Effect of time of day and day of the week on congestion duration and breakdown A case study at a bottleneck in Salem, NH Q3 ww sciencedirect com 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2[.]
Trang 1Original research paper
Effect of time-of-day and day-of-the-week on
congestion duration and breakdown: A case study
at a bottleneck in Salem, NH
Q3 Eric M Laflammea,*, Paul J Ossenbruggenb
aDepartment of Mathematics, Plymouth State University, Plymouth, NH 03264, USA
b
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA
h i g h l i g h t s
Features of recurrent congestion and recovery are analyzed via regression models
Probability of recurrent congestion increases between AM- and PM-rush periods
Effect of time-of-day on congestion duration depends on the day-of-the-week
a r t i c l e i n f o
Article history:
Available online xxx
Keywords:
Stochastic models
Ordinary least squares regression
Binary logistic regression
Congestion duration
Breakdown
a b s t r a c t
This work uses regression models to analyze two characteristics of recurrent congestion:
breakdown, the transition from freely flowing conditions to a congested state, and duration, the time between the onset and clearance of recurrent congestion First, we apply a binary logistic regression model where a continuous measurement for traffic flow and a dichoto-mous categorical variable for time-of-day (AM- or PM-rush hours) is used to predict the probability of breakdown Second, we apply an ordinary least squares regression model where categorical variables for time-of-day (AM- or PM-rush hours) and day-of-the-week (MondayeThursday or Friday) are used to predict recurrent congestion duration Models are fitted to data collected from a bottleneck on I-93 in Salem, NH, over a period of 9 months
Results from the breakdown model, predict probabilities of recurrent congestion, are consistent with observed traffic and illustrate an upshift in breakdown probabilities between the AM- and PM-rush periods Results from the regression model for congestion duration reveal the presences of significant interaction between time-of-day and day-of-the-week
Thus, the effect of time-of-day on congestion duration depends on the day-of-the-week This work provides a simplification of recurrent congestion and recovery, very noisy processes
Simplification, conveying complex relationships with simple statistical summaries-facts, is
a practical and powerful tool for traffic administrators to use in the decision-making process
© 2017 Periodical Offices of Chang'an University Publishing services by Elsevier B.V on behalf of Owner This is an open access article under the CC BY-NC-ND license (http://
creativecommons.org/licenses/by-nc-nd/4.0/)
* Corresponding author Tel.: þ1 603 724 5336
E-mail addresses:emlaflamme@plymouth.edu(E.M Laflamme),pjo@unh.edu(P.J Ossenbruggen)
Peer review under responsibility of Periodical Offices of Chang'an University
Available online at www.sciencedirect.com
ScienceDirect
journal homepage:w ww.elsevier.com/locat e/jtte
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http://dx.doi.org/10.1016/j.jtte.2016.08.004
2095-7564/© 2017 Periodical Offices of Chang'an University Publishing services by Elsevier B.V on behalf of Owner This is an open
access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Trang 21 Introduction
Traffic congestion costs American drivers billions of dollars in
wasted fuel and loss of productivity Among all types of
congestion, recurrent congestion, congestion occurring every
day, accounts for roughly half of the congestion experienced
by Americans (U.S Department of Transportation, 2016)
Furthermore, these recurring congestion events on highways
account for 40% of the total delay, more than the delay from
construction and traffic incidents (non-recurring events)
combined (Paniati, 2003)
Clearly, recurrent congestion is a problem and methods
must be developed to alleviate the situation But, before
expensive construction or remedial measures are employed, it
is critical that administrators identify trends/characteristics
of recurring congestion on their roadways As stated byRao
and Rao (2012), identifying characteristics of congestion
events will serve as a guide to administrators to help them
choose appropriate measures to mitigate such congestion
Along these same lines,Hao et al (2007)stated that in order
to control or alleviate congestion effectively, researchers
must investigate its key features
So, what are these characteristics of recurrent congestion,
and the key features of these events pursued by researchers?
Key features may refer to the underlying causes of congestion
For example, both high flow during peak-hours (Downs, 2004)
and physical structures such as bottlenecks (Ban et al., 2007)
have been shown to be responsible for recurrent congestion
Other features include congestion dynamics, the evolvement
of the congestion event such as queue propagation (Newell,
1993) and shock-wave analysis (Bertini and Cassidy, 2002;
Kerner, 2004) Other common features of congestion are the
stochastic nature of breakdown, the transition from freely
flowing traffic to a congested state, and the factors that
trigger these events (Elefteriadou et al., 1995) Other
researches are devoted to measures/metrics that quantify
their extent, duration, and intensity of congestion (Shaw,
2003) Such measures include level-of-service, congestion
duration (the time between onset of breakdown and
clearance of congestion event, or, alternatively, the time that
the travel rate indicates congested travel on a segment),
travel time index, etc In all cases, no matter the
characteristic, researchers aim to better understand
recurrent congestion and identify the underlying dynamics
of the process
With these works in mind, authors focus on two particular
characteristics of recurrent congestion: breakdown and
duration Authors will only focus on recurrent events, and,
going forward, “congestion” will always means “recurrent
congestion” As mentioned above, breakdown refers to the
transition from a freely flowing traffic state to a congested
state This is the standard definition and commonly used
phrase of breakdown given by Kerner (2009) Of course,
identifying breakdown relies on how we define “freely
flowing” and “congested” The precise criteria for identifying
freeflow and congestion, and thus how we define
breakdown, will be discussed later in Section2.4 Congestion
duration refers to the time between the onset of breakdown
and clearance of a congested event This too relies on how
we define a congested traffic state This definition and the precise criteria used to identify congestion duration will be discussed later in Section2.5
Why did authors choose these particular congestion characteristics? First, congestion duration is chosen because,
as a“time-based” measure of congestion, it is in keeping with the common perception of the congestion problem (Rao and Rao, 2012), yet the explicit investigation of congestion duration's statistical characteristics has attracted only limited attention in the literature Vlahogianni et al (2011)
state that the dynamics of congestion duration may contain useful information about intraday traffic operations and should be further explored Second, breakdown was chosen because, despite receiving ample treatment in the literature (Elefteriadou et al., 1995; Persaud et al., 1998), it remains a controversial topic That is, while the stochastic nature of traffic breakdown has been verified (Elefteriadou et al., 1995), the mechanism or trigger of breakdown is still mysterious It
is our goal to gain some insight into these two aspects of recurrent congestion
In this work, authors will use two separate regression an-alyses where breakdown and congestion duration are used as respective response variables We will then introduce explanatory variables derived from traffic stream data to identify underlying factors associated with both breakdown and duration Ideally, from our models, identifying significant predictors of breakdown and congestion duration will lead to
a better understanding of recurrent congestion
The remainder of this paper is organized as follows Sec-tion 2 presents authors' materials and methods including statistical (regression) model forms, the raw traffic stream data from authors' collection site, preprocessing procedures, data aggregation, and extraction of the requisite variables for model fitting Section 3 presents authors' results from model-fitting and interpretation of these results Lastly, section4is the conclusion of authors' study
2 Materials and methods
2.1 Statistical models: binary logistic regression model for breakdown probability
A traditional approach to identifying probability of breakdown
is the use of a generalized linear model (GLM) GLMs have the basic form for random response variable Y
gðmiÞ ¼ Xib (1) where the mean value mi is given by E(Yi), g($) is a smooth monotonic“link” function, Xiis the ith row of model matrix X, andb is a vector of unknown model parameters GLMs assume that the Yiare independent and Yi~ some exponential family distribution To model breakdown probability, the authors let
Y represent the traffic state where Y¼ 1 denotes congested traffic and Y ¼ 0 denotes freely flowing traffic Then, mi is interpreted as the probability,p, of Yitaking on the value of one, Pr(Y¼ 1), and the authors use the canonical link function with the form of Eq.(1)
gðpÞ ¼ ln p
1 p
(2)
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Trang 3Such models are known as binary logistic regression
models For further details regarding logistic models and
GLMs, the reader is directed toHosmer et al (1989)
Several analyses have been performed in which probability
of breakdown is identified as a function of flow
measure-ments.Athol and Bullen (1973)suggested that the expected
time to breakdown is a declining function of flow Lorenz
and Elefteriadou (2001)illustrated the probabilistic nature of
breakdown and showed empirical evidence that probability
of congestion increases with increasing flow rate Other
studies have similarly shown that congestion breakdown
occurs with some probability at various flow rates, that
breakdown is, in fact, a stochastic process (Dong and
Mahmassani, 2009a,b; Elefteriadou et al., 1995; Evans et al.,
2001).Persaud et al (1998)explored the relationship between
flow and the probability of breakdown empirically, by visual
assessment Ossenbruggen (2016), used stochastic
differential equations to predict breakdown probability
based on average flows for discretized time-of-day periods
Following these analyses, traffic flow (volume) will be used
as the primary predictor of breakdown
In addition to flow, and because breakdown probability is
likely influenced by time-of-day, a time variable will be
introduced into authors' regression model In the spirit of
simplification, our time-of-day measure will be a categorical
variable distinguishing between two time sectors: AM-rush
(7 a.m.e2 p.m.) and PM-rush (2 p.m.e7 p.m.) These periods
were not chosen arbitrarily, but based on several factors First,
and most simply, the vast majority, more than 90%, of
congestion events occur between 7 a.m and 7 p.m Second,
within these 12 h, the AM-rush and PM-rush periods were
distinguished/identified based on a visual comparison of
average flows throughout the day (Fig 1) This figure
illustrates how average flows within each period are similar
in terms of magnitude and variability, yet dissimilar to one
another That is, average flows from 2 p.m to 7 p.m
(PM-rush) are higher and have more variability as compared to
flows from 7 a.m to 2 p.m (AM-rush) Third, identification
of these periods was further based on a previous analysis
where a piecewise linear function was fitted to daily average
flows In this investigation, there was a distinct and sharp
increase in average flow at 2 p.m., lending credence to the choice of 2 p.m as a break point between AM- and PM-rush periods Furthermore, this analysis revealed a nearly flat trend in average flow from around 7 a.m up to 2 p.m., while flows after 2 p.m were more volatile and higher until about
7 p.m These estimated trends reinforced the AM- and PM-rush periods identified visually InOssenbruggen (2016), this piecewise linear modeling approach and these exact time periods were used to calibrate a stochastic differential equation model used to predict traffic breakdown In fact, these same AM- and PM-rush time periods were identified
as significant predictors of breakdown Fourth, in a study of both recurrent and non-recurrent congestion events,
Hallenbeck et al (2003) identified 3 p.m as a natural break between midday and afternoon peak periods While this work does not perfectly agree with our choice of 2 p.m as break point, it is very similar and we feel our choice is justified based on trends in average flows Also, this analysis supports our decision to simplify time-of-day in terms of a categorical variable with just a few variables Fifth, in an analysis of recurring congestion, Mazzenga and Demetsky (2009) identified AM- and PM-peak periods between 5 a.m.Q1
and 8 a.m and between 3 p.m and 7 p.m., respectively
Again, this does not agree with our periods exactly, but supports the use of simplified time-of-day categories The authors are confident that the time periods capture the recurring events and distinguish the morning and afternoon breakdowns As a final note, the authors acknowledge that time-of-day likely serves as a proxy for more complicated,
“lurking,” secondary traffic characteristics (aggressive driving, weaving, etc.) that cannot be extracted from the traffic stream data
InFig 1, vertical lines represent breaks between AM-rush and PM-rush periods Fig 1illustrates the homogeneity of traffic flows within each time period
In addition to a flow main effect and a time sector dichot-omous variable, the interaction between the time and the continuous flow variable will be included in our model spec-ification This will allow for a possible shift (change in inter-cept) and a change in slope (actually, a change in the“log odds”) between the two time sectors
Fig 1e Average flow and variability by time-of-day
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Trang 4So, for our breakdown model, the probability of breakdown
Pr(Y¼ 1), denoted p, may be expressed as follow
p ¼ expðb0þ b1qþ b2dþ b3qdÞ
1þ expðb0þ b1qþ b2dþ b3qdÞ (3)
where q is the flow, d is the dichotomous variable
dis-tinguishing AM- or PM-rush hours, qd is the interaction
vari-able The procedure for identifying our binary response
variable, freeflow or congested state, and the associated
pre-dictor variables will be discussed later in Section2.4
In terms of statistical methodology,Persaud et al (2001)is
especially relevant as the authors use logistic regression
models based on 3-lane, 1-min flow averages to predict
congestion events Also,Ossenbruggen (2016)uses a logistic
form for breakdown probability as a component of a
stochastic differential equation to explain the congestion
process To our knowledge, no statistical analyses have been
performed using our methodology and exact combination of
predictors
2.2 Statistical models: ordinary least squares regression
for congestion duration
To analyze congestion duration, we use an ordinary least
squares (OLS) regression platform Specifically, our response,
W is congestion duration in minutes, a continuous variable
measured from the onset of breakdown to the clearance of the
congestion event The authors chose two explanatory
vari-ables in model: time-of-day and day-of-the-week Since most
drivers have seen some association between time-of-day and
the presence/severity of the congestion, considering
time-of-day as a potential predictor of congestion duration makes
intuitive sense Furthermore, since flow follows a distinct
pattern daily, time-of-day may be considered a simple proxy
for traffic flow InTebaldi et al (2002), day-of-the-week was
used as a primary predictor of traffic flow in their
hierarchical regression models In the literature, however,
day-of-the-week was not commonly used to distinguish
traffic stream characteristics In fact, most analyses used
data collected from weekday traffic interchangeably
Germane to this analysis, Falcocchio and Levinson (2015)
summarized trends in recurrent congestion duration by both
time-of-day and day-of-the-week The authors found
end-of-week traffic more severe, a perspective supported by the
own driving experience at the collection site and from an
initial data investigation
Under the OLS regression format, a congestion duration
length W then has the following form
W¼ b0þ b1xþ b2dþ b3xdþ e (4)
where x is the categorical variable distinguishing
day-of-the-week, xd is the interaction variable, ande Nð0; seÞ and
in-dependent The procedure for identifying congestion duration
and the associated predictor variables will be discussed later
in Section2.5 Such a model (continuous response with two
categorical predictors) could also be analyzed using an
analysis of variance (ANOVA) format In fact, with such
dichotomous predictors, the two model forms and their
corresponding hypothesis tests produce equivalent results
The study of non-recurrent congestion duration (incident duration) has received ample treatment in the literature An-alyses byGarib et al (1997), Giuliano (1989), andSullivan (1997)
used lognormal distributions to describe freeway incident duration Conditional models for incident duration have been pursued by Jones et al (1991), for example.Nam and Mannering (2000) used hazard-based models to find the likelihood that an incident will end in the next short time period given its continuing duration Similarly,Stathopoulos and Karlaftis (2002) used a probabilistic log-logistic functional form to describe incident durations A number of works have used OLS regression models to investigate the association between incident duration and certain traffic stream variables (Garib et al., 1997; Gomez, 2005) To our knowledge, however, no statistical analyses have been performed using our methodology to analyze recurrent congestion duration
To fit the regression models, the authors use real-world traffic stream data collected by the New Hampshire (NH) Depart-ment of Transportation at a collection site in Salem, NH, along the northbound lane of I-93 just north of an off-ramp, exit 1, and just south of an on-ramp A bottleneck occurs here as, immediately north of this location (downstream), I-93 is physically constricted from three to two lanes (Fig 2) In addition to this physical bottleneck, traffic volumes at this site exceed 100,000 vehicles per day (VPD) which far surpass the 60,000e70,000 VPD that the roadway was designed to accommodate (U.S Department of Transportation, 2016)
Because of this heavy, daily flow at the bottleneck, recurrent congestion occurs here As stated byBrilon et al (2005), such sites are ideal for the collection of congestion data
Data collection occurred between April 1 and November 30,
2010 During this time, side-fire radar devices intermittently measured traffic at irregular but frequent time periods about
1 min apart Data observations (raw data) consist of the following measurements: vehicle counts, average speed, oc-cupancy, and speed (spot speed) of individual vehicles observed over the interval Since incidents of recurrent congestion are the sole focus, data obtained during weekends and holidays are omitted from the analysis Also, because of scheduled maintenance as well as unscheduled“gaps” where the radar devices stopped collecting data (some lasted for several days), many other days were omitted In the end, 128 complete days of quality data were retained Without any
Fig 2e Illustration of collection site structure along northbound lanes of I-93
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Trang 5records of non-recurrent events during the collection period,
the authors assume that all congestion events observed on
these days are recurrent
Next, because radar data were collected over very short,
irregular time intervals, these measurements were aggregated
into uniform intervals of 15 min 15-min intervals are
rec-ommended to ensure“stable” flow rates that are especially
suitable for macroscopic/speed-flow analyses (Smith and
Ulmer, 2003; TRB, 2010) Specifically, harmonic averages were
calculated from flow and spot speed observations within each
15-min interval to produce an aggregated flow rate (q) and
aggregated speed (u) in units of vehicles per minute (vpm) and
miles per hour (mph), respectively (Daganzo, 1997) Thus, for
each day, aggregation yields ut and qt (speed and flow,
respectively) where t represents time-of-day with t¼ 1, 2, …,
96.Fig 3illustrates the speed and flow aggregates produced
for one week in April Also, Fig 3illustrates the stochastic
nature of breakdown, how high flows typically, but do not
necessarily, result in breakdown (and high, sustained flows
are more likely to result in breakdown) Lastly, the authors
note that the figure includes the flow profiles during a
weekend (the third and fourth mounds) where, as is
typically the case, no breakdowns occur This supports the
decision to remove weekend days from the analysis of
recurrent congestion
InFig 3, black dots indicate speeds less than 50 mph, the
critical threshold speed Notice that breakdown corresponds
to sustained, high traffic flows on the first, fifth, and seventh
days Also, notice that no breakdown occurs on the third
and fourth days, which is a weekend
2.4 Extraction of variables for binary logistic regression
model
From the speed and flow aggregates, the variables required by
the regression models can be easily extracted
First, to identify a response variable Y to distinguish
be-tween congested and freely flowing traffic (to identify a traffic
breakdown), a fixed speed threshold u*is typically chosen to
identify the transition between them (Banks, 2006; Brilon
et al., 2005; Geistefeldt and Brilon, 2009; Habbib-Mattar et al.,
2009; Lorenz and Elefteriadou, 2001; Yeon et al., 2009) No
standard approach exists for identifying u*, but based on
bimodal speed aggregates, u*¼ 50 mph was chose This
tran-sitional threshold is similar to those of Brilon et al (2005),
Geistefeldt and Brilon (2009), and Lorenz and Elefteriadou
(2001), Yeon et al (2009), who used fixed values of 47 mph (75 kph), 50 mph, 43 mph (70 kph), and 56 mph, respectively So, for some time t, ut> u*and utþ1> u*indicates breakdown at time t and thus Yt¼ 1 Among others,Lorenz and Elefteriadou (2001) used a similar approach to identify breakdown from traffic stream data If, on the other hand, ut > u* and
utþ1> u*
, no breakdown occurs at time t and thus Yt ¼ 0
This simple rule was applied to create a response Ytfor each time t in the data
Because the GLM form assumes the independent response values, we must select an independent set of Y values for model fitting First, since congested observations are typically separated by long time intervals and periods of freeflow, the authors may assume that the congested responses are, by their rare nature, independent of one another Next, the au-thors take a random sample of uncongested responses This random sampling“removes” the serial correlation from these data and destroys any time structure that exists in the aggregated values The authors then combine the two sets of responses, the congested and sampled uncongested values, to form an independent set of Y values suitable for our model form
Next, for each response Ytconsidered, the corresponding flow aggregate is observed at time t So, when breakdown occurs at time interval t, qt represents the flow observed immediately prior to breakdown (commonly called a “break-down flow”).Kondyli et al (2013), who pursued a variety of breakdown identification methods (speed-based, occupancy-based, and volume-occupancy/correlation-based), identified these “breakdown flows” in a similar situation When breakdown does not occur at time t, qt represents a traffic flow under freely flowing conditions
Lastly, for each response Yt, a corresponding categorical variable for time-of-day is simply identified based on time markers retained from the aggregation process
Despite starting with a time-ordered, correlated set of aggregated values, we have extracted a set of data that con-tains no temporal structure By considering an independent set of response values (see above) and their corresponding flow and time-of-day values, we have extracted an indepen-dent set of data that is suitable for a regression analysis
2.5 Extraction of variables for duration model
For our duration modeling, the authors must identify a response variable W that represents the time between onset of
Fig 3e Speed and flow aggregates for one week (April 1, 2010eApril 7, 2010)
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Trang 6breakdown and clearance of breakdown, the time required by
the system to clear a congestion event As previously
dis-cussed, time of breakdown is first identified via transition
from sustained speeds above to below the threshold u* Then,
clearance is similarly identified via transition from sustained
speeds below to above the threshold ut< u* and utþ1 > u*
indicate a clearance at time t Then, for each congestion event,
a duration W is simply calculated by the time between onset
and clearance This definition agrees withRao and Rao (2012)
who stated that the duration of congestion can be determined
by measuring reduced travel speeds over a period of time
Other more complicated techniques exist for duration
estimation by Elefteriadou et al (2011a,b) who used a
wavelet transform method to identify the start and end time
of congestion event occurrence, but this simpler method
was applied
For all congestion events, a histogram of congestion
du-rations is given below (Fig 4) This figure reveals a very
right-skewed distribution with some very long congestion events
The median congestion duration is 120 min
Lastly, based on time and date markers retained from the
raw data, time-of-day and day-of-the-week categorical
vari-ables were identified for each congestion event These
cate-gorical variables were matched with each duration amount to
create a dataset to be used for our duration regression model
Because there were one or two congestion events per day,
which were separated by a period of freeflow, each duration
measurement was assumed to be independent In an initial
data investigation, no temporal dependence between
dura-tions was observed Thus, this data is appropriate for our
regression model
3 Results and discussion
Based on the variables extracted from the I-93 data, the
lo-gistic and OLS regression models for breakdown and duration,
respectively, model fitting via maximum likelihood
estima-tion was performed using R statistical software
3.1 Binary logistic regression model for probability of breakdown
The binary logistic regression model was fit to the following data: flow, the continuous explanatory variable; time sector designation, the dichotomous categorical explanatory vari-able; and congestion state, the binary response Based on in-dividual p-value analyses (using a 5% significance level) and Chi-squared tests for change in deviance, both the flow and the dichotomous time variables were found to be highly sig-nificant explanatory variables in the prediction of breakdown
The interaction term, however, was not found to be significant and thus omitted from the model Thus, in terms of the probability curve for the two time sectors (AM- and PM-rush), (1) there is a significant shift, and (2) there is no significant change in trend Residual deviances indicate no significant evidence of lack-of-fit of the model
Based on our fitted models, an “S” shaped curve repre-senting probability of breakdown for observed flows bp was produced for both time periods (Fig 5) 95% confidence bands were included to illustrate uncertainty associated with probabilities at certain flow values
95% confidence bands (light and dark gray bands) are included to illustrate variability associated with logistic curves
First, the authors analyze the time-of-day variable The fitted coefficient associated with this dichotomous variable was found to be highly significant (z¼ 6.750, p z 0) The odds ratio, given by dOR¼ expðbb2Þ, is the relative increase in odds of congestion when going from AM-rush (d ¼ 0) to PM-rush (d¼ 1) The fitted parameter estimate of bb2¼ 2.17 corresponds
to an odds ratio of 8.76, or about an 8-fold increase in the odds
of breakdown when going from AM-to PM-rush When considering the 95% confidence interval for bb2, (1.54, 2.80), this odds ratio is between 4.66 and 16.44 in 95% confident, clearly a significant (non-zero) increase in odds between time periods
From the plot of the logistic curves for AM- and PM-rush pe-riods (Fig 5), this change in odds is observed in the clear shift
Fig 4e Frequency distribution (histogram) for congestion
durations
Fig 5e Fitted logistic curves for AM- and PM-rush time periods
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Trang 7between the two curves While some overlap between
confidence bands occurs at high flows, this is simply
because the uncertainty associated with the AM-rush period
(dark bands indicate 95% confidence for AM-rush period)
increases at high flows where few observations exist
It can be concluded that, for any flow, the probability of
breakdown during the PM-rush is higher than that during the
AM-rush (that PM-rush traffic behavior is generally more
likely to be congested) Observed traffic behavior supports this
finding as congestion is much more common during the
PM-rush period than during the AM-PM-rush In fact, of the
conges-tion events considered, about 80% are observed during the
PM-rush period Certainly, flow is the most important driver of
congestion, but our result suggests that time-of-day may play
a role The authors suspect that driver behavior during these
periods may contribute to this discrepancy, that PM-rush
commuters likely drive more aggressively In any event, the
results warrant further investigation and analysis Lastly,
steps to minimize congestion duration for the AM-rush, say,
may not be effective for the PM-rush
This result illustrates the limitation of a single, fixed
ca-pacity value as prescribed by the Highway Caca-pacity Manual
(HCM) According to the HCM guidelines, the freeway capacity
for a single lane of traffic is 2000 vehicles per hour (vph) Since
our traffic observations were taken across three traffic lanes,
the HCM capacity for these three lanes is 6000 vph or 100
ve-hicles per minute (vpm) At that flow, for flows of 100 vpm, the
probability of breakdown is somewhat low (around 20%)
during the AM-rush period, while probability of breakdown
during the PM-rush period for the same flow is about 70%
Since the interaction term was not significant and omitted
from our model (b3was not found to be significantly different
from zero), it is assumed that both the AM- and PM-rush
pe-riods share a common trend That is, the change in odds of
breakdown for a one unit increase in flow is the same for both
AM- and PM-rush periods This shared change in odds is the
odds ratio associated withb1, given by dOR¼ expðbb1Þ From our
fitted model results, the coefficient estimate of bb1¼ 0.03
cor-responds to an odds ratio of 1.031 Thus, for both periods, the
probability of breakdown increases 3.1% for every one
addi-tional vehicle per minute From the plot above, both periods
have probabilities that progress at the same rate relative to
flow (Fig 5)
From an administrator's point of view, there may be a
practical application of these results Most simply, results
from the breakdown model suggest that measures to remedy
recurrent congestion be focused on PM-rush periods, when
congestion is more likely For example, hard shoulder running
may be implemented during these congestion-prone hours to
mitigate expected congestion This practice is a proven
tech-nique for congestion mitigation in Northern Virginia
(Mazzenga and Demetsky, 2009), and the extension of its use is
recommended (Bauer et al., 2004) Furthermore, this practice
has been shown to effectively ease congestion without
increasing accident rates (ITS International, 2013) Or,
administrators may opt to introduce variable speed limits
(VSLs) during these congestion-prone hours Such VSLs are
used extensively in Europe with great success (Mazzenga
and Demetsky, 2009) As a final example, administrators
may implement ramp metering during the PM-rush It has
been shown that ramp metering is effective at limiting the number of vehicles entering a freeway and can help to prevent recurrent congestion (Texas A&M Transportation Institute, 2016)
3.2 Ordinary least squares regression model for congestion duration
The least squares regression model was fit to the following data: time-of-day, a dichotomous explanatory variable dis-tinguishing AM- and PM-rush time sectors; day-of-the-week, a categorical explanatory variable; and congestion duration, the continuous response Initially, the categorical variable repre-senting day-of-the-week contained a level (category) for each weekday, Monday through Friday However, after several model-fitting exercises, it was determined that most days were not significantly different from one another In fact, based on an analysis of variance, data collected from Monday, Tuesday, Wednesday, and Thursday are nearly identical
However, it was found that MondayeThursday data was significantly different from Friday data, so a dichotomous variable was used in lieu of a categorical variable for day-of-week Also, initial models suggested a lack of normality among the residuals To remedy this, a transformation of the response was pursued Because duration times are highly right-skewed, and because variance seems to increase with duration, a log-transformation was applied (Fig 4) Therefore, the slightly modified model is as follow
logðWÞ ¼ b0þ b1d1þ b2d2þ b3d1d2þ e (5) where d1is a dichotomous variable distinguishing AM- and PM-rush periods, d2is a dichotomous variable distinguishing Fridays from all other days (d2¼ 1 if Friday, d2¼ 0 if Monday, Tuesday, Wednesday, or Thursday), d1d2 is the interaction term Similarly, Garib et al (1997) used linear regression models to predict the log of duration of traffic incidents observed from California freeways (non-recurrent events)
While the model fits the data surprisingly well (R2suggests the model form explains about 67% of the total variation observed in log(Duration) measurements), a “better” model was not pursued While attempting to find a model that maximizes the explained variation is a useful exercise, our aim is to identify significant explanatory variables and their implications on congestion duration
Of primary importance from our fitted result is the statis-tical significance of the interaction term (t ¼ 3.277, p-value¼ 0.00189) Both main effects, time-of-day and day-of-the-week, are also found to be statistically significant (t¼ 4.323, p-value ¼ 7.15e-05; t ¼ 4.455, p-value ¼ 4.61e-05, respectively), although main effects are typically ignored in the presence of interaction That is, if two terms interact, changes in both explanatory variables will have an effect on the response outcome In our case, the effect on the mean outcome, log(Duration), from a change in time-of-day de-pends on the level of day-of-the-week This effect is most easily seen in the plots below (Fig 6) Interaction lines calculated from the fitted regression model are included
The two plots above represent the same phenomenon from two different perspectives: grouping the data by day-of-the-week (Fig 6(a)) and time-of-day (Fig 6(b)) In either case, a
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Trang 8ordinal interaction is marked by the clearly non-parallel
interaction lines.Fig 6(b) is probably the most interpretable
as the moving along the x-axis represents transition from
AM-to rush periods Here, when going from AM-to
PM-rush periods (x-axis grouping), the change in log(Duration)
depends on day-of-the-week For MondayeThursday, there
is an increase in log(Duration), while there is a decrease in
log(Duration) for Fridays Analysis of the fitted parameters
allows us to precisely quantify this effect After converting
back to the original scale of measurement (duration in
minutes instead of log(Duration)), calculations reveal that
for MondayeThursday traffic, transitioning from AM-rush
period to PM-rush period results in more than a 2-fold
increase (273%) in duration On the other hand, for Friday
traffic, transitioning from AM-to PM-rush periods results in a
59% decrease in duration Similarly, fromFig 6(a), there is a
distinct change in log(Duration) when going from Monday to
Thursday to Fri categories, and the magnitude of the change
depends on the time-of-day For the AM-rush, this change
(from Monday to Thursday to Fri.) is a steep increase in
log(Duration) For the PM-rush period, this change is
increasing as well, but more modestly
Results from our duration modeling are a bit surprising
The authors can speculate that Friday“evening” commutes
may be occurring earlier in the day Since our analysis did not
identify a significant number of congestion events in the early
afternoon, though, this idea was abandoned Perhaps the
evening commute is spread across a wide time span and
re-sults in fewer long congestion events This conjecture would
agree with the results presented in this analysis In any event,
this result may warrant a more day-specific analysis of
congestion events If our results are confirmed, and
Mon-dayeThursday traffic experiences longer congestion events,
perhaps administrators should implement congestion
miti-gation techniques during these days or possibly extend these
mitigation techniques later into the evening, beyond
tradi-tional“rush hours” As mentioned above, these methods to
remedy such congestion may be hard shoulder running, VSLs,
or ramp metering
Lastly, the regression/model fitting diagnostics suggest
that the regression assumptions are all met (that our model
specification is appropriate for this data) Plots of residuals versus fitted values show constant variance and normality plots show the residuals to be normally distributed Also, there was no indication of correlated errors, further evidence that the duration response values are completely uncorrelated and suitable for the OLS format
4 Conclusions
Recurrent congestion is a complicated process that is likely triggered by a variety of interconnected conditions That said, understanding the frequency and duration of these events is critical to highway administrators and decision-makers The goal of this study is not to create a real-time forecasting tool, but rather to identify model structures that reveal the importance, driving mechanisms of the congestion process
Such structures will contain information that is easily inter-preted and understood
In our binary logistic regression model to predict proba-bility of breakdown by flow, results show a distinct shift by time-of-day For any flow, the PM-rush period has higher probability of breakdown than the AM-rush period Overall, based on the parameter estimate associated with the time-of-day, there is an estimated 8-fold increase in the odds of breakdown when going from AM-to PM-rush So, while flow is the primary predictor of congestion, our result indicates that time-of-day is likely a factor as well From an administrator's point of view, this may suggest that measures to remedy recurrent congestion should be focused on PM-rush periods
As previously mentioned, these remedial measures may include hard shoulder running, variable speed limits (VSLs), or ramp metering
By considering time-of-day as a simple proxy for secondary traffic characteristics, the result from our logistic regression analysis may suggest that driver behavior affects congestion and roadway capacity That is, one could speculate that there are more aggressive drivers during the PM-rush hour, and these aggressive drivers are prone to excessive weaving, etc
With more detailed data, future work could investigate these
Fig 6e Boxplot for congestion duration (a) MondayeThursday and Friday traffic by time-of-day (b) AM- and PM-rush
periods for day-of-the-week
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Trang 9specific, underlying factors rather than time-of-day
categories
In our duration model, an ordinary least squares regression
model is used to predict congestion duration using
time-of-day and time-of-day-of-the-week categorical variables After an initial
investigation, it suffices to use dichotomous variables for
time-of-day, AM- or PM-rush, and day-of-the-week,
Mon-dayeThursday or Friday The primary result from this model
is the statistically significant interaction term This suggests
that when going from AM-to PM-rush periods, the change in
congestion duration depends on day-of-the-week In fact, this
change is increasing for MondayeThursday and decreasing
for Fridays This surprising result may suggest the need for
more day-specific analysis of congestion events Ultimately,
administrators may choose to focus congestion mitigation
techniques (hard shoulder running, VSLs, ramp metering, etc.)
during MondayeThursday, evening commute hours
This work represents an empirical study on breakdown
and delay at a known bottleneck in Salem, NH Our respective
model specifications successfully identify significant
pre-dictors that shed light upon the congestion process A
limi-tation of this analysis, however, is that our conclusions are
restricted to the one location from which data was collected
That is, our results are not transferable to other locations,
even those very near our location or in other directions Data
from additional bottlenecks will allow for transferable results,
a general model that could explain the congestion process
across diverse locations That says, the results here are
justi-fied and illuminate some congestion phenomena that are
likely not site-specific
Uncited reference
Texas A and M Transportation Institute, 2015
Q2
r e f e r e n c e s
Athol, P.J., Bullen, A., 1973 Multiple ramp control for a freeway
bottleneck Highway Research Record 456, 50e54
Ban, X., Chu, L., Benouar, H., 2007 Bottleneck identification and
calibration for corridor management planning Transportation
Research Record 1999, 40e53
Banks, J., 2006 New Approach to Bottleneck Capacity Analysis:
Final Report UCB-ITS-PRR-2006-13 California PATH, Institute
of Transportation Studies, University of California, Berkeley
Bauer, K., Harwood, D., Hughes, W., et al., 2004 Safety effects of
narrow lanes and shoulder-use lanes to increase capacity of
urban freeways Transportation Research Record 1897, 71e80
Bertini, R.L., Cassidy, M.J., 2002 Some observed queue discharge
features at a freeway bottleneck downstream of a merge
Transportation Research Part A: Policy and Practice 36 (8),
683e697
Brilon, W., Geistefeld, J., Regler, M., 2005 Reliability of freeway
traffic flow: a stochastic concept of capacity In: 16th
International Symposium of Transportation and Traffic
Theory, College Park, 2005
Daganzo, C., 1997 Fundamentals of Transportation and Traffic
Operations Pergamon, Oxford
Dong, J., Mahmassani, H.S., 2009a Flow breakdown, travel
reliability and real-time information in route choice behavior
In: 18th International Symposium on Transportation and Traffic Theory, Hong Kong, 2009
Dong, J., Mahmassani, H.S., 2009b Flow breakdown and travel time reliability Transportation Research Record 2124, 203e212
Downs, A., 2004 Still Stuck in Traffic: Coping with Peak-Hour Traffic Congestion The Brookings Institution Press, Washington DC
Elefteriadou, L., Kondyli, A., Washbum, S., et al., 2011a
Proactive ramp management under the threat of freeway flow breakdown Procedia-Social and Behavioral Sciences 16, 4e14
Elefteriadou, L., Martin, B., Simmerman, T., et al., 2011b Using Micro-Simulation to Evaluate the Effects of Advanced Vehicle Technologies on Congestion 2009-06 University of Florida Transportation Research Center, Gainesville
Elefteriadou, L., Roess, R.P., McShane, W.R., 1995 Probabilistic nature
of breakdown at freeway merge junctions Transportation Research Record 1484, 80e89
Evans, J., Elefteriadou, L., Gautam, N., 2001 Determination of the probability of breakdown on a freeway based on zonal merging probabilities Transportation Research Part B:
Methodological 35, 237e254
Falcocchio, J.S., Levinson, H.S., 2015 Road Traffic Congestion: a Concise Guide Springer International, Switzerland
Garib, A., Radwan, A.E., Al-Deek, H., 1997 Estimating magnitude and duration of incident delays ASCE Journal of Transportation Engineering 123 (6), 459e466
Geistefeldt, J., Brilon, W., 2009 A comparative assessment of stochastic capacity estimation methods In: 18th International Symposium on Transportation and Traffic Theory, Hong Kong, 2009
Giuliano, G., 1989 Incident characteristics, frequency, and duration
on a high volume urban freeway Transportation Research Part A: General 23 (5), 387e396
Gomez, N.M., 2005 Incident Duration Model and Secondary Incident Causation Model Based on Archived Traffic Management Center Data (Master thesis) Vanderbilt University, Nashville
Habbib-Mattar, C., Abishai, P., Cohen, M.A., 2009 Analysis of the breakdown process on congested freeways Transportation Research Record 2124, 58e66
Hao, Y., Xu, T., Sun, L., 2007 Analysis and control of recurrent traffic congestion on urban expressway In: International Conference on Transportation Engineering, Chengdu, 2007
Hallenbeck, M.F., Ishimaru, J.M., Nee, J., 2003 Measurement of Recurring Versus Non-Recurring Congestion: Technical Report
WA-RD 568.1 Washington State Department of Transportation, Seattle
Hosmer, D.W., Lemeshow, S., Sturdivant, R.X., 1989 Applied Logistic Regression John Wiley, New York
ITS International, 2013 Traffic Monitoring and Hard Shoulder Running Available at: http://www.itsinternational.com/
categories/detection-monitoring-machine-vision/features/
traffic-monitoring-and-hard-shoulder-running/ (Accessed 1 May 2016)
Jones, B., Janssen, L., Mannering, F., 1991 Analysis of the frequency and duration of freeway accidents in seattle
Accident Analysis and Prevention 23 (4), 239e255
Kerner, B.S., 2004 The Physics of Traffic Springer, New York
Kerner, B.S., 2009 Introduction to Modern Traffic Flow Theory and Control Springer, New York
Kondyli, A., Elefteriadou, L., Brilon, W., et al., 2013 Development and evaluation of methods for constructing breakdown probability models ASCE Journal of Transportation Engineering 139 (9), 931e940
Lorenz, M., Elefteriadou, L., 2001 Defining freeway capacity as function of breakdown probability Transportation Research Record 1776, 43e51
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
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52
53
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65
66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130
Trang 10Mazzenga, N.J., Demetsky, M.J., 2009 Investigation of Solutions to
Recurring Congestion on Freeways Virginia Transportation
Research Council, Charlottesville, 2009
Nam, D., Mannering, F., 2000 An exploratory hazard-based
analysis of highway incident duration Transportation
Research Part A: Policy and Practice 34 (2), 85e102
Newell, G.F., 1993 A simplified theory of kinematic waves in
highway traffic I: general theory II: queuing at freeway
bottlenecks III: multi-destination flows Transportation
Research Part B: Methodological 27 (4), 289e303
Ossenbruggen, P., 2016 Assessing freeway breakdown and
recovery: a stochastic model ASCE Journal of Transportation
Engineering 142 (7), 04016025
Paniati, J.F., 2003 Using Intelligent Transportation Systems (ITS)
Technologies and Strategies to Better Manage Congestion
ITS Joint Program Office, Washington DC
Persaud, B., Yagar, S., Brownlee, R., 1998 Exploration of the
breakdown phenomenon in freeway traffic Transportation
Research Record 1634, 64e69
Persaud, B., Yagar, S., Tsui, D., et al., 2001 Study of
breakdown-related capacity for a freeway with ramp metering
Transportation Research Record 1748, 110e115
Rao, A.M., Rao, K.R., 2012 Measuring urban traffic congestion: a
review International Journal for Traffic and Transport
Engineering 2 (4), 286e305
Shaw, T., 2003 NCHRP Synthesis 311: Performance Measures of
Operational Effectiveness for Highway Segments and
Systems Transportation Research Board, Washington DC
Smith, B.L., Ulmer, J.M., 2003 Freeway traffic flow rate
measurement: an investigation into the impact of the
measurement time interval Journal of Transportation
Engineering 129 (3), 223e229
Stathopoulos, A., Karlaftis, M.G., 2002 Modeling duration of
urban traffic congestion ASCE Journal of Transportation
Engineering 128 (6), 587e590
Sullivan, E.C., 1997 New model for predicting freeway incidents
and incident delays ASCE Journal of Transportation
Engineering 123 (4), 267e275
Tebaldi, C., West, M., Karr, A.F., 2002 Statistical analyses of
freeway traffic flows Journal of Forecasting 21 (1), 39e68
Texas A&M Transportation Institute, 2015 Traffic Gridlock Sets New Records for Traveller Misery Available at: http://
mobility.tamu.edu/ums/media-information/press-release/
(Accessed 1 July 2015)
Texas A&M Transportation Institute, 2016 Ramp flow control e Urban Mobility Information Available at: http://mobility
tamu.edu/mip/strategies-pdfs/active-traffic/technical-summary/Ramp-Flow-Control-4-Pg.pdf (Accessed 1 June 2016)
TRB, 2010 Highway Capacity Manual Transportation Research Board, Washington DC
U.S Department of Transportation, 2016a Bridging Multiple Communities: New Hampshire's I-93 Improvement Project
Available at:http://www.fhwa.dot.gov/construction/accelerated/
wsnh06.pdf(Accessed 1 May 2016)
U.S Department of Transportation, Federal Highway Administration, 2016b Reducing Recurring Congestion
Available at: http://ops.fhwa.dot.gov/program_areas/reduce-recur-cong.htm(Accessed 1 May 2016)
Vlahogianni, E., Karlaftis, M.G., Kepaptsoglou, K., 2011 Nonlinear autoregressive conditional duration models for traffic congestion estimation Journal of Probability and Statistics
2011, 798953
Yeon, J., Hernandez, S., Elefteriadou, L., 2009 Differences in freeway capacity by day-of-the-week, time-of-day, and segment type ASCE Journal of Transportation Engineering
135 (7), 416e426
Eric M Laflamme is a professor of Mathe-matics at Plymouth State University in Ply-mouth, NH He received a Master's degree in Applied Statistics from Cornell University and a PhD in Mathematics from the Uni-versity of New Hampshire His areas of research are transportation and extreme value theory related to climate change projections
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