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Popuri, Ben-Akiva and Proussaloglou 1
Time of Day Modeling in a Tour-Based Context:
The Tel-Aviv Experience
Yasasvi Popuri
*
Associate, Cambridge Systematics, Inc.
115 South LaSalle Street, Suite 2200, Chicago, IL 60603
Tel: 312-346-9907, Fax: 312-346-9908, E-mail: ypopuri@camsys.com
Moshe Ben-Akiva
Massachusetts Institute of Technology
77 Massachusetts Avenue, Room 1-181, Cambridge, MA 02139
Tel: 617-253-5324, Fax: 617-253-0082, E-mail: mba@mit.edu
Kimon Proussaloglou
Principal, Cambridge Systematics, Inc.
115 South LaSalle Street, Suite 2200, Chicago, IL 60603
Tel: 312-346-9907, Fax: 312-346-9908, E-mail: kproussaloglou@camsys.com
Total Word Count: 7,627 (Includes 2 Tables, 4 Figures)
Final Submission Date: April 1, 2008
Original Submission Date: August 1, 2007
* Corresponding Author
Popuri, Ben-Akiva and Proussaloglou 2
ABSTRACT
Insights into the time of day preferences of individuals are crucial to accurately quantify vehicle
emissions and understand time shifting behavior in response to traffic congestion and road
pricing. Most urban models account for travel demand by time of day using fixed factors or by
incorporating the choice among aggregate time periods such as AM, PM and off-peak. These
methods are inadequate for examining traveler response to congestion mitigation strategies. To
address some of these concerns, the Federal Highway Administration (FHWA) recently
conducted a research project to develop innovative methods of modeling travel by time of day.
This paper extends the FHWA methodology to the Tel-Aviv tour-based model system that
Cambridge Systematics is currently developing for Israel’s Ministry of Transport. The purpose
of this paper is three-fold: first, it discusses the modeling framework used in Tel-Aviv; second, it
describes the data inputs required for modeling time of day decisions; and finally, it describes the
model estimation procedure and empirical results from Tel-Aviv. The main features of the
modeling approach include using half-hour time intervals, accounting for schedule delay in the
absence of desired arrival and departure time data, and modeling the 24-hour cycle. This paper
serves as a proof of concept for the FHWA methodology and demonstrates that using the
commonly available household survey data and some basic level-of-service data, it is possible to
develop time of day models that are more detailed and better suited for policy testing.
Key Words: Time of day choice, tour-based models, travel demand models, schedule delay,
discrete choice models.
Word Count: 239
Popuri, Ben-Akiva and Proussaloglou 3
1. BACKGROUND AND MOTIVATION
An individual’s travel behavior is characterized by a series of important decisions including, but
not limited to, the choice of frequency, location, mode and route of travel. Most urban travel
models explicitly address these four dimensions through the four-step model system: trip
generation, distribution, mode choice and assignment. However, time of day has traditionally
received little or no attention. This is in part because travel demand models were originally
developed to evaluate the impact of highway improvements on overall daily traffic volumes (1).
Over the past two decades, the emphasis has gradually shifted from the evaluation of long-term
improvements to understanding the impact of shorter term congestion mitigation strategies such
as road pricing. The need for modeling mobile source emissions and air quality has also
necessitated higher temporal resolution in travel demand forecasts.
Many urban models account for time of day using fixed factors that apportion daily travel
demand to different time periods. These factors are typically derived from household travel
surveys and have the advantage of being specific to the urban area under consideration.
However, the fixed factor approach cannot capture shifts in time of travel due to traffic
congestion and is not sensitive to the effect of traffic management strategies such as congestion
pricing (2). Furthermore, this approach assumes that the time of day factors remain the same for
both the base and forecast years and does not account for the impact of changes in demographic
and employment characteristics over a long period of time.
An improvement over the fixed factor approach is to model the “choice” of trip departure
time periods, usually represented as large contiguous time windows such as AM peak, PM peak
and off-peak periods (3-5). These models have the advantage of being sensitive to level-of-
service measures and individual socio-economic characteristics. However, the large size of the
time periods precludes the finer temporal analysis that is required for air quality modeling and
evaluation of congestion mitigation strategies. Moreover, time period models are based on a pre-
supposed partitioning of the day into large time intervals. It is, therefore, possible that different
definitions of the time periods may lead to different model results (6).
Researchers have widely acknowledged these model deficiencies and proposed several
methodological improvements that vary either by the level of temporal analysis or the
complexity of the model structure. Notable improvements in the level of temporal analysis
include the representation of time of day on a continuous time scale (6-8). Alternative model
structures proposed to model time of day include nested logit (9), ordered generalized extreme
value (10), multinomial probit (11), and error components logit models (12, 13). These models
are mathematically complex and are not always easy to implement, resulting in the limited
implementation of this research in regular urban travel models.
The recent TRB Special Report 288, Metropolitan Travel Forecasting: Current Practice
and Future Direction (14), emphasized the need for more advanced time of day modeling in
practice. The report recognized that representing the various pricing outcomes would require the
shift to more advanced tools such as activity or tour-based models (15). While some models have
addressed time of day decisions within a tour-based context (16), these studies have largely
analyzed aggregate time periods. Some other studies that used more detailed time periods, have
Popuri, Ben-Akiva and Proussaloglou 4
not addressed the cyclical nature of time of day decisions, that is, the recurrent features in time of
day decisions from one 24-hour period to the next (17).
Many of these limitations were addressed by a recent research project conducted by the
FHWA (18-20). This project contributed to the existing body of literature in several ways: first,
it developed a framework for modeling time of day decisions at all stops on a given tour; second,
it used half-hour time intervals, allowing for a detailed temporal analysis of travel demand; third,
it proposed a way of accounting for schedule delay when data on desired departure and arrival
times are unavailable; and fourth, it established a way of dealing with the 24-hour cycle of
departure and arrival time decisions.
This paper extends the FHWA methodology to the Tel-Aviv tour-based model system
that Cambridge Systematics is currently developing for Israel’s Ministry of Transport. The
purpose of this paper is three-fold: first, it describes the time of day modeling framework used in
Tel-Aviv; second, it identifies data issues and describes a way of generating travel times that are
necessary for modeling half-hour time intervals; and finally, it shares the model estimation
experience from Tel-Aviv. The focus of this paper is on the estimation of detailed time of day
models; the application process and results will be addressed in future papers.
The remainder of this paper is organized as follows. Section 2 discusses the time of day
modeling framework adopted in Tel-Aviv. Section 3 describes a way of generating travel times
necessary for modeling half-hour time intervals. Section 4 presents the model estimation process
and the empirical results. Finally, Section 5 presents the conclusions from this study.
2. TEL-AVIV TIME OF DAY MODELING FRAMEWORK
The Tel-Aviv tour-based model system is being developed as part of a new transportation master
plan for the city. The model is geared towards evaluating traveler response to transportation
policies such as parking restrictions and road pricing in addition to new highway and transit
services. The Israel National Travel Habits Survey (NTHS) conducted in 1996 is the primary
data source for model development. The survey sampled nearly one percent of the population in
Tel-Aviv and included travel information from nearly 8,000 households. The NTHS was later
supplemented by additional travel data from a rail corridor survey, intended to capture the travel
behavior of households along an increasingly important rail corridor.
The tour-based model system comprises a hierarchy of logit and nested logit models for
the main stop of the tour, namely, activity type, time of day, destination and mode. The
intermediate stops are subsequently modeled conditional on the main activity stop models. The
Tel-Aviv model accounts for up to two tours for each person: the most important tour of the day,
referred to as the primary tour, and the second most important tour of the day, or the secondary
tour (see (21) for a detailed discussion of the Tel-Aviv model structure).
Time of day features at two different levels in the Tel-Aviv model hierarchy. First, a logit
model accounts for the joint choice of time period of departure from home and from the main
activity stop of the tour. The time periods modeled include early morning, AM peak, mid-day,
PM peak and late evening. Apart from the socio-economic and geographic variables, this model
uses logsum measures from the destination and mode choice models. The purpose of this model
Popuri, Ben-Akiva and Proussaloglou 5
is primarily to determine the most likely departure time periods from home and main activity
stop, which will subsequently be used to determine the appropriate level of service data for the
mode and destination choice models. The second level time of day model is a post-processor to
the travel demand model system and is designed to be a policy analysis tool. This model looks at
the time of day choices at a greater level of temporal resolution for auto modes and will be the
primary focus of this paper. Owing to space limitations, only primary work tour models are
discussed in this paper.
The Tel-Aviv model recognizes that the time of day decisions at various stops in a tour
are interconnected. At the main activity stop, the model addresses the joint choice of arrival and
departure times. The travel day starts at 3:00 AM in the morning and ends at 3:00 AM the next
day. A total of 36 time slots are used to discretize the continuous time scale for modeling arrival
and departure times. The first time slot extends from 3:00 AM to 5:00 AM and the last time slot
extends from 10:00 PM to 3:00 AM the next day. The remaining 17 hours between 5:00 AM and
10:00 PM are divided into 34 half-hour time slots.
If
a
and
d
represent the arrival and departure time slots respectively, the joint utility
),U( da
is modeled as:
dada
main
d
home
main
main
a
main
home
main
a
main
d
main
d
main
a
TtTt
TTTTda
,21
21
)ln(*)ln(*
)|(*)|(*
)()()(),U(
εδδ
ττ
γβα
+∆+∆
++
+−++=
(1)
Where,
),U( da represents the utility of arriving at the main stop in time slot
a
and departing from the
main stop in time slot
d
, such that
a
,
d
}36,...,2,1{
∈
(the 36 time slots described above)
,
and
da
≤
(the arrival time at the main stop needs to precede the departure time from the main stop).
Thus, there are a total of 666 (=36 * (36+1)/2) utilities corresponding to the various arrival and
departure time combinations;
main
a
T
and
main
d
T
represent the mid-points of the arrival and departure time slots, measured in
hours from an arbitrary time such as midnight;
)(
main
a
T
α
and )(
main
d
T
β
represent the arrival and departure time functions that capture, among
other things, the effect of schedule delay in the absence of data on desired time of arrival and
departure. Schedule delay is defined as the disutility caused by traveling at times other than the
desired times of travel (
22,23
). The form of the arrival and departure time functions and the
estimation of the parameters that define them are discussed in greater detail in Section 4;
)(
main
a
main
d
TT −
γ
represents the duration function for the primary work tour, discussed in greater
detail in Section 4. The duration function represents the utility associated with the time spent at
the destination. The duration at the destination itself is calculated as the difference in hours
between the mid-point of the departure and arrival time slots;
Popuri, Ben-Akiva and Proussaloglou 6
main
a
main
home
Tt |
is the travel time from home to the main stop for the arrival time
main
a
T
at the main
stop, and
main
d
home
main
Tt | is the travel time from the main stop to home for the departure time
main
d
Tfrom the main stop.
1
τ
and
2
τ
are multiplicative parameters that need to be estimated;
)ln(
a
∆
and )ln(
d
∆
are the log-size measures that are introduced to account for the unequal size of
the 36 time slots that are used to discretize continuous time.
a
∆
and
d
∆
are defined as the
number of half-hour intervals in the arrival and departure time slots respectively.
1
δ
and
2
δ
are
multiplicative parameters that need to be estimated. The size terms are essentially correction
terms applied to the first and the last time slots, which are both larger than 30 minutes in
duration;
finally,
da
,
ε
are the error components assumed to be independent and identically Gumbel
distributed.
The probability of an individual arriving at the main stop in time slot
a
and departing
from the main stop in time slot
d
is given by the logit formula (
24
):
∑
≤∈∀
=
''''
''
},36,..2,1{,
),V(
),V(
),(
dada
da
da
e
e
daP
(2)
Where,
),V( da
represents the systematic (non-random) component of the utility function defined
in equation (1).
The time of day choices at the intermediate stops are conditional on the arrival and
departure time choices at the main stop. For the intermediate stop on the half-tour from home to
the main stop, the departure time
bef
d
T
may be determined as:
)|(
main
a
main
bef
main
a
bef
d
TtTT −=
(3)
Where,
main
a
main
bef
Tt |
is the travel time between the before-main stop and the main stop for the
arrival time
main
a
T
at the main stop. Therefore, at the before-main stop, only the arrival time
bef
a
T
is modeled.
Similarly, for the intermediate stop on the half-tour from the main stop to home, the
arrival time
aft
a
T
may be determined as:
)|(
main
d
aft
main
main
d
aft
a
TtTT +=
(4)
Popuri, Ben-Akiva and Proussaloglou 7
Where,
main
d
aft
main
Tt | is the travel time between the main stop and the after-main stop for the
departure time
main
d
Tfrom the main stop. Therefore, at the after-main stop, only the departure
time
aft
d
Tis modeled.
Thus, both the intermediate stop time of day models have a maximum of 36 alternatives.
The form of the utility expressions is generally very similar to that shown in equation (1), with
the exception that only the relevant arrival or departure terms feature in the utility expression.
Given the higher importance of the time of day decisions at the main activity stop of the tour,
this paper limits the discussion to the time of day decisions at the main stop.
As evident from equations (1), (3) and (4), a key data input in modeling time of day is the
pertinent origin-destination trip travel time (such as
main
a
main
home
Tt | and
main
d
home
main
Tt | for the main stop
models). Because the Tel-Aviv model uses 36 time periods to discretize a travel day, the travel
times are also desired at the same temporal resolution for model estimation. Travel time
estimates are not readily available for each detailed time slot and must be generated from the
available network travel times for two or three time periods. The next section describes the
generation of travel times for the detailed time slots.
3. TRAVEL TIME GENERATION
The Tel-Aviv model produces preliminary network travel times for three periods: AM (between
6:30 and 8:30 AM), PM (between 3:00 and 8:00 PM) and off-peak (rest of the day). This
information had to be translated into travel times for each origin-destination pair for all the 36
time slots used for modeling. As with the other travel diary surveys, the NTHS recorded the
travel time of each trip in a respondent’s itinerary. The reported travel times were the only source
of travel time information that was specific to the exact time slot of travel. Therefore, a
regression methodology was developed to relate the reported times to the network travel times.
The rationale behind this procedure was to develop a formula that allowed the calculation of
travel time for any origin-destination pair and for any of the 36 time slots, given the network
travel times for the three time periods and the geographic characteristics of the origin and
destination zones. The reported travel times are often subject to random error due to respondent
rounding. To reduce this error, the data were thoroughly screened for outliers.
It must be noted that the expression in equation (1) uses two travel times,
main
a
main
home
Tt | and
main
d
home
main
Tt | . Therefore, two types of travel times needed to be estimated: first, the trip travel time
given the arrival time at the destination of the trip; and second, the trip travel time given the
departure time from the origin of the trip. A wide array of regressions was tested before
finalizing the expression shown in equation (5) below.
Popuri, Ben-Akiva and Proussaloglou 8
( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
ε
βββ
βββ
βββ
ββββ
β
ββ
ββ
ββ
+
++++
++++
++++
++++
+
+
++
++
++=
Delay*
ndestinatiosuburban
ndestinatiourban ndestinatio CBD
originsuburban originurban
origin CBDdistancelnIntercept
Speed Freeflow
Speed Reported
ln
4
424
2
422421
4
320
2
318317
4
216
2
214213
4
112
2
110198
7
65
43
21
TgTgTg
TgTgTg
TgTgTg
TgTgTg
L
L
L
L
(5)
The dependent variable in the regression was the logarithm of the ratio of reported speed
to the freeflow speed. Speeds were used instead of travel times to incorporate the distance effect.
The reported speed was calculated as the distance between the origin and the destination divided
by the reported travel time from the survey. Because reported travel times typically include out-
of-vehicle components as well, the freeflow speed was calculated as the distance between the
origin and destination divided by the corresponding network freeflow travel time plus any out-
of-vehicle components such as parking search and walk times. This ensured consistency between
the reported and freeflow speeds.
The ln(distance) variable was used to capture the effect of trip length on speed. The
expectation was that the speeds for longer trips would be higher than speeds associated with
shorter trips. The area type indicators (CBD, urban, suburban) were introduced with the rural
area type as the base variable. The coefficients on all of these indicators were expected to be
negative implying that, ceteris paribus, travel speeds are lower if either the origin or the
destination, or both are in the CBD, urban areas, or suburban areas.
Perhaps the most important variable in the regression is the delay term, defined as
follows:
−= speed freeflownetwork
speedpeak network
1Delay
(6)
Network peak speed was defined as the AM peak speed if the departure (or arrival) time
was in the AM peak, PM peak speed if the departure (or arrival) time was in the PM peak, and
the minimum of AM and PM peak speeds, if the departure (or arrival) time was in the off-peak
period. Network freeflow speed was directly obtained from the off-peak travel speeds. The delay
variable captures the congestion effects of the network speeds and when interacted with a
cyclical function of arrival or departure time T (where T is measured as hours elapsed from an
arbitrary time, such as midnight), replicates the daily pattern of travel speeds. The functions
(
)
Tg
1
,
(
)
Tg
2
,
(
)
Tg3 and
(
)
Tg
4
are cyclical with a period of 24 hours, given by:
( )
=24
2
sinexp
1
T
Tg
π
,
( )
=24
2
cosexp
2
T
Tg
π
,
Popuri, Ben-Akiva and Proussaloglou 9
( )
=24
4
sinexp
3
T
Tg
π
,
( )
=24
4
cosexp
4
T
Tg
π
(7)
The regression shown in equation (5) was estimated using the screened trip-level data set
described earlier. The estimation results are shown in Table 1. The parameter on the ln(distance)
variable was positive and significant in both the regressions, corroborating the apriori
expectation that the speeds would be higher for longer trips. The area type dummy variables
were included to control for the effect of land use on travel times. The rural zone indicators were
used as the base variables for estimation. All the area type indicators except the suburban
destination variables, showed negative and significant coefficients, indicating lower speeds for
all CBD and urban trips and, for all trips originating from the suburban areas.
While some of the delay variables in the regression were not significantly different from
zero, the overall profile of the delay terms closely replicated observed speed patterns. Figure 1
shows the profiles of the delay terms for the arrival time and departure time-based regressions.
The figure indicates that the delay coefficients, and thereby the predicted travel speeds, are the
lowest between 7:30 and 8:00 AM in the morning and between 4:30 and 5:00 PM in the evening.
The figure also indicates a “lag” of about 30 minutes between the arrival and departure time
delay profiles during the peak hours of the day. This implies that the average travel time across
all types of trips is about 30 minutes in the peak periods, a statistic that is corroborated by the
NTHS data.
Using simply the NTHS data on reported travel times and the basic network travel time
data, the regression methodology allowed for the calculation of speeds and trip travel times,
based on the arrival time at the destination as well as the departure time from the origin, for each
of the 36 time slots. These travel time estimates were subsequently used to model the arrival and
departure time choice at the main stop, as indicated in equation (1).
It must be noted that the travel time regression methodology presented here provides an
approximation of reported times. It is merely a substitute until fully dynamic models are
developed. The regression model could be improved to better reflect the change in the travel time
variation across the day resulting from congestion pricing policies.
Data preparation for the joint modeling of arrival and departure time choice at the main
stop involved several steps. First, for each tour record, information on home zone, main stop
zone and socio-economic characteristics was assembled from the NTHS data. Second, using the
home and main stop zones, network travel times were attached from the skim matrices. Third,
the travel times were converted to travel speeds after accounting for the out of vehicle time
components. Finally, the parameters in Table 1 were used to calculate two sets of “reported
speeds” for the 36 time slots. The first set, based on the arrival times, simply indicates the travel
speed (or the time) between home and the main stop that would have been reported had the
respondent arrived at the main stop in a given time slot. The second set, based on departure
times, indicates the travel speed between the main stop and home that would have been reported
had the respondent departed from the main destination in a given time slot. These speeds were
then converted back to travel times; because there are a total of 666 combinations of arrival and
departure time slots, there are also 666 combinations of travel times from home to main stop and
main stop to home. The joint arrival and departure time choice model included all the 666
Popuri, Ben-Akiva and Proussaloglou 10
combinations without sampling, and was estimated using ALOGIT software. The modeling
procedure and results are detailed in the next section.
4. MODEL ESTIMATION METHODOLOGY AND RESULTS
The model structure for the joint choice of arrival and departure times at and from the main stop
has been detailed in Section 2. Since time is continuous, the effect of any variable that is
included in the utility equations should also be continuous as a function of time. The use of
alternative-specific constants and alternative-specific variables (such as socio-economic dummy
variables) would result in a discontinuous utility function. For this reason, equation (1) includes
continuous arrival time, departure time, and duration functions interacted with socio-economic
variables. Using the continuous time functions instead of alternative-specific constants also
reduces the number of parameters that need to be estimated.
Ben-Akiva and Abou Zeid (20) show that the continuous arrival and departure time
functions also capture the effect of schedule delay, which is the disutility caused by traveling at
times other than the desired times of arrival or departure. They point out that when desired
arrival and departure time data are not available (as is usually the case with most travel diary
surveys), if the desired times are assumed to be the same for a given market segment, then the
continuous arrival and departure time functions reflect, among other things, the effect of
schedule delay for that market segment.
Because most urban trips have a cycle length of 24 hours, the arrival and departure time
function terms
)(
main
a
T
α
and )(
main
d
T
β
in equation (1) must also satisfy the condition that:
)24()( +=
main
a
main
a
TT
αα
and )24()( +=
main
a
main
a
TT
ββ
(8)
The Tel-Aviv model adopts a trigonometric functional form for the arrival time function
as shown below:
(
)
[
]
i
i
main
ai
main
a
STT
∑
×=
''
)( λα
α
(9)
Where,
i
indicates the
th
i market segment;
'
i
α
is a row vector of parameters that need to
be estimated for the market segment
i
, given by:
( ) ( )
(
)
KiiKiKKiii
22121
αααααα
KK
++
; and
(
)
main
a
Tλ is a row vector of trigonometric terms given by:
24
2
cos
24
4
cos
24
2
cos
24
2
sin
24
4
sin
24
2
sin main
a
main
a
main
a
main
a
main
a
main
aTKTTTKTT
ππππππ
KK
;
i
Srepresents the socio-economic variable indicating the
th
i market segment. The Tel-Aviv
model used several socio-economic variables including the full-time worker dummy variable,
service sector employment dummy variable, gender, age, number of vehicles in the household,
household size, the distance between home and main stop and the CBD destination dummy
variable.
Popuri, Ben-Akiva and Proussaloglou 11
Similarly, the departure time function is defined as:
(
)
[
]
i
i
main
d
main
d
STT
∑
×=
''
i
λβ)(
β
(10)
Where,
'
i
β
is a row vector of parameters that need to be estimated for the market segment
i
,
given by:
( ) ( )
(
)
KiiKiKKiii 22121
ββββββ
KK
++
; and
(
)
main
d
Tλ
is a row vector of trigonometric terms
given by:
24
2
cos
24
4
cos
24
2
cos
24
2
sin
24
4
sin
24
2
sin main
d
main
d
main
d
main
d
main
d
main
dTKTTTKTT
ππππππ
KK
Thus, both the arrival and departure time functions include
2K
parameters for each
market segment:
K
parameters for the sine terms and
K
for the cosine terms. The sine and cosine
terms are used to ensure that the utility value is unique for each arrival and departure time in a
given day. Multiple
π
2 cycles are used to enable a better model fit as compared to just a single
π
2 cycle. The exact value of
K
is determined through a trial and error procedure of estimating
models with various cycles and testing the log-likelihood value of each model in comparison to
the others. In addition, the reasonableness of the arrival and departure time profiles is also used
to determine the number of cycles.
The duration function
)(
main
a
main
d
TT −
γ
captures the duration effect at the main stop, and
together with the arrival and departure time functions, explains the systematic bias to a given
arrival and departure time combination. The Tel-Aviv model uses a polynomial form for the
duration function as shown below:
Dmain
a
main
dD
main
a
main
d
main
a
main
d
main
a
main
d
TTTTTTTT )(*)(*)(*)(
2
21
−++−+−=−
γγγγ
L (11)
D
γγ
,,
1
L
are parameters that need to be estimated. The value of D is once again determined
through a trial and error procedure based on model fit as well as the ability of the duration profile
to replicate observed patterns.
The )ln(
a
∆and )ln(
d
∆terms are the log-size measures that are introduced to account for
the unequal size of the time slots used to discretize continuous time. Ben-Akiva and Abou Zeid
(20) prove mathematically that in a logit model, time periods of unequal length can be accounted
for by adding the natural logarithm of the length of the period to its systematic utility and
constraining the coefficient of the size variable to 1. For this reason, both
1
δ
and
2
δ
were
constrained to 1 during model estimation.
The travel time variables described in the previous section vary across the alternatives.
Therefore, a single coefficient was used for the travel times from home to main stop conditional
on the arrival time at destination, and a single coefficient for the travel times from main stop to
home conditional on the departure time from the destination.
Popuri, Ben-Akiva and Proussaloglou 12
Table 2 summarizes the most important model estimation results (for want of space, only
key parameter estimates have been presented). The travel time coefficients were negative and
marginally significant, indicating that all else being equal, individuals tend to choose their arrival
and departure times so as to minimize their overall travel time. The travel time coefficients for
the home to main stop and main stop to home were constrained to be equal based on statistical
testing. One possible reason for the relatively low significance of travel time coefficients is the
lack of enough variables to explain time of day choice, such as work flexibility, for instance. If
such data were available in the NTHS and included in the model, then it will be possible to
uncover the disutility of longer travel times in a “stronger” fashion.
A quadratic function was used to represent the effect of duration, as shown in Figure 2.
The parameters on the duration function implied that the maximum utility occurs at a duration of
9.5 hours. This was generally consistent with the observed duration profiles for work tours from
the NTHS data, which suggested that the mean duration of work activities was about 7.5 hours
with a standard deviation of almost three hours. Nearly 95 percent of the observations in the
estimation sample showed work durations between two hours and 13 hours. Because of minimal
estimation data beyond these two duration extremes, the model-predicted duration profile must
be interpreted and used carefully for long duration tours.
To interpret the parameters on the arrival and departure time functions shown in Table 2,
the function values were first calculated for each market segment. The arrival time function
values were then divided by the utility at 8:00 AM, and the departure time values divided by the
value at 5:00 PM. This allowed for understanding the relative arrival and departure time
preferences of various market segments. Figure 3 shows the normalized arrival time utility
profiles for four market segments. The part-time workers indicated a higher propensity to arrive
at work after 8:00 AM than the full-time workers. This may be because the part-time jobs start
later in the day, may be even later in the afternoon. Female full-time workers with kids in the
household also showed a higher propensity to arrive after 8:00 AM than their other full-time
counterparts. Women tend to be the primary care takers of children at home and may prefer later
arrivals. Full-time workers with two or more vehicles in the household also showed a propensity
to arrive later in the day. It is possible that individuals with more vehicles tend to hold higher
positions at work and, therefore, have greater work flexibility.
Figure 4 shows the departure time utility profiles for the four market segments discussed
above. The part-time workers showed a much higher propensity to leave from work before 5:00
PM than the full-time workers. This may be because most part-time jobs tend to be centered
towards the middle of the day. The departure time profile for female full-time workers with kids
showed a peak at 3:00 PM. It is likely that women prefer to leave early from work either to pick
up their kids from school or to be at home by the time the kids return from school. Full-time
workers with two or more vehicles in the household have a flatter and higher utility distribution
around 5 PM than other full-time workers, perhaps due to the greater flexibility available to
them.
The arrival time utility profiles in Figure 3 indicate two peaks, one in the morning and
one in the evening, for all the market segments. While the morning peak is understandable, the
evening peak is not expected, except for part-time workers. The estimation data indicated that 90
percent of all observations had arrival times between 5:00 AM and noon. Because of minimal
Popuri, Ben-Akiva and Proussaloglou 13
estimation data with arrival times in the late afternoon, the model-predicted utility profiles will
need to be interpreted judiciously for arrival times in the late afternoon and beyond. A similar
observation extends to the departure utility profiles in Figure 4. Over 90 percent of the
observations in the estimation sample had departure times between noon and 8:00 PM.
5. CONCLUSIONS
This paper discussed the time-of-day model framework for Tel-Aviv. A joint model of arrival
and departure time choice at the main activity stop of the tour was presented. The model
benefited from the use of 36 time slots to represent a travel day. It used travel times generated
from a regression methodology that related reported travel times from the NTHS to network
skims for three large time periods. The regression methodology allowed for the prediction of
door-to-door travel speeds and times for arrival or departure in each of the 36 time slots. Such a
methodology provides an approach to generate detailed travel time estimates using data that are
readily available in most model areas.
The work tour time of day models discussed in this paper used continuous trigonometric
functions of arrival and departure time instead of alternative-specific variables and constants.
This allowed for smooth and cyclic arrival and departure time profiles for each market segment.
Analyzing these profiles indicated the relative preferences of arrival and departure times for
various segments of the population.
The authors acknowledge that the 666 combinations of arrival and departure times
modeled here may be correlated, thereby violating the underlying assumption of independent and
identically distributed error terms. However, the focus was to develop a model that was as
advanced as possible within the realm of popularly understood discrete choice model systems
such as the multinomial logit (MNL) model. The model is geared towards finding a middle
ground between the simplified time of day models being used in practice, and the mathematically
complex models available in the literature.
This paper focused on the estimation details of the time of day model. Model application
is currently underway and will be addressed in future papers. The model discussed here can
easily be used to evaluate policy scenarios such as congestion pricing. So, for instance, if a toll
were to be imposed on a given corridor, this can be converted to equivalent travel time minutes
for each market segment using the corresponding values of time. The updated travel times can
then be input into the model application routine to predict the re-adjustment of arrival and
departure times for each market segment. Similarly, the impact of highway and/or transit
improvements on travel demand by time of day can also be evaluated.
ACKNOWLEDGMENTS
The work presented in this paper was part of the tour-based transportation model development
for Israel’s Ministry of Transport. The authors thank Ruti Amir, the project manager for the
Ministry, for providing data and valuable suggestions throughout the model development
process. The authors would also like to thank Shlomo Bekhor, Leonid Kheifits, Yossi Prashker,
Yoram Shiftan and Marcos Szeinuk for their comments and suggestions during the model
Popuri, Ben-Akiva and Proussaloglou 14
development process. Finally, the authors would like to thank five anonymous reviewers, whose
comments have added substantial value to this paper.
Popuri, Ben-Akiva and Proussaloglou 15
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Popuri, Ben-Akiva and Proussaloglou 17
LIST OF TABLES
TABLE 1 Travel Speed Regression Summary
TABLE 2 Joint Arrival and Departure Time Choice for Primary Work Tours
LIST OF FIGURES
FIGURE 1 Delay Profiles from Travel Speed Regressions.
FIGURE 2 Duration Profile for Work Tours
FIGURE 3 Arrival Time Profiles for Primary Work Tours
FIGURE 4 Departure Time Profiles for Primary Work Tours
Popuri, Ben-Akiva and Proussaloglou 18
TABLE 1 Travel Speed Regression Summary
Variable
Trip Speed Regression
Given the Arrival Time
at Destination of Trip
Trip Speed Regression
Given the Departure
Time from Origin of Trip
Parameter t-stat Parameter t-stat
Intercept 0.0317 3.2 0.0291 2.9
Ln(distance) 0.0072 2.9 0.0093 3.8
delay -1.4865 -0.2 -17.3535 -2.6
delay
T*
24
2
sinexp
π
-11.2967 -1.0 12.5456 1.0
delay
T*
24
2
sinexp
2
π
12.5130 1.2 -9.4571 -0.9
delay
T*
24
2
sinexp
3
π
-5.6582 -1.6 2.6285 0.7
delay
T*
24
2
sinexp
4
π
0.9182 2.1 -0.2404 -0.5
delay
T*
24
2
cosexp
π
7.0780 0.7 19.9279 1.8
delay
T*
24
2
cosexp
2
π
-5.6571 -0.6 -15.3596 -1.6
delay
T*
24
2
cosexp
3
π
1.8769 0.6 4.5522 1.4
delay
T*
24
2
cosexp
4
π
-0.2359 -0.6 -0.4790 -1.2
delay
T*
24
4
sinexp
π
-1.9606 -1.1 3.5345 2.0
delay
T*
24
4
sinexp
2
π
1.7074 1.1 -3.2051 -2.0
delay
T*
24
4
sinexp
3
π
-0.5433 -0.9 1.2742 2.1
delay
T*
24
4
sinexp
4
π
0.0630 0.8 -0.1915 -2.4
delay
T*
24
4
cosexp
π
2.3598 1.1 10.2721 4.5
delay
T*
24
4
cosexp
2
π
-0.7701 -0.5 -8.0543 -4.7
delay
T*
24
4
cosexp
3
π
0.1102 0.2 2.6979 4.4
delay
T*
24
4
cosexp
4
π
0.0018 0.0 -0.3379 -4.2
CBD Origin Dummy -0.2093 -20.0 -0.2052 -19.5
Other Urban Origin Dummy -0.1304 -19.4 -0.1261 -18.7
Suburban Origin Dummy -0.1518 -19.8 -0.1492 -19.4
CBD Dest Dummy -0.0341 -3.3 -0.0304 -3.0
Other Urban Dest Dummy -0.0429 -6.5 -0.0456 -6.9
Suburban Destination Dummy 0.0015 0.2 -0.0009 -0.1
R-Squared 0.1522 0.1572
Adjusted R-Squared 0.1521 0.1565
Total Observations 41,130 41,130
Degrees of Freedom 24 24
Popuri, Ben-Akiva and Proussaloglou 19
TABLE 2 Joint Arrival and Departure Time Choice for Primary Work Tours
Variable Parameter (t-stat)
Travel Times
Round Trip Travel Time from Home to Main and Main to Home -0.0093 (-1.5)
Duration at Main Activity
Duration ^ 1 0.551 (13.2)
Duration ^ 2 -0.0290 (-24.7)
Size of Intervals
(# Half Hours in Arrival Time Slots) 1.00 (*)
(# Half Hours in Departure Time Slots) 1.00 (*)
Arrival and Departure Time Functions
Sin(2*pi()*t
arr
/24) -2.94 (-8.8)
Sin(4*pi()*t
arr
/24) -2.31 (-11.3)
Cos(2*pi()*t
arr
/24) -4.52 (-7.5)
Cos(4*pi()*t
arr
/24) -2.79 (-9.1)
Sin(2*pi()*t
dep
/24) -1.07 (-2.8)
Sin(4*pi()*t
dep
/24) -0.365 (-2.3)
Cos(2*pi()*t
dep
/24) -1.48 (-4.7)
Cos(4*pi()*t
dep
/24) -0.0275 (-0.1)
Full-Time Worker Dummy interacted with Arrival and Departure time functions
Full Time * Sin(2*pi()*t
arr
/24) 2.00 (17.4)
Full Time * Sin(4*pi()*t
arr
/24) 1.02 (10.9)
Full Time * Cos(2*pi()*t
arr
/24) 0.738 (2.5)
Full Time * Cos(4*pi()*t
arr
/24) 0.158 (1.0)
Full Time * Sin(2*pi()*t
dep
/24) 0.978 (5.1)
Full Time * Sin(4*pi()*t
dep
/24) 0.543 (7.6)
Full Time * Cos(2*pi()*t
dep
/24) 1.31 (13.4)
Full Time * Cos(4*pi()*t
dep
/24) -1.20 (-11.4)
Female with Kids Dummy interacted with Arrival and Departure time functions
Female with Kids * Sin(2*pi()*t
arr
/24) -1.34 (-7.4)
Female with Kids * Sin(4*pi()*t
arr
/24) -1.32 (-8.3)
Female with Kids * Cos(2*pi()*t
arr
/24) -4.12 (-6.2)
Female with Kids * Cos(4*pi()*t
arr
/24) -2.14 (-7.0)
Female with Kids * Sin(2*pi()*t
dep
/24) -0.120 (-0.4)
Female with Kids * Sin(4*pi()*t
dep
/24) 0.718 (6.7)
Female with Kids * Cos(2*pi()*t
dep
/24) -0.582 (-4.3)
Female with Kids * Cos(4*pi()*t
dep
/24) 0.374 (2.3)
Two Vehicle Dummy interacted with Arrival and Departure time functions
Two Vehicles * Sin(2*pi()*t
arr
/24) -0.240 (-2.7)
Two Vehicles * Sin(4*pi()*t
arr
/24) -0.0991 (-1.4)
Two Vehicles * Cos(2*pi()*t
arr
/24) -0.699 (-3.2)
Two Vehicles * Cos(4*pi()*t
arr
/24) -0.154 (-1.4)
Two Vehicles * Sin(2*pi()*t
dep
/24) -0.388 (-2.8)
Two Vehicles * Sin(4*pi()*t
dep
/24) -0.201 (-3.6)
Two Vehicles * Cos(2*pi()*t
dep
/24) -0.191 (-2.6)
Two Vehicles * Cos(4*pi()*t
dep
/24) 0.205 (2.7)
Observations 7,764
Final log (L) -42547.9
Initial log(L) -50476
Number of Parameters 91
Adjusted Rho Squared (0) 0.155
Popuri, Ben-Akiva and Proussaloglou 20
-1.300
-1.200
-1.100
-1.000
-0.900
-0.800
-0.700
-0.600
-0.500
-0.400
-0.300
-0.200
-0.100
0.000
0300-0459
0530-0559
0630-0659
0730-0759
0830-0859
0930-0959
1030-1059
1130-1159
1230-1259
1330-1359
1430-1459
1530-1559
1630-1659
1730-1759
1830-1859
1930-1959
2030-2059
2130-2159
0300-0459
Delay Coefficient
Reported Speed Regression given Arrival Time at Destination
Reported Speed Regression given Departure Time from Origin
FIGURE 1 Delay Profiles from Travel Speed Regressions
Popuri, Ben-Akiva and Proussaloglou 21
Duration Profiles
-4.000
-3.000
-2.000
-1.000
0.000
1.000
2.000
3.000
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
14.00
15.00
16.00
17.00
18.00
19.00
20.00
21.00
22.00
23.00
24.00
Duration
Utility
Duration Function
FIGURE 2 Duration Profile for Primary Work Tours
Popuri, Ben-Akiva and Proussaloglou 22
Normalized Arrival Profiles
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
3:00 AM
4:00 AM
5:00 AM
6:00 AM
7:00 AM
8:00 AM
9:00 AM
10:00 AM
11:00 AM
12:00 PM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
5:00 PM
6:00 PM
7:00 PM
8:00 PM
9:00 PM
10:00 PM
11:00 PM
12:00 AM
1:00 AM
2:00 AM
3:00 AM
Arrival Time
Normalized Arrival Function Value
Full-Time Worker
Part-Time Worker
Female Full-Time Worker w ith Kids
Full-Time w ith Tw o or More Vehicles in HH
FIGURE 3 Arrival Time Profiles for Primary Work Tours
Popuri, Ben-Akiva and Proussaloglou 23
Normalized Departure Profiles
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3:00 AM
4:00 AM
5:00 AM
6:00 AM
7:00 AM
8:00 AM
9:00 AM
10:00 AM
11:00 AM
12:00 PM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
5:00 PM
6:00 PM
7:00 PM
8:00 PM
9:00 PM
10:00 PM
11:00 PM
12:00 AM
1:00 AM
2:00 AM
3:00 AM
Departure Time
Normalized Departure Function Value
Full-Time Worker
Part-Time Worker
Female Full-Time Worker w ith Kids
Full-Time w ith Two or More Vehicles in HH
FIGURE 4 Departure Time Profiles for Primary Work Tours
