is not Case 2. Then we test the

variation of speed over time. For a positive variation of speed over time, we conclude that

this is the case where traffic hysteresis takes place and speed is recovering. (Case 4)

For a negative variation of speed over time, we test the variation of downstream

flow over time and if this variation is positive, we conclude that higher flows are taking

place downstream with time and the present section must be a bottleneck (Case 3a). If

however, the derivative of downstream flow with respect to time is negative, then we

conclude that the downstream section is a region of active bottleneck formation and the

present section is affected by the bottleneck downstream and we call this Case 3d.

35

The typology can thus be represented as:

Relationship between queuing diagram, the cases and traffic flow phases

It is easy to see that the five cases relate to the phases of traffic flow and queuing

diagram and the importance of understanding these relationships cannot be over-

emphasized. This section describes these relationships.

�

!v

!t

> 0 < 0

Hysteresis (Case 4)

�

qd

qu<1 (Case 3d)

�

qd

qu>1

(Case 3a: Active bottleneck)

36

Figure 4.5: Traffic phase diagram: relationship with cases

Looking at Figure 4.5, we see that while cars move at free-flow speed in Phase 1,

the density increases and that is when flow remains uncongested and we encounter Case

1. After a critical density is reached (point A) we move over to Phase 2, speeds drop but

flow does not and we end up with Case 2. When we move over to Phase 3 (point B), both

speed and flow drop and bottleneck formation takes place. This bottleneck may be a

result of a downstream bottleneck (Case 3d) or the section may it itself be the active

bottleneck (Case 3a). Last comes Phase 4 wherein speeds recover and hysteresis takes

place (Case 4).

Flow

Density

Phase 1, Case 1

Phase 2, Case 2

Phase 3, Case 3a or Case 3d

Phase 4, Case 4

A B

37

These relationships can also be seen with respect to the queuing diagram as

shown in Figure 4.6.

Figure 4.6: Queuing diagram – relationships with phases and cases.

From figure 4.6, it is easy to understand how the queuing diagram can explain the

occurrence of phases and cases. During the uncongested period, Phase 1 and Case 1

occur. Congestion takes place when density increases beyond the critical density and that

is when speed drops and bottleneck formation may also take place. So it represents both

Number of cars

Time

Phase 1, Case 1

Phase 2 and Phase 3 Case 2, Case 3a and

Case 3d

Phase 4, Case 4

Arrival curve

Departure curve

Phase 2 Case 2

Phase 3 Case 3a and Case 3d

Phase 4 Case 4

38

Phase 2 and Phase 3 and the accompanying Case2, Case 3a and Case 3d. Hysteresis

(Phase 4 and Case 4) is seen when speed recovers and the queue dissipates. Something

important to note are the different arrival and departure curves. It has been claimed in the

beginning of this chapter that speed drops at the bottleneck to such an extent that the

bottleneck cannot sustain the free-flow departure rate and that shows up in the departure

curve which branches out. But this is only true in case of bottleneck formation at the

section or downstream (Case 3a and Case 3d). If it is Case 2, then the departure curve

remains the same (shown by dotted line). Following this is the hysteresis in Phase 4

(Case 4). Also note the separation between departure and arrival curves even before the

start of Case 2 and after Case 4, which reflects the flow that is already present in the

section and even after the effects of queuing are over.

4.4 Tracking bottleneck formation and propagation:

In order to track bottleneck formation, various techniques are adopted. The first

plots what are called ‘Transformed Cumulative Flow’ curves. This includes transforming

the cumulative flow curves by subtracting a constant flow measure, every fifth minute

interval, that was decided based on observation.

Nt = N x,t( ) ! q0* t'

The above equation was first proposed by Cassidy and Bertini (2000) and it

shows the transformation methodology. N(x,t) is the present cumulative count and q0 is

the fixed count of vehicles subtracted for each time interval past the initial time,

�

t' . Nt is

the transformed cumulative flow value. Plotting such transformed cumulative flow curves

39

for successive freeway sections give us a diagram much like the queuing diagram. The

transformation as shown above is necessary because the cumulative flow counts are in

the order of tens of thousands for any day and the differences between cumulative flows

over successive freeway sections cannot be easily discerned unless those differences are

blown up. The transformation does this job. The choice of q0 should be of little concern

as it is solely used to expand the vertical separation of the cumulative curves. In such

graphs, the downstream detector station for any section acts as the portion of the freeway

that determines the departure rate and the rate of arrival on the upstream detector station

determines the arrival curve of the queuing diagram. This is consistent for each pair of

upstream and downstream detector station, that is for each freeway section.

The other method that is applied in order to detect and track the progression of

bottlenecks (in the form of shockwaves) is by finding the differences in cumulative flows

between the starting and the ending detector station for each successive freeway section.

Such differences give the number of cars within each freeway section at any point of time

and indicate the relative flow peaking patterns over successive detector stations, which in

turn indicate possible bottleneck formations and their progression. This approach in fact

developed from the previous one. Instead of transforming the cumulative counts and

plotting the successive detector stations, we simply deduct the downstream detector

station flow.

Mathematically,

�

Nst

= Qin

0

t

! " Qout

0

t

!

where:

Nst = Number of cars stored in any section after a time, ‘t’.

40

Qin = Flow entering the section

Qout = Flow exiting the section

4.5 Density Calculation (II):

With the corrections applied as described in Chapter 2, we plot the differences

between the starting and ending detector stations over time for each successive freeway

section. This difference gives us number of cars stored in each freeway section. Each

such freeway section will have a known number of lanes. And each such detector station

will have detectors that differ to the extent to which each measures traffic flows (For

example, the leftmost lane will presumably have fewer vehicles passing over it than the

middle lane and so the corresponding detector will count fewer vehicles than the middle

counterpart). So in the ideal case, the number of cars stored in each freeway section is

divided in proportion to detector counts in the station that is present in that section.

Dividing this flow by the length of the freeway section (which will be same for all the

detectors placed in parallel) gives the density over that detector. We denote such

densities, “K”. The densities calculated this way could be converted to occupancies as

described earlier for comparison purposes.

A freeway section is defined by the starting freeway/on-ramp detectors at one end

of it (the ‘head’) and the ending freeway/off-ramp detectors at the other end (the ‘tail’).

The densities calculated are space measurements while detectors are points on the

freeway section that define separate sections. So we cannot ‘allocate’ this density to any

particular detector involved in a certain section. To resolve this issue, we allocate the

densities as follows:

41

For the first freeway detector station, we assume that the ‘head’ detector station

will have densities for each detector in it same as the densities for each lane.

For subsequent sections, the densities for detectors in each detector station will be the

average of corresponding lane densities both upstream and downstream of the detector

station. This is illustrated in Figure 4.7. The densities for detectors ‘a’, ‘b’ and ‘c’ are

assumed to be same as the lane densities for the corresponding lanes over the length L1,

however the densities for detectors ‘d’, ‘e’ and ‘f’ are calculated as averages for the

corresponding lanes over the lengths L1 and L2. Again for the last section, the densities

for the individual detectors in the last detector station is assumed to be equal to the lane

densities for the last section.

Figure 4.7: Density calculation for individual detectors on freeway sections.

Mathematically, after a time, ‘t’,

�

(Ka)t

=(Q

L1)t

L1

* ra

where,

(Ka)t = Density over detector ‘a’ at time ‘t’.

a

b

c

d

e

f

L1 L2

42

�

(QL1)= Flow over length L1, on the lane where ‘a’ lies after time, ‘t’

L1 = Length of section, L1 and

�

ra

=a

a + b + c gives the ratio of cumulative flows over detector ‘a’ over cumulative

flows on the entire station.

Similarly we can find the densities for ‘b’ and ‘c’. However for detector ‘d’, we have the

following mathematical equation:

�

Kd( )t

=

{Ql1( )

t

L1

* ra +QL2( )

t

L2

* rd}

2

where

(Kd)t = Density over detector ‘d’ at time ‘t’.

�

(QL2) = Flow over length L2, on the lane where ‘d’ lies after time, ‘t’

L2 = Length of section, L2 and

�

rd =d

d + e + f gives the ratio of cumulative flows on detector ‘d’ over cumulative

flows on the entire station.

Densities calculated slightly differently can also be used to indicate shocks and

their propagation. The number of cars present in each freeway section divided by the

number of lanes on the freeway gives us the average number of cars present on each

freeway lane. Dividing this average number of cars present at any point of time per lane

by length of that freeway section gives the density in cars per lane mile for that section.

43

This gives a generic density value for the freeway section. We call such densities

“Normalized Densities (Kn)”. Waxing and waning in Normalized Density patterns

indicate where queues might be forming and bottlenecks may be propagating.

Plotting the densities just described against the corrected flows will give us a

revised version of Q-K curve which should be closer to reality than the ones calculated

from detector occupancies and flows. Moreover we get the appropriate values of speeds

from the above analysis by dividing the corrected flows over each detector by the

densities over them.

Densities, calculated as described above, depend upon accumulated flows.

However there are times when the downstream detector station of any freeway section

(the ‘tail’) registers higher flows than the upstream detector station (the ‘head’) of the

freeway section. Such occurrences result in a negative value of the differences (upstream

flow minus the downstream flow). As a result, the densities and hence the speeds turn out

to be negative and do not make sense. But such results are rare and mostly occur during

off-peak hours. So we mostly consider the results obtained for the period from 6 am to 7

pm. During peak hours, there is far less chance for detectors to over-count or under-count

and the variance in detected flow from actual flow that occurs is less. However, for data

correction/calibration purposes, the entire 24-hour period starting from 3:30 am to 3:30

am next day was used.

One of the benefits of the analysis done as described above is the discovery of

faulty detector stations. Even after corrections have been applied, there are cases where

in a succession of three detector stations the middle one may be incompatible with either

of the peripheral ones. At the same time, the peripheral ones may be compatible with

44

each other. This becomes clear after plotting the cumulative flows and such, as we have

already controlled for flow conservations. Then it becomes clear that the middle detector

station consists of faulty detectors. Such detector stations are normally excluded from the

analysis and the ‘peripheral’ detector stations are compared. This however stretches the

‘section’ under consideration and may minimize some queuing and shockwave

propagation effects.

45

CHAPTER 5

RESULTS

The results of the analyses detailed in earlier sections are illustrated here with the help of

examples. The first section shows the tracking of shockwaves, the second section shows

the detection of bottleneck formation and backward propagation of downstream queues,

the third section details some of the results in calculating the densities and the fourth and

last section identifies active bottlenecks.

5.1 Bottleneck formation and propagation

The results from the analyses are shown for trunk highway (TH) 169 in Figures

5.1, 5.2 and 5.3. The Figures show the way cumulative differences in flows vary across

time over the freeway section considered.

46

Figure 5.1: Cumulative flow differences across time on TH 169 (Nov 7th, 2000)

Figure 5.1 indicates possible bottleneck formation. Note that ‘Difference 1’is

upstream of ‘Difference 2’ and so on. In the Figure, the peaking for ‘Difference 2’ occurs

before that for ‘Difference 3’ during the morning peak which indicates that cars started

piling up upstream before the platoon arrives downstream. Also ‘Difference 2’ leads

‘Difference 1’ which indicates some backward propagation in the afternoon. ‘Difference

3’ leads ‘Difference 2’ in the early stages of afternoon bottleneck formation which again

indicates some backward propagation taking place. This can be seen in the blown up

version of the afternoon peak period in Figure 5.2.

Cumulative flow difference

across time

-50

0

50

100

150

200

250

300

0:05

0:45

1:25

2:05

2:45

3:25

4:05

4:45

5:25

6:05

6:45

7:25

8:05

8:45

9:25

10:05

10:45

11:25

12:05

12:45

13:25

14:05

14:45

15:25

16:05

16:45

17:25

18:05

18:45

19:25

20:05

20:45

21:25

22:05

22:45

23:25

5 Min intervals

Difference 1 Difference 2 Difference 3

47

Figure 5.2: Afternoon peak period, cumulative flow differences on TH 169.

Toward the end of the afternoon peak, ‘Difference 1’ falls before ‘Difference 2’ and

‘Difference 3’, which shows that the platoon traveled downstream before getting

dissipated completely.

5.2 Shockwave tracking using transformed cumulative curves

Another technique that was used to track possible shocks and resultant queuing

was transforming the cumulative flows and then plotting successive graphs. The

transformations were done by subtracting a constant flow measure, every fifth minute

interval, the subtraction having been decided based on observation. The premise behind

this approach is that increasing separation between upstream detector station and the

Cumulative differences

in flow

-50

0

50

100

150

200

250

300

14:20

14:30

14:40

14:50

15:00

15:10

15:20

15:30

15:40

15:50

16:00

16:10

16:20

16:30

16:40

16:50

17:00

17:10

17:20

17:30

17:40

17:50

18:00

18:10

18:20

18:30

18:40

18:50

19:00

Time

Difference 1 Difference 2 Difference 3

48

downstream detector station indicate the creation of a queue and the closing in of the two

indicate a dissipation of any queue previously present there.

Figure 5.3:Transformed cumulative curves for successive detector stations on TH 169

(Between Valley View Road and TH 62)

Figure 5.3 shows a typical representation of transformed cumulative flow curves

for a freeway section on TH 169. The subtraction done for purpose of transforming only

moved the curves with respect to the Y-axis. However, we notice that the curves almost

superimpose over one another over time. This is because of the high magnitude of the

transformed cumulative flow values on the Y-axis. But a close look at the curves reveal

Transformed cumulative graphs

for successive stations

-6000

-4000

-2000

0

2000

4000

6000

0:05

0:45

1:25

2:05

2:45

3:25

4:05

4:45

5:25

6:05

6:45

7:25

8:05

8:45

9:25

10:05

10:45

11:25

12:05

12:45

13:25

14:05

14:45

15:25

16:05

16:45

17:25

18:05

18:45

19:25

20:05

20:45

21:25

22:05

22:45

23:25

5 min intervals

Transformed1a Transformed1b

49

that there is some kind of separation between the two during the afternoon peak period

and that is what we expand and illustrate in the Figure 6.4.

Figure 5.4: Afternoon queuing on TH 169, transformed cumulative flow curves.

Figure 5.4 shows a queuing process occurring on the freeway section. The dotted line

shows the arrival curve at the upstream of the section and the solid one denotes the

departure curve at the downstream of the section. Clearly, we can see that such a

separation occurs at about 3:25 pm and starts dissipating around 4:45 pm. Something to

notice here is that the departure curve is not a straight line. This is because freeway

queues behave quite differently from a controlled queuing process. Departure rate

depends on various factors like discharge rate of the vehicles downstream and the

geometry of the freeway section and may change temporally.

Transformed cumulative flow

difference curves (blown up)

1500

2000

2500

3000

3500

4000

15:00

15:05

15:10

15:15

15:20

15:25

15:30

15:35

15:40

15:45

15:50

15:55

16:00

16:05

16:10

16:15

16:20

16:25

16:30

16:35

16:40

16:45

16:50

16:55

17:00

17:05

17:10

17:15

17:20

17:25

17:30

17:35

17:40

5 minutes

T1a T1b

50

The next two Figures (5.5(a) and 5.5(b)) show how queues propagate backward and how

this phenomenon is captured using transformed cumulative curves.

Figure 5.5(a): Downstream queuing on TH 169 (Between Excelsior Boulevard and TH 7)

Queuing process downstream

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

14:45

14:55

15:05

15:15

15:25

15:35

15:45

15:55

16:05

16:15

16:25

16:35

16:45

16:55

17:05

17:15

17:25

17:35

17:45

17:55

18:05

18:15

18:25

18:35

18:45

18:55

5 minutes

Transformed cumulative flows

Queuing process upstream

-500

0

500

1000

1500

2000

2500

3000

3500

4000

14:50

15:05

15:20

15:35

15:50

16:05

16:20

16:35

16:50

17:05

17:20

17:35

17:50

18:05

18:20

18:35

18:50

19:05

19:20

19:35

5 minutes

Transformed cumulative flows

Figure 5.5(b): Upstream queuing taking place on TH 169 (Between Lincoln Drive and Excelsior Boulevard)

51

From Figures 5.5(a) and 5.5(b), we can track a possible queue taking place.

Figure 5.5(a) shows first signs of the departure and arrival curves separating at around

14:45 pm and the upstream queuing (Figure 5.5(b)) starting to take place around 14:55

pm. There is a similar lag between the two curves closing in for the downstream and

upstream freeway sections. This indicates a possible back-propagation of the queue from

the downstream to the upstream section.

5.3 Density calculations:

One of the approaches, which is adopted in order to show possible shockwaves

propagating backwards, is by plotting ‘Normalized Density’ curves. These are nothing

but the curves obtained by dividing the differences in cumulative flows by the distances

of the freeway sections. Earlier in the section it was explained how plotting the

differences in cumulative flows over successive detector stations may be indicative of

possible queuing and shockwave propagation taking place. However, simply the

differences in cumulative flows may not be sufficient to determine queue formation and

propagation. This is particularly true for freeway sections of varying lengths. A longer

section may store more vehicles by virtue of its length and not due to a high density. This

is where using normalized densities (Kn), becomes crucial. While doing this, we are not

exactly finding densities over individual detectors, but we are computing the aggregate

number of vehicles stored in the particular freeway section over all its lanes.

52

Normalized density curves should give better insights into queuing formation and

propagation process. An example is shown in Figure 5.6.

Figure 5.6: Sample normalized density curve on I 94 East Bound.

In Figure 5.6, ‘Norm_den1’ is upstream of ‘Norm_den2’, which is upstream of

‘Norm_den3’. We can see a clear case of high density occurring over the third freeway

section before it seems to have propagated backward onto the second section during the

morning peak. Platoon formation and its propagation from the first to the second freeway

section can also be seen. To note here is the long time that third freeway section remains

congested. This indicates a sustained presence of the queue, resulting form a platoon

traveling from the first and second sections, which dissipated gradually.

Normalized densities

0

50

100

150

200

250

3:05

4:00

4:55

5:50

6:45

7:40

8:35

9:30

10:25

11:20

12:15

13:10

14:05

15:00

15:55

16:50

17:45

18:40

19:35

20:30

21:25

22:20

23:15

0:10

1:05

2:00

2:55

5 minutes

Norm_den1 Norm_den2 Norm_den3

53

5.4 Active Bottleneck Identification

A significant contribution of this research is an ability to frame a queuing based model to

detect the presence of ‘Active Bottlenecks’. Active bottlenecks on a freeway are places

where bottleneck formation takes place. These are not places where bottleneck activity

arose as a result of a backward propagating queue from downstream. Identifying active

bottlenecks on freeways is essential to effectively control traffic on them and optimize

system performance. Knowing the location of active bottlenecks and controlling traffic

flowing onto them (using ramp meters and other demand management strategies), rather

than the sections that are ‘susceptible to active bottlenecks’ will ensure that right places

on freeways are metered and not just ‘any’ section.

The following steps describe the process of finding active bottlenecks:

1) We obtain the speeds over detectors from the flow-density-speed relationship,

with flow and density calculated from queuing theory.

2) Calculate 5-interval moving averages of flow, density and speed, each interval

being 5 minutes long. This is done to smooth the transitions of each traffic flow

characteristic.

3) Select a representative detector out of any station. However consistency was

maintained in selecting detectors. Normally, a middle lane detector was chosen

and the subsequent detectors downstream would be selected from the same lane.

4) Select two periods to analyze: morning peak (beginning around 5:30 am and

continuing till around 10:30 am) and evening peak period (beginning around 1:30

pm and continuing till around 6 pm). This is done keeping in view the fact that

54

during the intermediate periods, data indicate that no queuing is taking place.

Activity patterns are seen most distinctly during the morning and afternoon peaks.

5) Conduct statistical analysis of flow, density and speed relationships to identify

active bottlenecks for each selected detector.

6) Plot flows and speeds over time for each successive detector going upstream to

downstream to support the results of the statistical analyses.

Then it is a matter of watching how the curves behave over time.

Here we present exemplars of each case:

Case 1: No bottleneck formation, neither speeds nor flows drop.

Regression results of speed over density (Detector 52 on Interstate 94 East Bound,

November 3rd, 2000) are shown in Table 5.1.

Dependent Variable: Speed Variable Hypothesis Coefficient P > |Z| Constant +S 47.89 1.96E-14 Density NS 0.862 0.58

Significance based on 90% confidence interval Number of observations = 43 R-squared = 0.0075

Table 5.1: Regression results (speed over density) for detector 52 on I 94

Results indicate that density is not a significant variable and so it is case 1, where neither

flow nor speed drops and no queuing or bottleneck formation/propagation takes place.

This is corroborated by the Flow-Density relationship curve shown in Figure 5.7.

55

Figure 5.7: Flow-density relationship for detector 52 (Lowry Avenue), on I 94 EB on

November 3rd, 2000

Case 2: Speed drops but flow does not.

When it is not Case 1 for a detector, we find the standard deviation of speeds for that

detector and weed out the data for it for which speeds do not fall by at least one standard

deviation. Then for the rest of the data, we find the statistical significance of density

while regressing flow over density. Results for detector 83 on Interstate 94 (42nd Avenue)

are shown in Table 5.2.

Flow

0

20

40

60

80

100

120

140

1606 9

13

17

18

18

20

22

22

24

26

29

34

38

41

41

41

42

42

44

44

45

Density

56

Dependent Variable: Flow Variable Hypothesis Coefficient P > |Z| Constant +S 68.23 1.96E-14 Density NS 0.127 0.58

Significance based on 90% confidence interval

Number of observations = 49

R-squared = 0.011

Table 5.2: Regression results (flow over density, for speeds below free-flow speed) for

detector 83, near 42nd Avenue on I 94 on November 2nd, 2000

We see the insignificance of density in the above table and conclude that it is case 2 and

speed should drop while flow remains almost same.

This is corroborated by the following Q-K-V relationship in Figure 5.8. Speed is shown

by the dashed line while the solid line shows flow variation across density. Dotted lines

are superimposed on both flow and speed curves for emphasis.

57

Figure 5.8: Flow and speed varying by density on detector 83 on November 2nd 2000

Case 3a, Case 3d and Case 4:

If it is neither Case 1 nor Case 2, we investigate further. We discover that some detectors

show a mix of cases 3a, 3d and 4. This indicates that, at the location of the detector,

sometimes active bottleneck formation takes place, sometimes speed recovers and

hysteresis takes place and at other times, bottleneck formation is a result of bottlenecks

forming downstream. This is natural on a freeway with stochastic traffic flow. However

often, only Cases 3a and 4 take place (or only Cases 3d and 4 take place) which suggest

that the section may be considered an active bottleneck (subject to downstream

bottleneck). There are however instances where all three cases occur on one section as

will be illustrated here.

Flow and speed

0

10

20

30

40

50

60

70

80

90

7 7 8

10

12

14

15

15

15

16

16

16

17

18

21

26

27

28

29

29

30

30

30

31

31

32

32

32

34

35

36

Density in cars per lane mile

Flow Speed

58

Example of Phase 3a and Phase 4:

Time Phase Time Phase 14:05 Phase 3a 16:05 Phase 4 14:10 Phase 3a 16:10 Phase 3a 14:15 Phase 3a 16:15 Phase 3a 14:20 Phase 4 16:20 Phase 4 14:25 Phase 4 16:25 Phase 4 14:30 Phase 4 16:30 Phase 3a 14:35 Phase 3a 16:35 Phase 4 14:40 Phase 3a 16:40 Phase 3a 14:45 Phase 3a 16:45 Phase 3a 14:50 Phase 3a 16:50 Phase 4 14:55 Phase 3a 16:55 Phase 4 15:00 Phase 4 16:05 Phase 4 15:05 Phase 4 17:00 Phase 3a 15:10 Phase 4 17:05 Phase 4 15:15 Phase 4 17:10 Phase 4 15:20 Phase 4 17:15 Phase 3a 15:25 Phase 3a 17:20 Phase 4 15:30 Phase 3a 17:25 Phase 4 15:35 Phase 4 17:30 Phase 3a 15:40 Phase 4 17:35 Phase 4 15:45 Phase 3a 17:40 Phase 3a 15:50 Phase 4 17:45 Phase 3a 15:55 Phase 4 17:50 Phase 4 16:00 Phase 3a 17:55 Phase 4

Table 5.3: Cases for detector 600 on I 94 for November 1st, 2000 (Near Dupont Avenue)

From Table 5.3, we can see that the section fluctuates between the states of active

bottleneck and recovery.

59

Example of Case 3d and Case 4:

Time Phase Time Phase 14:05 Phase 3d 16:00 Phase 3d 14:10 Phase 3d 16:05 Phase 3d 14:15 Phase 4 16:10 Phase 4 14:20 Phase 4 16:15 Phase 3d 14:25 Phase 4 16:20 Phase 4 14:30 Phase 3d 16:25 Phase 4 14:35 Phase 4 16:30 Phase 3d 14:40 Phase 4 16:35 Phase 3d 14:45 Phase 3d 16:40 Phase 3d 14:50 Phase 3d 16:45 Phase 3d 14:55 Phase 4 16:50 Phase 3d 15:00 Phase 3d 16:55 Phase 3d 15:05 Phase 4 17:00 Phase 3d 15:10 Phase 4 17:05 Phase 4 15:15 Phase 4 17:10 Phase 4 15:20 Phase 4 17:15 Phase 3d 15:25 Phase 4 17:20 Phase 4 15:30 Phase 3d 17:25 Phase 4 15:35 Phase 3d 17:30 Phase 3d 15:40 Phase 4 17:35 Phase 3d 15:45 Phase 3d 17:40 Phase 4 15:50 Phase 3d 17:45 Phase 3d 15:55 Phase 4

Table 5.4: Cases for detector 83 (Dowling Avenue) on I 94 on November 1st, 2000

We can see how the section defined by detector 83 is largely affected by

downstream bottleneck, but is itself not an active bottleneck.

60

Example of Case 3a, Case 3b and Case 4:

Time Phase Time Phase 5:30 Phase 3a 7:50 Phase 3a 5:35 Phase 3a 7:55 Phase 3a 5:40 Phase 3a 8:00 Phase 4 5:45 Phase 4 8:05 Phase 4 5:50 Phase 3a 8:10 Phase 3a 5:55 Phase 3a 8:15 Phase 4 6:00 Phase 4 8:20 Phase 4 6:05 Phase 3a 8:25 Phase 4 6:10 Phase 3a 8:30 6:15 Phase 4 8:35 6:20 Phase 3a 8:40 6:25 Phase 3a 8:45 Phase 3d 6:30 Phase 4 8:50 Phase 3a 6:35 Phase 3a 8:55 Phase 3d 6:40 Phase 3a 9:00 Phase 4 6:45 Phase 4 9:05 Phase 3d 6:50 Phase 3a 9:10 Phase 4 6:55 Phase 3a 9:15 Phase 3d 7:00 Phase 3a 9:20 Phase 3d 7:05 Phase 3a 9:25 Phase 3d 7:10 Phase 3a 9:30 Phase 3d 7:15 Phase 4 9:35 Phase 3d 7:20 Phase 4 9:40 Phase 4 7:25 Phase 4 9:45 Phase 4 7:30 Phase 4 9:50 Phase 4 7:35 Phase 4 9:55 Phase 3d 7:40 Phase 3a 7:45 Phase 4

Table 5.5: Cases for detector 666 (Shingle Creek parkway) on I 94 on November 2, 2000

We can see from Table 5.5 that for the early part of the morning peak, the section

behaved mostly as an active bottleneck though during the later part, a downstream

61

bottleneck was affecting it. This example shows how all three cases can co-exist for a

single freeway section. Also note that at a few places, neither of the three cases appear.

This is because those times are when it is either Case 1 or Case 2. Any one section cannot

be labeled as an ‘active bottleneck’ in generic terms for this reason. The Q-K relationship

in Figure 5.9 illustrates this.

Figure 5.9: Flow density relationship for detector 688 (49th/53rd Avenue on I 94) on

November 3rd, 2000

The summary table which shows the bottleneck properties for each section on Interstate

94 for the period from November 1st to November 6th is included in Table 5.6.

Flow

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60

Density in cars per lane mile

62

Table 5.6: Cases for individual sections on I 94 for the period from November 1st to November 6th, 2000

The asterisked (*) fields are cases where either the detector was not functioning properly or the case could not be ascertained due to

non-availability of data for downstream sections. Note the cases where only case 3d and 4 occur and not 3a and the ones where cases

3a and 4 occur but not 3d.

Date(s)

Morning peak Sections -->

Shingle Creek

Parkway

Dupont

Avenue 57th Avenue

49th/53rd

Avenue 42nd Avenue

Dowling

Avenue

Lowry

Avenue

Broadway

Avenue

Plymouth

Avenue

1-Nov-00 3a,4 3a, 3d, 4 1 1 3a, 3d, 4 1 1 1 *

2-Nov-00 3a, 3d, 4 1 1 1 1 1 3d, 4 3a, 3d, 4 2

3-Nov-00 3a, 3d, 4 1 1 1 1 1 1 3a, 3d, 4 2

4-Nov-00 1 1 1 * 1 3d, 4 3d, 4 3a, 3d, 4 *

5-Nov-00 3a, 3d, 4 * 1 1 * * 1 * *

6-Nov-00 1 1 3d, 4 3,5 3d, 4 3d, 4 3d, 4 * *

Evening peak

1-Nov-00 2 3a, 4 1 1 3d, 4 1 1 2 *

2-Nov-00 2 2 1 1 2 1 2 2 2

3-Nov-00 * 2 2 3a, 3d, 4 3a, 3d, 4 3a, 3d, 4 1 2 2

4-Nov-00 1 1 3a, 3d, 4 3a, 3d, 4 3d, 4 3d, 4 3d, 4 3a, 4 2

5-Nov-00 3d, 4 2 1 1 2 3d, 4 2 2 *

6-Nov-00 3d, 4 2 1 1 3d, 4 1 1 * *

63

CHAPTER 6

SUMMARY AND CONCLUSIONS

This thesis analyzes the traffic flow parameters (flow, density and speed) over various

freeway sections on the Twin Cities freeway network. Queuing theory has been

extensively applied to find densities, to predict and track bottleneck formation and their

propagation and to detect active bottleneck locations.

The major contributions of this work can be summarized as:

1) Balance flows

An algorithm is developed to balance flows on any freeway section ensuring flow

conservation. This algorithm searches for the region of least error and then

calibrates the other freeway section based on this ‘least-error’ section, called a

‘truth couplet’

2) Apply queuing theory to traffic flow parameters

Queuing theory has been extensively applied to analyze the traffic flow

characteristics. While errors in flows given by detectors have been minimized by

the calibration procedure, we also do not depend upon the occupancy data from

loop detectors for calculating densities. Assumptions of vehicles lengths are not

required either.

3) Develop a typology to determine the location of active bottlenecks on the

freeways based on statistical analysis and their correspondence to visual

64

relationships between flow, density and speed, this typology helps us identify the

locations which are prone to bottleneck formation and those which are prone to

being affected by downstream bottlenecks.

4) Apply the typology to data.

We test our typology to data gathered from freeways Trunk Highway 169 and

Interstate 94 and find active bottleneck locations. A sample of six days for I 94 for

morning and evening peak periods show that the same section cannot be

characterized as an ‘active bottleneck’ location always. Even normally active

bottlenecks may be constrained by downstream events. Traffic flow

characteristics change and that leads to changing situations on each freeway

section.

5) Develop methodology to track queue propagation.

Graphical methods to track shockwaves have been explained and they have been

applied to real life data in order to track the way queues propagate backward

affecting the upstream sections of freeways.

Apart from these, the analysis also helps identify bad detectors on different days.

Freeway bottleneck prediction and their minimization has been a big focus of

recent research and the work in this thesis presents new methods in analyzing traffic flow

parameters.

65

Reference:

1) Gerlough, D. and M. Huber, “Traffic flow theory: A monograph”, TRB special

report 165, 1975.

2) Daganzo, C. K. Smilowitz, M. Cassidy, R. Bertini, “Some observations of

highway traffic in long queues” Transportation Research Record 1678, pages 225-

233, 1999.

3) Lovell, D. and W. Levine, “The Freeway Access Control Problem – A Survey of

Successes and Continuing Challenges”, IEEE Transportation Systems Conference

Proceedings, Oakland, CA, August 2001

4) Zhang, H. and T. Kim, “A car-following theory for multiphase vehicular traffic

flow”. ITS e-news, University of California, Davis, 2000

5) Jia, Z., C. Chen, B. Coifman and P. Varaiya, “The PeMS algorithms for accurate,

real-time estimates of g-factors and speeds from single-loop detectors” IEEE

Transportation Systems Conference Proceedings, Oakland, CA, August 2001

6) Zhang, H. and W. Lin, “Some recent developments in traffic flow theory”’, IEEE

Transportation Systems Conference Proceedings, Oakland, CA, August 2001

7) Cassidy, M., S. Anani and J. Haigwood, “Study of Freeway Traffic Near and Off-

Ramp”, Transportation Research Part A: Policy and Practice Volume: 36 (6)

pages 563 – 572, 2002

8) Cassidy, M. and R. Bertini, “Some Traffic Features at Freeway Bottlenecks”,

Transportation Research, Part B, Volume: 33, Number 1, Elsevier, pages 25-42,

January 1999

9) Cassidy, M., and R. Bertini, “Some Observed Queue Discharge Features at a

66

Freeway Bottleneck Downstream of a Merge”, Transportation Research, Part A,

Volume 36, Number 8, Elsevier, October 2002, pages 683-697.

10)Persaud, B. and D. Tsui, “Study Of Breakdown-Related Capacity For A Freeway

With Ramp Metering” Transportation Research Board publication, paper 01-2636,

2001.

11) Cassidy, M and R. Bertini, “Some traffic features at freeway bottlenecks”,

Transportation Research Record, 33B, 25-42, August 2001

12) Lawson, T., D. Lovell and C. Daganzo, “Using the Input-Output Diagram to

Determine the Spatial and Temporal Extents of a Queue Upstream of a

Bottleneck” Transportation Research Record 1572, pages 140-147, 1997

13) Cassidy, M. “Recent Findings on Simple Attributes of Freeway Formation and

Propagation”, IEEE Transportation Systems Conference Proceedings, Oakland,

CA, August 2001

14) All detector report, Traffic Management Center, Minneapolis, MN

15) www. mapquest.com

16) California PATH website (www.path.eecs.berkeley.edu), Accessed by Shantanu

Das on May 30, 2002

17) Minnesota Department of Transportation website (www.dot.state.mn.us), May

2002

18) Barbour and Fricker, “Balancing link counts at nodes using a variety of criteria:

an application of local area traffic assignment”, Transportation Research Record,

1220. Transportation Research Board, National Research Council, p. 33-39, 1990.

19) P. Weiss, Stop-and-Go Science, science news online (www.sciencenews.org), July 1999.

67

Appendix: A.1 Example of flow calibration A.2 Brief description of computer program used to achieve flow

conservation

68

Appendix A.1 Example of flow calibration: Lets say we have four counts of cumulative flows: A, B, C and D. Let A and C be the incoming and outgoing freeway flows and B and D be the on-ramp and off-ramp flows. In the ideal case, A+B = C+D Lets assume, A=2000 (We have arrived at this corrected value after calibration) B=100 C=1600 D=300 Based on the value of A, we have to adjust the values of B, C and D.

A+B = C+D A = C+D-B 2000 = 1600+300-100 which is not true. Difference between the Left Hand Side (LHS) and Right Hand Side (RHS) (LHS – RHS) equals 200. We split this differences into C, D and B such that flow is conserved.

For C

Increase required =

�

1600

2000* diff =

1600

2000*200 =160

For D

Increase required =

�

300

2000* diff =

300

2000*200 = 30

For B

Decrease required =

�

100

2000* diff =

100

2000*200 =10

Now A+B = 2000+90 = 2090 C+D = 1760+330 = 2090 thus satisfying the flow conservation criterion.

69

Appendix A.2 A BRIEF DESCRIPTION OF THE COMPUTER PROGRAM USED FOR

CALIBRATION OF DETECTOR COUNTS

The program to automate the analysis reads from a text file the total flows on

successive freeway or ramp detectors. While inputting, we indicate whether the flow is a

freeway flow by ‘f’, whether it’s an input or on-ramp flow by ‘i’ and if it’s an off-ramp

flow by ‘o’. It allocates positive signs to the input freeway and on-ramp flows and

negative signs to output freeway and off-ramp flows. It considers the first freeway entry

of a section as the ‘head’ and the last freeway entry as a ‘tail’. Until it encounters the next

freeway entry, it will keep allocating the signs (as just described). Once it is done with

one freeway section, it jumps to the next and then it will allocate the ‘head’ status to the

‘tail’ of the previous freeway section and the ‘tail’ status to the ending freeway count of

the present freeway section. It continues this all the way until it completes all the freeway

sections. Also it calculates the individual section input output difference and labels the

one with least difference as the ‘truth couplet’. Moreover, it also adjusts the truth couplet,

splitting whatever error may be present into all the detectors that are involved in the truth

couplet weighing them based on their total individual flows. After that it calibrates the

freeway section as described before.