Download Connecting Networkwide Travel Time Reliability and the Network

Connecting Networkwide Travel Time
Reliability and the Network Fundamental
Diagram of Traffic Flow
Hani S. Mahmassani, Tian Hou, and Meead Saberi
of the network. One application of NFD is for estimating the traffic
state of a neighborhood in relation to its space mean speed. However,
another important aspect that reflects traffic quality, the travel time
reliability, is missing.
Travelers often expect regular congestion delays and can accommodate them by early departure, while drivers are most frustrated by
unexpected delays. This situation has motivated a growing number
of studies on the topic of travel time reliability. Although the concept
of reliability is relatively new in the area of transportation compared
with some other engineering fields, several definitions and measures
of reliability have been developed. Turner et al. defines “trip time
reliability” as the range of travel times experienced during a large
number of daily trips (8). The 1998 California Transportation Plan
defines “reliability” as the variability between the expected travel
time that is based on the scheduled or average travel time and the
actual travel time because of the effects of nonrecurrent congestion
(9). FHWA defines “reliability” as the consistency or dependability in
travel times, which can be measured from day to day, across different
times of the day, or both (10). Performance measures of travel time
reliability include 95th-percentile travel time, buffer index, planning
index, frequency of congestion, and the like.
Although the definitions and measures of travel time reliability
vary in different contexts, they are all closely related to the variation
of travel time. Lomax et al. first recognized that, while reliability
and variability are interchangeable in many contexts, they are different in their focuses (11). More precisely, variability represents
the amount of inconsistency and can be used to measure the degree
of unreliability. FHWA identified seven major sources of travel time
variation: incidents, work zones, weather, fluctuations in demand,
special events, traffic control devices, and inadequate base capacity
(12). Many research studies have examined estimation and prediction
of travel time variability, on both the segment–link level (13, 14) and
the path level (15, 16). Some of these studies show that travel time
distribution is affected by average demand level and the speed–flow
relationship (i.e., travel time variability depends on both the traffic
density of the road and traffic flow rates). However, these studies focus
on the link and path levels only, and research on networkwide travel
time variability is limited.
Given the knowledge of NFD and travel time reliability performance measures, the objectives of this research are to establish a
bridge between network traffic flow theory and travel time reliability
and to extend travel time variability models from the link and path
levels to the network level. This study establishes an important result
in the networkwide characterization of traffic flow, relating the overall
reliability of travel in a given network to commonly used performance
descriptors.
The existence of the network fundamental diagram (NFD) has been
established at the urban network scale. It relates three traffic descriptors:
speed, density, and flow. However, its deterministic nature does not
convey the underlying variability within the network. In contrast, travel
time reliability as a network performance descriptor is of growing
concern to both the traveling public and traffic managers and policy
makers. The objectives of this paper were to extend travel time reliability
modeling from the link–path level to the network level and to connect
overall network variability to NFD. Robust relationships between travel
time variability and network density and flow rate were analytically
derived, investigated, and validated with both simulated and real-world
trajectory data. The distance-weighted standard deviation of travel time
rate, as a measure of travel time variability, was found to increase monotonically with network density. A maximum network flow rate existed
beyond which network travel time reliability deteriorated at a much
faster pace. The results also suggest that these relationships are inherent
network properties (signature) that are independent of demand level.
The effects of en route information on the proposed relationships were
also studied. The results showed that en route information reduced network travel time variability. The findings provide a strong connection
between NFD and travel time variability, and this connection can be used
further for modeling of network travel time reliability and assessment of
measures intended to improve reliability of travel in a network.
The most commonly used variables to characterize traffic streams are
flow, density, and speed. The relationship between these variables in
individual facilities is often referred to as the fundamental diagram
(FD). Similarly, the existence of a reproducible and well-defined
relationship between networkwide average flow, average density,
and average speed has been established in the literature since the
1970s (1–7). Such a networkwide relationship has more recently been
referred to as a macroscopic FD (MFD) or a network FD (NFD).
Lately, Geroliminis and Daganzo retheorized this idea and proposed
analytical approximations for NFD of urban traffic (7). They recognized that, if network-level macroscopic relationships are insensitive to
origin–destination (O-D) demand, they could be viewed as properties
H. S. Mahmassani, Transportation Center, 215 Chambers Hall, and T. Hou and
M. Saberi, Department of Civil Engineering, Northwestern University, 600 Foster
Street, Evanston, IL 60208. Corresponding author: H. S. Mahmassani, masmah@
northwestern.edu.
Transportation Research Record: Journal of the Transportation Research Board,
No. 2391, Transportation Research Board of the National Academies, Washington,
D.C., 2013, pp. 80–91.
DOI: 10.3141/2391-08
80
Mahmassani, Hou, and Saberi
81
Theoretical Background
Description of NFD
In the late 1970s, a two-fluid theory of town traffic was developed
by Herman and Prigogine, who proposed a relationship between the
average fraction of vehicles moving in a street network and their
average speed, in the following manner (1):
Vr = Vm (1 − fs )
n
(1)
where
Vr=average speed of moving (running) vehicles in network over
observation period,
Vm=average maximum running speed, and
fs=fraction of vehicles stopped.
In the 1980s and 1990s, several studies extended the network-level
variables, to relate average speed, flow, and concentration by taking
averages over all vehicles in the network over a specified period
(2–6). Mahmassani et al. simulated an isolated hypothetical network
with a fixed number of vehicles circulating within the network on
the basis of some microscopic rules (2). Their results indicated that
the network-level variables have an interrelationship similar to that
captured by the traffic models established for individual roads.
Daganzo revisited this idea and retheorized the network-level
macro­scopic relationships for single neighborhoods and for systems
of interconnected neighborhoods (17). Recent results from field experiments in Yokohama, Japan (7); Toulouse, France (18); the Twin Cities,
Minnesota (19); and Portland, Oregon (20); and simulated data for
the San Francisco, California (7); Amsterdam, Netherlands (21); and
Nairobi, Kenya (22); networks have revealed useful insights about
the properties of such relationship for urban traffic. A c­ omprehensive
background can be found in Saberi and Mahmassani (20).
Travel Time Variability
Modeling travel time reliability requires characterizing travel time
distributions. Lomax et al. proposed to calculate buffer time by using
the difference between 95th-percentile travel time and average travel
time for a trip as a measure of extra time needed to allow a traveler to
arrive on time (11). The planning time index also involves travel time
distribution, which is calculated as the ratio between the 95th-percentile
travel time and average travel time. Ideally, the distribution function
of travel time should be estimated by using historical data; however,
most of the time, this is not possible because of either unavailability or
insufficiency of data. Alternatively, the distribution of travel time can
be approximated if some of the statistics are known. Two key statistics
commonly used to describe a distribution are the mean and the standard deviation, with one depicting the central tendency and the other
describing the dispersion. Lam and Small used the standard deviation
of travel time to represent variability and included it into the route
choice utility function to study the value of reliability (23). Sen et al.
proposed a mean–variance multiobjective model to study traveler’s
route choice behavior, in which travel time variability is expressed
as the variance of travel time (24).
Of those two statistics (mean and standard deviation), the mean of
the travel time is usually relatively easy to obtain. In contrast, obtaining or predicting the standard deviation is more challenging. However,
the kinetic theory of traffic flow by Prigogine and Herman suggests
that a strong correlation between the magnitude of mean travel time
and its fluctuation exists (25). According to those researchers, short
mean travel times often prevail together with low standard deviation;
this corresponds to the uncongested traffic condition, when drivers
are able to maintain fairly constant high speeds. As congestion
increases, average speed decreases and drivers experience increased
interactions with other drivers in their vicinities. These interactions
cause increased fluctuation in driving speed and thus affect travel
time. Herman and Lam proposed that, when travel times on different links are independent and identically distributed, the standard
deviation is proportional to the average travel time; they verified
this relationship through data collected by circulating vehicles in a
network (26). Taylor confirmed and extended the theory of Herman
and Lam by using collected data from public transit in Paris (26, 27).
He also introduced a measure of variability that could be related to
the traffic congestion level. In general, this theoretical relationship
between mean travel time and its standard deviation can be expressed
by Equation 1:
σ ( t ′ ) = θ I + θ2µ ( t ′ ) + ε
(1)
where
t′ =
t
d
t′=travel time per unit distance,
t=travel time,
d=travel distance,
σ(t′)=standard deviation of t′,
µ(t′)=mean value of t′,
θ1, θ2=coefficients, and
ε=random error.
In Equation 1, travel time per unit distance or travel time rate (t′)
is used instead of travel time (t) alone. The advantage of using t′ is
that it helps in excluding the source of variability coming from trip
distance and focuses on the travel time variability caused by variation of speed. Richardson and Taylor first suggested using the unit
travel time (travel time divided by the trip distance) and studied the
relationship between congestion level and average unit travel time
(28). Later, Jones et al. studied trip travel time variability at the link
level by using commuting data in Texas and verified the correlation
between the standard deviation of travel time and the mean travel time,
both taken on a per unit distance basis (29). Recently, Mahmassani
et al. calibrated this linear relationship by means of both simulated
and real vehicle trajectory data and found it to be a robust relationship whose validity can be extended to different aggregation levels
and different scales of networks (30). Application of this model is not
restricted to predicting the standard deviation of travel time per mile
when the average travel time is available but can be widely extended
to the research framework related to travel time reliability.
Data Description and Study Area
Previous studies have generally relied on traffic data obtained from
loop detectors to study the evolution of networkwide traffic states
(18–20). Those data contain traffic density, flow rate, and average
speed at the link–segment level. However, the coverage of loop
detectors is usually limited to freeways only, a situation that may
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Transportation Research Record 2391
give a biased representation of the entire network without adding
similar data on arterials. Other limitations of loop detector data are
that the software to which loop detectors connect usually provides
aggregated average speed only and the variation across individual
vehicles is often not recorded. According to a review by Lomax et al.,
the best alternative to measuring the travel time variability is by using
a review of probe vehicles (11). Although probe data give travel time
of individual vehicles that can be further used to estimate travel time
variability, they provide no or only partial information on density and
flow. Therefore, both detector and probe data have their respective
advantages and drawbacks. To accommodate the need to study both
NFD and travel time variability, traffic data simulated by a dynamic
traffic assignment program, DYNASMART, were used here. Two sets
of traffic data were used in this study:
The expression to calculate the distance-weighted mean of the travel
time rate follows:
1. Trajectory data that contain itinerary information of each
vehicle circulating in the network, such as departure time, origin,
all intermediate nodes visited, node exit times, link travel time, and
accumulated stop time, and
2. Link density and flow on each link at different intervals within
the simulation time window.
σ=
Five major quantities were used in this study: space mean speed,
network density, network flow rate, distance-weighted travel time
rate, and distance-weighted standard deviation of travel time rate.
The mathematical expressions of these quantities are shown in
Equations 2 to 6.
The space mean speed, u–s, is calculated as
n
us =
∑d
i
i =1
n
(2)
∑ ti
i =1
n
i i
µ=
=
i =1
n
∑d
i
=
i =1
n
∑d
i
i =1
i
1
us
(5)
i =1
where t i′ is the travel time rate of vehicle i (minutes per mile) and ti
is the travel time of vehicle i.
The distance-weighted standard deviation of travel time rate is
calculated as follows:
n
∑ d (t ′− µ)
i
i =1
2
i
n
∑ di
(6)
i =1
NFD mainly considers the first three variables, while travel time
variability models use the other two quantities. The connection
between these two sets of variables consists of the space mean speed
(u–s) and the distance-weighted travel time rate (µ), which are exactly
the inverse of each other.
Four road networks were used in this study: Chicago, Illinois;
Baltimore, Maryland; Salt Lake City, Utah; and CHART. The CHART
network refers to the Baltimore–Washington, D.C., corridor, iden­
tified by Maryland’s Coordinated Highways Action Response
Team (CHART). Illustrations of these four networks are presented
in Figure 1. Detailed descriptions of network configurations are
provided in Table 1.
Experimental Results
where
i=vehicle index,
n=number of vehicles, and
di=travel distance of vehicle i.
–
The average network density, k , and flow, q–, are calculated,
respectively, as
m
∑l k
j
k=
n
∑ d t′ ∑ t
j
j =1
m
(3)
∑ lj
j =1
Dynamic traffic assignment simulations were performed on the four
selected networks, at each of three demand levels (i.e., normal, low,
and high). The normal demand level is calibrated in accordance with
historical static O-D matrix and time-dependent traffic counts on
observation links. Low demand level refers to the demand pattern
that has the same distribution as normal level but 25% fewer vehicles.
Similarly, high demand level also has the same distribution but has
50% more vehicles than the normal level. The entire planning horizon
or simulation period was divided into series of 5-min intervals. For
each 5-min interval, the space mean speed, network density, network
flow rate, distance-weighted travel time rate, and distance-weighted
standard deviation of travel time rate were computed with Equations 2
to 6 by using simulation outputs.
m
∑l q
j
q=
j
j =1
m
∑ lj
j =1
where
m=number of links,
j=link index,
lj=length of link j,
kj=density of link j (vehicles per mile per lane), and
qj=flow rate of link j (vehicles per hour).
(4)
Discussion of NFD
The first step of this study was to verify the existence of NFD by
using simulated traffic data. As shown by Geroliminis and Daganzo,
when homogeneity conditions hold, NFD exhibits smooth curves
that have less scatter than individual links, and it is reproducible
under different demand conditions (7). For the current study, Figure 2
presents graphs of space mean speed versus network density and
network density versus network flow rate at 5-min intervals for all
four networks. The first three 5-min intervals were excluded from
the analysis, as they accounted for the warm-up period of simulation
Mahmassani, Hou, and Saberi
83
(a)
(b)
(c)
(d)
FIGURE 1 Snapshot of study areas: (a) Chicago, (b) Baltimore, (c) Salt Lake City, and (d) CHART.
when the networks changed dramatically from empty to steady state.
Figure 2, a, c, e, and g, shows the existence of a smooth declining
curve that relates space mean speed and network density, regardless of the demand level, even when homogeneity conditions do not
strictly hold. Although, within a single network, the speed–density
curves cover different ranges at different demand levels, these curves
overlap and follow the same trend. Similar observations are found
in the flow–density diagrams (Figure 2, b, d, f, and h). For the various demand levels tested in this study, the O-D matrix structure
remains the same. The high degree of overlap in the produced curves
suggests that the shape of NFD is an inherent property of the network
and is not particularly sensitive to the overall demand levels when the
O-D matrix structure remains unchanged. As well, the speed–density
and flow–density graphs are plotted in such a way that only the loading periods of the networks are presented, and thus no recovery and
hysteresis loop is present in Figure 2.
TABLE 1 Network Configurations
Network
Chicago
Baltimore
Salt Lake
City
Chart
Number of nodes
Number of links
Number of vehicles
Demand duration (h)
1,578
4,805
805,275
6
6,825
14,317
898,671
2
8,022
17,947
937,483
3
2,182
3,387
151,973
2
Relationship Between Weighted Mean Travel
Time Rate and Its Standard Deviation
According to the kinetic theory of traffic flow, the variability of travel
time increases with the mean travel time when network congestion
reduces the average speed and causes more interaction between
drivers. This part of the theory is verified in Figure 3, where the
0.6
0.5
0.4
0.3
0.2
0.1
0
Spac e M ean Speed (m ile/m in)
low demand
normal demand
high demand
0.6
0.5
0.4
0.3
0.2
0.1
0
Space Mean Speed (mile/min)
low demand
normal demand
high demand
0.4
0.3
0.2
0.1
0
300
200
low demand
normal demand
high demand
100
0
0
1000
800
600
400
low demand
high demand
normal
demand
200
0
0
500
400
300
200
low demand
normal demand
high demand
100
0
0
(e)
0.4
0.3
0.2
0.1
0
10
20
30
40
50
Network Density (veh/mile/lane)
(g)
1200
1000
Network Fow (veh/hr)
Spac e M ean Speed (m ile/m in)
low demand
normal demand
high demand
0.5
0
20
40
60
80
Network Density (veh/mile/lane)
(f)
0.8
0.6
20
40
60
Network Density (veh/mile/lane)
(d)
600
20
40
60
80
Network Density (veh/mile/lane)
0.7
20
40
60
80
Network Density (veh/mile/lane)
(b)
1200
20
40
60
Network Density (veh/mile/lane)
(c)
0.5
0
400
20
40
60
80
Network Density (veh/mile/lane)
(a)
0.7
0
500
Network Flow (veh/hr)
0
600
Network Flow (veh/hr)
low demand
normal demand
high demand
0.7
Network Flow (veh/hr)
Spac e M ean Speed (m ile/m in)
0.8
800
600
400
low demand
normal demand
high demand
200
0
0
10
20
30
40
50
Network Density (veh/mile/lane)
(h)
FIGURE 2 Network speed–density and flow–density diagrams, respectively, for (a and b) Chicago,
(c and d) Baltimore, (e and f) Salt Lake City, and (g and h) CHART (veh = vehicles).
85
80
Weighted SD of Travel Time Rate
(min/mile)
Weighted SD of Travel Time Rate
(min/mile)
Mahmassani, Hou, and Saberi
low demand
70
normal demand
60
high demand
50
40
30
20
10
0
0
5
10
15
20
Weighted Mean Travel Time Rate
(min/mile)
80
low demand
70
normal demand
60
high demand
50
40
30
20
10
0
0
5
10
15
20
Weighted Mean Travel Time Rate
(min/mile)
(b)
80
Weighted SD of Travel Time Rate
(min/mile)
Weighted SD of Travel Time Rate
(min/mile)
(a)
low demand
70
normal demand
60
high demand
50
40
30
20
10
0
0
5
10
15
20
80
low demand
70
normal demand
60
high demand
50
40
30
20
10
0
0
Weighted Mean Travel Time Rate
(min/mile)
5
10
15
20
Weighted Mean Travel Time Rate
(min/mile)
(c)
(d)
FIGURE 3 Weighted standard deviation of travel time rate versus weighted mean travel time rate
for (a) Chicago, (b) Baltimore, (c) Salt Lake City, and (d) CHART (SD = standard deviation).
distance-weighted standard deviation of travel time rate is plotted
against the distance-weighted mean travel rate under various demand
conditions for the four networks. The graphs suggest that a strong
positive correlation exists between the weighted mean travel time and
its standard deviation. In a manner similar to what is observed in NFD,
the curves produced at different demand levels follow the same trend,
with a curve for the high demand level having the greatest extent. The
linear model in Equation 1 was used to fit the data. The fitted trend
lines are also plotted in Figure 3, and the calibration results of the
model parameters are presented in Table 2.
The magnitude of the slope term (θ2) reflects the extent to which
the standard deviation will increase when the weighted-average travel
time rate increases by one unit. A greater θ2 value means that travel time
variability is introduced more easily as the mean travel speed decreases.
TABLE 2 Calibration Results of Model of Travel Time Variability
θ1
θ2
Study Area
Estimate
Standard
Error
Estimate
Standard
Error
−θ1/θ2a
(min/mi)
−60 θ2/θ1b
(mph)
Chicago
Baltimore
Salt Lake City
CHART
−6.3639
−2.8416
−5.3319
−2.1929
0.1565
0.1271
0.4038
0.0781
4.0274
2.6731
4.1227
2.4412
0.0262
0.0284
0.0573
0.0241
1.58
1.06
1.29
0.90
37.97
56.44
46.39
66.79
a
Minimum travel time per unit distance.
Maximum speed.
b
R2
.9928
.9931
.9811
.9939
86
Transportation Research Record 2391
Connection Between NFD
and Travel Time Variability
Chicago
Baltimore
Salt Lake City
CHART
70
60
50
40
30
20
10
0
0
low demand
normal demand
high demand
60
40
20
0
0
20
40
60
80
Network Density (veh/mile/lane)
low demand
normal demand
high demand
30
25
20
15
10
5
0
0
16
low demand
normal demand
high demand
60
40
20
0
0
20
40
60
80
Network Density (veh/mile/lane)
(c)
20
40
60
80
Network Density (veh/mile/lane)
(b)
Weighted SD of Travel Time
Rate (min/mile)
Weighted SD of Travel Time
Rate (min/mile)
(a)
80
20
analytical relations must exist between travel time variability and
network density and flow rate as well. That is, if NFD exists, the
space mean speed is directly related to network density and flow;
then, if the travel time variability is correlated to mean travel time
rate (the inverse of space mean speed), it is also directly related to
network density and flow. These relationships are explored by plotting weighted standard deviation of travel time rate against network
density (Figure 5) and network flow rate (Figure 6). Figure 5 shows
that travel time variability increases with network density. The relationship is reproducible in different networks under various demand
levels. Although higher demand levels tend to increase network den-
35
80
5
10
15
Weighted Mean Travel Time Rate (min/mile)
FIGURE 4 Comparison of calibrated linear travel time variability
models of studied networks.
Weighted SD of Travel Time
Rate (min/mile)
Weighted SD of Travel Time
Rate (min/mile)
As it has been verified in the previous sections that NFD exists
and that the linear travel time variability model is valid, certain
80
Weighted SD of Travel
Time Rate (min/mile)
The negative ratio between θ1 and θ2 [−θ1/θ2 (i.e., the x-intercept)]
represents the reciprocal of the free-flow speed of the network. The
calibration results in Table 2 show that the Salt Lake City network
has the greatest θ2 value, which means that it is the most vulnerable
to congestion-induced deterioration in reliability and is followed in
order by Chicago, Baltimore, and CHART. CHART has the smallest
x-intercept value, 0.898 min/mi, of the four studied networks; this
value corresponds to the greatest free-flow speed, 67 mph. The model
of the Chicago network has the greatest x-intercept and thus indicates
that Chicago has the lowest free-flow speed, 38 mph. The differences
in free-flow speeds are in part attributable to the relative proportions
of arterial versus freeway lane miles included in the simulated networks. For example, the simulated CHART network does not include
as many arterials as the simulated Chicago network. Overall, results
suggest that the relationship between the weighted mean travel time
rate and its standard deviation is a property of a network that rep­
resents its inherent reliability. Such a property seems to be independent of demand and only dependent on network structure and control.
Figure 4 presents the calibrated linear models of the four study areas
in a single graph and compares the characteristics of travel time
variability of these networks.
low demand
normal demand
high demand
14
12
10
8
6
4
2
0
0
20
40
60
Network Density (veh/mile/lane)
(d)
FIGURE 5 Weighted standard deviation of travel time rate versus network density obtained
from (a) Chicago, (b) Baltimore, (c) Salt Lake City, and (d) CHART.
Mahmassani, Hou, and Saberi
87
35
Weighted SD of Travel Time
Rate (min/mile)
Weighted SD of Travel Time
Rate (min/mile)
80
60
40
low demand
normal demand
high demand
20
0
0
200
400
600
Network Flow (veh/hr)
(a)
30
25
20
10
0
200 400 600 800 1000 1200
Network Flow (veh/hr)
(b)
16
Weighted SD of Travel Time
Rate (min/mile)
Weighted SD of Travel Time
Rate (min/mile)
5
0
800
80
60
40
low demand
normal demand
high demand
20
0
low demand
normal demand
high demand
15
0
200
400
Network Flow (veh/hr)
600
14
12
10
low demand
normal demand
high demand
8
6
4
2
0
0
500
1000
Network Flow (veh/hr)
(c)
(d)
FIGURE 6 Weighted standard deviation of travel time rate versus network flow obtained
from (a) Chicago, (b) Baltimore, (c) Salt Lake City, and (d) CHART.
sity, they do not cause any change in the observed trend. Changes
in travel time variability related to network density follow the same
pattern under different demand levels. Moreover, the relationship
between travel time variability and network flow rate is not monotonic. At the beginning of simulation, when vehicles are loaded
onto the empty network, the standard deviation of travel time rate
increases with network flow rate. Flow rate reaches a maximum
point beyond which the network flow starts to drop while travel
time variability continues to increase. The results show that, at the
maximum flow point, the corresponding weighted standard deviation
of the travel time rate is between 5 to 10 min/mi for the four studied
networks. Travel time variability has been found to increase at a much
faster pace when the network flow rate starts to decrease compared
with the period in which travel time variability and network flow
increase together.
The graphs in Figures 5 and 6 intuitively convey the underlying
relationships between travel time variability and network density or
flow rate. However, from the modeling perspective, development of a
consistent set of equations that can replicate these results is desirable.
This development can be achieved when the analytical expressions
of NFD are known. Daganzo and Geroliminis proposed an analytical
approximation for NFD under the assumptions of variational theory
and successfully applied the formulae to model the network traffic
state in San Francisco (31). In this study, equations relating travel time
variability and network density or flow are analytically derived by
using the Greenshields et al. model (32). The procedure is generally
applicable to other given mathematical expressions of NFD.
Many traffic flow models in the literature describe how vehicle
speed changes with density. These were developed on the basis of
empirical data, theoretical considerations, or both. The seminal work
on this topic is probably the 1935 paper by Greenshields et al., in which
the following linear speed–density relationship was derived (32):
k
us =  1 −  u f
kj 

(7)
where kj is jam density and uf is free-flow speed.
By assuming that this relationship is also valid at the network
level, by the fundamental identity q = ku–s, a parabolic relationship
between network flow and density is obtained, as in Equation 8:
k2 

q =  k −  uf
kj 

(8)
On the basis of Equation 8, the network flow rate reaches its maximum
value qmax at its optimal density k0, where k0 = 1/2kj.
As noted earlier, the linkage between NFD and travel time variability is provided by the inverse relationship between space mean
speed and distance-weighted mean travel time rate (i.e., µ = 1/u–s).
88
Transportation Research Record 2391
By substituting that and the travel time variability model σ = θ1 + θ2µ
into Equation 7, the weighted standard deviation of the travel time
rate can then be expressed in relation to network density and other
constant parameters:
θ2 i kj
(9)
( k j − k ) uf
Given the flow–density relationship (Equation 8), the network flow
can also be expressed as a function of the weighted standard deviation
of travel time rate:
q=
θ22 i kj
θ 2 i kj
−
( σ − θ1 ) ( σ − θ1 )2uf
(10)
in which at the optimal density k0,
σ = θ1 +
2θ2
uf
(11)
Because the x-intercept of the linear travel time variability model can
be approximated by the minimum travel time rate (i.e., −θ1/θ2 ≈ 1/uf),
replacing uf in Equation 11 by θ1 and θ2, gives σ ≈ −θ1.
Other speed–density models, like the modified Greenshields, as
reported in Chang et al. (33), Greenberg (34), and others, can also be
used to derive the analytical relationship between travel time variability and NFD, by following the procedure described above. However, most of the models are not as simple as the Greenshields et al.
model, and numerical methods may be required to find the critical
value for standard deviation value of travel time at the maximum
network flow rate.
Effect of En Route Information
Advanced traveler information systems are a key component of intelligence transportation systems, which deliver real-time information
on prevailing traffic conditions to road users. Various researchers
have shown that advanced traveler information systems affect driver
behavior and have the potential to improve overall network perfor-
Validation
The existence of NFD and linear travel time variability models
have been validated in the literature by using real-world traffic data.
Geroliminis and Daganzo used taxi trip data in Yokohama, Japan,
and estimated NFD (7). Mahmassani et al. used trajectory data in the
Seattle, Washington, area and validated the correctness of the linear
relationship between mean travel time rate and its standard deviation
on different aggregation levels (30). This paper used trajectory data
to validate the findings from the simulation experiments.
The trajectory data used in this study were collected from vehicles
in New York City equipped with Global Positioning Systems on a
selected weekday (May 6, 2010). In that region, 10,367 recorded trips
were identified from 6:00 a.m. to 6:00 p.m. The distance-weighted
mean travel time rate and standard deviation were computed for
every 5-min interval on the basis of these trips. Figure 9a shows
80
Weighted SD of Travel Time
Rate (min/mile)
40
30
20
normal demand
without en route
10
0
normal demand
with 20% en route
Weighted SD of Travel Time
Rate (min/mile)
σ = θ1 +
mance by reducing congestion and mean trip travel time (35, 36). To
study the effects of autonomous driver information and route guidance on network travel time reliability, simulation experiments were
conducted by using the Salt Lake City network, with 20% of drivers
having access to perfect en route information on prevailing traffic
conditions. Those 20% of vehicles were modeled as being capable of
receiving real-time en route information, reevaluating shortest paths
at every intersection, and implementing path switching on the basis
of two criteria: the indifference band and the threshold bound for
switching decisions under bounded rationality (37, 38).
Figures 7 and 8 compare the results of simulations with and without
en route information in the Salt Lake City network. Observations
show that the relationships investigated and analytically derived in
the preceding section between travel time variability and network
density or flow rate were still valid when some of the vehicles in
the network were guided by en route information. With en route
information, the weighted standard deviation of travel time rate, as
a measure of travel time variability, tended to be reduced; however,
its change with network density and flow rate followed closely the
same pattern obtained from the results simulated without en route
guidance, regardless of demand level. This result again suggests
that these relationships are likely inherent network properties that
are primarily governed by network structure and control.
70
high demand
without en route
60
high demand with
20% en route
50
40
30
20
10
0
0
20
40
60
80
Network Density (veh/mile/lane)
(a)
0
20
40
60
80
Network Density (veh/mile/lane)
(b)
FIGURE 7 Comparison of variability–density curves with and without vehicles receiving
en route information in Salt Lake City network at (a) normal and (b) high demand levels.
Mahmassani, Hou, and Saberi
89
80
30
normal demand
without en route
20
normal demand
with 20% en route
10
0
0
Weighted SD of Travel Time
Rate (min/mile)
Weighted SD of Travel Time
Rate (min/mile)
40
70
60
50
40
20
high demand
with 20% en route
10
0
200
400
600
Network Flow (veh/hr)
high demand
without en route
30
0
200
400
600
Network Flow (veh/hr)
(a)
(b)
FIGURE 8 Comparison of variability–flow curves with and without vehicles receiving
en route information in Salt Lake City network at (a) normal and (b) high demand levels.
6
5
y = 1.0483x -0.1469
R2 = 0.6239
4
3
2
1
0
0
1
2
3
4
5
6
Weighted Mean Travel Time Rate
(min/mile)
(a)
Conclusion
This paper established a connection between NFD and travel
time variability, which can be further used for both modeling of network travel time reliability and assessment of measures intended to
improve reliability of travel in a network. In addition to the wellestablished relationships between the networkwide averages of speed,
density, and flow, in the form of a reproducible NFD, this work connected a recently validated relation between the mean travel time
rate (per unit distance) and its standard deviation to derive a relationship between overall travel time reliability experienced by users in a
network and the networkwide averages of flow and density. The paper
validated the two sets of results involved in this derivation in relation
to the existence of a reproducible NFD on an urban scale as well as
a robust linear relationship between the mean travel time rate and
its standard deviation. Through use of simulation results from four
urban networks in the United States, this paper confirmed that both
sets of relationships are inherent network properties (signature) that
are independent of demand level.
Through the inverse relationship between space mean speed and
distance-weighted mean travel time rate, a connection between NFD
and travel time reliability was established. Observations of simulated
Trip Completion Rate (/5 min)
Weighted SD of Travel Time
Rate (min/mile)
that the linear trend obtained from simulation is valid, although the
data appear to be more scattered. The network density and flow rate
cannot be obtained directly from sample trajectory data; instead,
the number of vehicles traveling in the network and number of
trips completed within each 5-min interval are computed. Under the
assumption that the probe vehicles are roughly representative of
the entire vehicle population within the study area, the network
density and flow rate can be inferred by upward scaling of these
two observed quantities with a multiplication factor. That assumption also means that the relationship between these two quantities
reflects a similar pattern as network flow rate versus network density. Figure 9b provides further confirmation by showing a positive
correlation between the number of completed trips and the number
of vehicles in the network, which is consistent with the uncongested
part of NFD as presented in Figure 2. Figure 10 shows how the
weighted standard deviation of travel time rate changes with the
number of vehicles traveling (Figure 10a) and the number of completed trips (Figure 10b). Although the curves are not as smooth as
those obtained from simulated data, the uptrend obviously exists in
both graphs. This trend provides further support for the argument
that travel time variability increases with network density and flow
rate, at least before the maximum network flow rate is reached.
100
80
60
40
20
0
0
100
200
300
400
Number of Vehicles in the
Network
(b)
FIGURE 9 New York City data analysis results: (a) weighted SD of travel time rate
versus weighted mean travel time rate and (b) number of completed trips versus
number of vehicles traveling.
90
Transportation Research Record 2391
6
Weighted SD of Travel
Time Rate (min/mile)
Weighted SD of Travel
Time Rate (min/mile)
6
5
4
3
2
1
0
0
100
200
300
400
Number of Vehicles in the
Network
(a)
5
4
3
2
1
0
0
20
40
60
80 100
Trip Completion Rate (/5 min)
(b)
FIGURE 10 New York City data analysis results: weighted SD of travel time rate versus
number of (a) vehicles traveling and (b) completed trips.
traffic data showed that the distance-weighted standard deviation
of travel time rate, as a measure of travel time variability, increases
monotonically with network density. Network flow rate has a maximum value beyond which network travel time reliability deteriorates
at a much faster pace. Analytical expressions of the relationships
between travel time variability and network density and flow rate
were derived, given the mathematical expression of networkwide
speed–density. The effects of en route information on the proposed
relationships were studied. The results showed that en route information reduces network travel time variability, while the trends of the
relationships remain the same. Finally, the findings in this paper were
validated by using trajectory data collected vehicles from equipped
with Global Positioning Systems in the New York City network.
In summary, this paper filled the gap between network traffic flow
theory and travel time reliability. The relationships between travel time
variability and network density and flow rate were investigated. These
relationships appear to be universally valid and reproducible in different networks and independent of demand level. Analytical formulations have been derived to model these relationships and offer great
potential for further application to model and analyze travel time reliability at the network level. Of particular significance is that the established relationships would allow for the assessment of measures and
policies to improve travel time reliability in a given network without
the need for detailed microlevel analysis.
Acknowledgments
This research was supported in part by the National Science Foundation and by SHRP 2 Project L04, Incorporating Reliability Performance Measures in Operations and Planning Modeling Tools.
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The authors are responsible for all content of the paper.
The Traffic Flow Theory and Characteristics Committee peer-reviewed this paper.