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MACROSCOPIC FLOW MODELS

BY JAMES C. WILLIAMS9

9
Associate Professor, Department of Civil Engineering, University of Texas at Arlington, Box 19308,
Arlington, TX 76019-0308.
CHAPTER 6 - Frequently used Symbols
Note to reader: The symbols used in Chapter 6 are the same as those used in the original sources. Therefore, the reader is cautioned that
the same symbol may be used for different quantities in different sections of this chapter. The symbol definitions below include the sections
in which the symbols are used if the particular symbol definition changes within the chapter or is a definition particular to this chapter.
In each case, the symbols are defined as they are introduced within the text of the chapter. Symbol units are given only where they help
define the quantity; in most cases, the units may be in either English or metric units as necessary to be consistent with other units in a
relation.

A = area of town (Section 6.2.1)

c = capacity (vehicles per unit time per unit width of road) (Section 6.2.1)
D = delay per intersection (Section 6.2.2)
f = fraction of area devoted to major roads (Section 6.1.1)
f = fraction of area devoted to roads (Section 6.2.1)
f = number of signalized intersections per mile (Section 6.2.2)
fr = fraction of moving vehicles in a designated network (Section 6.3)
fs = fraction of stopped vehicles in a designated network (Section 6.3)
fs,min = minimum fraction of vehicles stopped in a network (Section 6.4)
I = total distance traveled per unit area, or traffic intensity (pcu/hour/km) (Sections 6.1.1 and 6.2.3)
J = fraction of roadways used for traffic movement (Section 6.2.1)
K = average network concentration (ratio of the number of vehicles in a network and the network length, Section 6.4)
Kj = jam network concentration (Section 6.4)
N = number of vehicles per unit time that can enter the CBD (Section 6.2.1)
n = quality of traffic indicator (two-fluid model parameter, Section 6.3)
Q = capacity (pcu/hr) (Section 6.2.2)
Q = average network flow, weighted average over all links in a designated network (Section 6.4)
q = average flow (pcu/hr)
R = road density, i.e., length or area of roads per unit area (Section 6.2.3)
r = distance from CBD
T = average travel time per unit distance, averaged over all vehicles in a designated network (Section 6.3)
Tm = average minimum trip time per unit distance (two-fluid model parameter, Section 6.3)
Tr = average moving (running) time per unit distance, averaged over all vehicles in a designated network (Section 6.3)
Ts = average stopped time per unit distance, averaged over all vehicles in a designated network (Section 6.3)
V = average network speed, averaged over all vehicles in a designated network (Section 6.4)
Vf = network free flow speed (Section 6.4)
Vm = average maximum running speed (Section 6.2.3)
Vr = average speed of moving (running) vehicles, averaged over all in a designated network (Section 6.3)
v = average speed
v = weighted space mean speed (Section 6.2.3)
vr = average running speed, i.e., average speed while moving (Section 6.2.2)
w = average street width
 = Zahavi’s network parameter (Section 6.2.3)
 = g/c time, i.e. ration of effective green to cycle length
6.
MACROSCOPIC FLOW MODELS

Mobility within an urban area is a major component of that area's level provides this measurement in terms of the three basic
quality of life and an important issue facing many cities as they variables of traffic flow: speed, flow (or volume), and
grow and their transportation facilities become congested. There concentration. These three variables, appropriately defined, can
is no shortage of techniques to improve traffic flow, ranging also be used to describe traffic at the network level. This
from traffic signal timing optimization (with elaborate, description must be one that can overcome the intractabilities of
computer-based routines as well as simpler, manual, heuristic existing flow theories when network component interactions are
methods) to minor physical changes, such as adding a lane by the taken into account.
elimination of parking. However, the difficulty lies in evaluating
the effectiveness of these techniques. A number of methods The work in this chapter views traffic in a network from a
currently in use, reflecting progress in traffic flow theory and macroscopic point of view. Microscopic analyses run into two
practice of the last thirty years, can effectively evaluate changes major difficulties when applied to a street network:
in the performance of an intersection or an arterial. But a
dilemma is created when these individual components, 1) Each street block (link) and intersection are modeled
connected to form the traffic network, are dealt with collectively. individually. A proper accounting of the interactions
between adjacent network components (particularly in the
The need, then, is for a consistent, reliable means to evaluate case of closely spaced traffic signals) quickly leads to
traffic performance in a network under various traffic and intractable problems.
geometric configurations. The development of such
performance models extends traffic flow theory into the network 2) Since the analysis is performed for each network
level and provides traffic engineers with a means to evaluate component, it is difficult to summarize the results in a
system-wide control strategies in urban areas. In addition, the meaningful fashion so that the overall network performance
quality of service provided to the motorists could be monitored can be evaluated.
to evaluate a city's ability to manage growth. For instance,
network performance models could also be used by a state Simulation can be used to resolve the first difficulty, but the
agency to compare traffic conditions between cities in order to second remains; traffic simulation is discussed in Chapter 10.
more equitably allocate funds for transportation system
improvements. The Highway Capacity Manual (Transportation Research Board
1994) is the basic reference used to evaluate the quality of traffic
The performance of a traffic system is the response of that service, yet does not address the problem at the network level.
system to given travel demand levels. The traffic system consists While some material is devoted to assessing the level of service
of the network topology (street width and configuration) and the on arterials, it is largely a summation of effects at individual
traffic control system (e.g., traffic signals, designation of one- intersections. Several travel time models, beginning with the
and two-way streets, and lane configuration). The number of travel time contour map, are briefly reviewed in the next section,
trips between origin and destination points, along with the followed by a description of general network models in Section
desired arrival and/or departure times comprise the travel 6.2. The two-fluid model of town traffic, also a general network
demand levels. The system response, i.e., the resulting flow model, is discussed separately in Section 6.3 due to the extent of
pattern, can be measured in terms of the level of service the model's development through analytical, field, and simulation
provided to the motorists. Traffic flow theory at the intersection studies. Extensions of the two-fluid model into general network
and arterial models are examined in Section 6.4, and the chapter references
are in the final section.

6.1 Travel Time Models

Travel time contour maps provide an overview of how well a dispatched away from a specified location in the network, and
street network is operating at a specific time. Vehicles can be each vehicle's time and position noted at desired intervals.


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Contours of equal travel time can be established, providing case, general model forms providing the best fit to the data were
information on the average travel times and mean speeds over selected. Traffic intensity (I, defined as the total distance
the network. However, the information is limited in that the traveled per unit area, with units of pcu/hour/km) tends to
travel times are related to a single point, and the study would decrease with increasing distance from the CBD,
likely have to be repeated for other locations. Also, substantial
resources are required to establish statistical significance. Most
importantly, though, is that it is difficult to capture network I
A exp r/a (6.1)
performance with only one variable (travel time or speed in this
case), as the network can be offering quite different levels of
service at the same speed. where r is the distance from the CBD, and A and a are
parameters. Each of the four cities had unique values of A and
This type of model has be generalized by several authors to a, while A was also found to vary between peak and off-peak
estimate average network travel times (per unit distance) or periods. The data from the four cities is shown in Figure 6.1.
speeds as a function of the distance from the central business A similar relation was found between the fraction of the area
district (CBD) of a city, unlike travel time contour maps which which is major road (f) and the distance from the CBD,
consider only travel times away from a specific point.

f
B exp r/b , (6.2)
6.1.1 General Traffic Characteristics
as a Function of the Distance
where b and B are parameters for each town. Traffic intensity
from the CBD and fraction of area which is major road were found to be
linearly related, as was average speed and distance from the
Vaughan, Ioannou, and Phylactou (1972) hypothesized several CBD. Since only traffic on major streets is considered, these
general models using data from four cities in England. In each

Figure 6.1
Total Vehicle Distance Traveled Per Unit Area on Major Roads as a
Function of the Distance from the Town Center (Vaughan et al. 1972).


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results are somewhat arbitrary, depending on the streets selected had been fitted to data from a single city (Angel and Hyman
as major. 1970). The negative exponential asymptotically approaches
some maximum average speed.

6.1.2 Average Speed as a Function of The fifth function, suggested by Lyman and Everall (1971),
Distance from the CBD 1b 2r 2
v
(6.7)
acb 2r 2
Branston (1974) investigated five functions relating average
speed (v) to the distance from the CBD (r) using data collected
by the Road Research Laboratory (RRL) in 1963 for six cities in
also suggested a finite maximum average speed at the city
England. The data was fitted to each function using least-
outskirts. It had originally be applied to data for radial and ring
squares regression for each city separately and for the aggregated
roads separately, but was used for all roads here.
data from all six cities combined. City centers were defined as
the point where the radial streets intersected, and the journey
Two of the functions were quickly discarded: The linear model
speed in the CBD was that found within 0.3 km of the selected
(Equation 6.5) overestimated the average speed in the CBDs by
center. Average speed for each route section was found by
3 to 4 km/h, reflecting an inability to predict the rapid rise in
dividing the section length by the actual travel time
average speed with increasing distance from the city center. The
(miles/minute). The five selected functions are described below,
modified power curve (Equation 6.4) estimated negative speeds
where a, b, and c are constants estimated for the data. A power
in the city centers for two of the cities, and a zero speed for the
curve,
aggregated data. While obtaining the second smallest sum of
v
arb (6.3) squares (negative exponential, Equation 6.6, had the smallest),
the original aim of using this model (to avoid the estimation of
a zero journey speed in the city center) was not achieved.
was drawn from Wardrop's work (1969), but predicts a zero
The fitted curves for the remaining three functions (negative
speed in the city center (at r = 0). Accordingly, Branston also
exponential, Equation 6.6; power curve, Equation 6.3; and
fitted a more general form,
Lyman and Everall, Equation 6.7) are shown for the data from
v
c  ar b , (6.4) Nottingham in Figure 6.2. All three functions realistically
predict a leveling off of average speed at the city outskirts, but
only the Lyman-Everall function indicates a leveling off in the
CBD. However, the power curve showed an overall better fit
where c represents the speed at the city center.
than the Lyman-Everall model, and was preferred.
Earlier work by Beimborn (1970) suggested a strictly linear
While the negative exponential function showed a somewhat
form, up to some maximum speed at the city edge, which was
better fit than the power curve, it was also rejected because of its
defined as the point where the average speed reached its
greater complexity in estimation (a feature shared with the
maximum (i.e., stopped increasing with increasing distance from
Lyman-Everall function). Truncating the power function at
the center). None of the cities in Branston's data set had a clear
measured downtown speeds was suggested to overcome its
maximum limit to average speed, so a strict linear function alone
drawback of estimating zero speeds in the city center. The
was tested:
complete data set for Nottingham is shown in Figure 6.3,
v
a  br . (6.5) showing the fitted power function and the truncation at r = 0.3
km.

A negative exponential function,

v
a b e cr , (6.6)


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Figure 6.2
Grouped Data for Nottingham Showing Fitted a) Power Curve,
b) Negative Exponential Curve, and c) Lyman-Everall Curve
(Branston 1974, Portions of Figures 1A, 1B, and 1C).

Figure 6.3
Complete Data Plot for Nottingham; Power Curve
Fitted to the Grouped Data (Branston 1974, Figure 3).


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If the data is broken down by individual radial routes, as shown Hutchinson (1974) used RRL data collected in 1967 from eight
in Figure 6.4, the relation between speed and distance from the cities in England to reexamine Equations 6.3 and 6.6 (power
city center is stronger than when the aggregated data is curve and negative exponential) with an eye towards simplifying
examined. them.

Figure 6.4
Data from Individual Radial Routes in Nottingham,
Best Fit Curve for Each Route is Shown (Branston 1974, Figure 4).


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The exponents of the power functions fitted by Branston (1974) city. Assuming that any speed between 50 and 75 km/h would
fell in the range 0.27 to 0.36, suggesting the following make little difference, Hutchinson selected 60 km/h, and
simplification
v
60 a e r/R . (6.9)
v
kr 1/3
(6.8)

Hutchinson found that this model raised the sum of squares by

When fitted to Branston's data, there was an average of 18 30 percent (on the average) over the general form used by
percent increase in the sum of squares. The other parameter, k, Branston. R was found to be strongly correlated with the city
was found to be significantly correlated with the city population, population, as well as showing different averages with peak and
with different values for peak and off-peak conditions. The off-peak conditions, while a was correlated with neither the city
parameter k was found to increase with increasing population, population nor the peak vs. off-peak conditions. The difference
and was 9 percent smaller in the peak than in the off-peak. in the Rs between peak and off-peak conditions (30 percent
higher during peaks) implies that low speeds spread out over
In considering the negative exponential model (Equation 6.6), more of the network during the peak, but that conditions in the
Hutchinson reasoned that average speed becomes less city center are not significantly different. Hutchinson (1974)
characteristic of a city with increasing r, and, as such, it would used RRL data collected in 1967 from eight cities in England to
be reasonable to select a single maximum limit for v for every reexamine Equations 6.3 and 6.6 (power curve and negative
exponential) with an eye towards simplifying them.

6.2 General Network Models

A number of models incorporating performance measures other where  is a constant. General relationships between f and
than speed have been proposed. Early work by Wardrop and (N/cA) for three general network types (Smeed 1965) are
Smeed (Wardrop 1952; Smeed 1968) dealt largely with the shown in Figure 6.5. Smeed estimated a value of c (capacity per
development of macroscopic models for arterials, which were unit width of road) by using one of Wardrop's speed-flow
later extended to general network models. equations for central London (Smeed and Wardrop 1964),
q
2440 0.220 v 3 , (6.11)
6.2.1 Network Capacity

Smeed (1966) considered the number of vehicles which can where v is the speed in kilometers/hour, and q the average flow
"usefully" enter the central area of a city, and defined N as the in pcus/hour, and divided by the average road width, 12.6
number of vehicles per unit time that can enter the city center. meters,
In general, N depends on the general design of the road network,
c
58.2 0.00524 v 3 . (6.12)
width of roads, type of intersection control, distribution of
destinations, and vehicle mix. The principle variables for towns
with similar networks, shapes, types of control, and vehicles are:
A, the area of the town; f, the fraction of area devoted to roads; A different speed-flow relation which provided a better fit for
and c, the capacity, expressed in vehicles per unit time per unit speeds below 16 km/h resulted in c = 68 -0.13 v2 (Smeed 1963).
width of road (assumed to be the same for all roads). These are
related as follows: Equation 6.12 is shown in Figure 6.6 for radial-arc, radial, and
ring type networks for speeds of 16 and 32 km/h. Data from
N
 f c A, (6.10) several cities, also plotted in Figure 6.6, suggests that c=30,


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Note: (e=excluding area of ring road, I=including area of ring road)

Figure 6.5
Theoretical Capacity of Urban Street Systems (Smeed 1966, Figure 2).

Figure 6.6
Vehicles Entering the CBDs of Towns Compared with the Corresponding
Theoretical Capacities of the Road Systems (Smeed 1966, Figure 4).


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and using the peak period speed of 16 km/h in central London, 6.2.2 Speed and Flow Relations
Equation 6.10 becomes
Thomson (1967b) used data from central London to develop a
N
33 0.003 v 3 f A , (6.13)
linear speed-flow model. The data had been collected once
every two years over a 14-year period by the RRL and the
Greater London Council. The data consisted of a network-wide
where v is in miles/hour and A in square feet. It should be noted average speed and flow each year it was collected. The average
that f represents the fraction of total area usefully devoted to speed was found by vehicles circulating through central London
roads. An alternate formulation (Smeed 1968) is on predetermined routes. Average flows were found by first
converting measured link flows into equivalent passenger
N
33 0.003 v 3 J f A (6.14)
carunits, then averaging the link flows weighted by their
respective link lengths. Two data points (each consisting of an
average speed and flow) were found for each of the eight years
where f is the fraction of area actually devoted to roads, while J the data was collected: peak and off-peak.
is the fraction of roadways used for traffic movement. J was
found to range between 0.22 and 0.46 in several cities in Plotting the two points for each year, Figure 6.7, resulted in a
England. The large fraction of unused roadway is mostly due to series of negatively sloped trends. Also, the speed-flow capacity
the uneven distribution of traffic on all streets. The number of (defined as the flow that can be moved at a given speed)
vehicles which can circulate in a town depends strongly on their gradually increased over the years, likely due to geometric and
average speed, and is directly proportional to the area of usable traffic control improvements and "more efficient vehicles." This
roadway. For a given area devoted to roads, the larger the indicated that the speed-flow curve had been gradually changing,
central city, the smaller the number of vehicles which can indicating that each year's speed and flow fell on different curves.
circulate in the network, suggesting that a widely dispersed town Two data points were inadequate to determine the shape of the
is not necessarily the most economical design. curve, so all sixteen data points were used by accounting for the

Figure 6.7
Speeds and Flows in Central London, 1952-1966,
Peak and Off-Peak (Thomson 1967b, Figure 11)


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changing capacity of the network, and scaling each year's flow The equation implies a free-flow speed of about 48.3 km/h
measurement to a selected base year. Using linear regression, however, there were no flows less than 2200 pcu/hour in the
the following equation was found: historical data.
v
30.2 0.0086 q (6.15)
Thomson used data collected on several subsequent Sundays
(Thomson 1967a) to get low flow data points. These are
reflected in the trend shown in Figure 6.9.Also shown is a curve
where v is the average speed in kilometers/hour and q is the developed by Smeed and Wardrop using data from a single year
average flow in pcu/hour. This relation is plotted in Figure 6.8. only.

Note: Scaled to 1964 equivalent flows.

Figure 6.8
Speeds and Scaled Flows, 1952-1966 (Thomson 1967b, Figure 2).

Figure 6.9
Estimated Speed-Flow Relations in Central London
(Main Road Network) (Thomson 1967b, Figure 4).


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The selected area of central London could be broken into inner

1
and outer zones, distinguished principally by traffic signal
1 fd , (6.18)
densities, respectively 7.5 and 3.6 traffic signals per route-mile. v vr
Speed and flow conditions were found to be significantly
different between the zones, as shown in Figure 6.10, and for the
inner zone, where v is the average speed in mi/h, vr, the running speed in
mi/h, d the delay per intersection in hours, and f the number of
v
24.3 0.0075 q , (6.16)
signalized intersections per mile. Assuming vr = a(1-q/Q) and
d = b/(1-q/s), where q is the flow in pcu/hr, Q is the capacity
in pcu/hr,  is the g/c time, and s is the saturation flow in pcu/hr,
and for the outer zone, and combining into Equation 6.18,
v
34.0 0.0092 q . (6.17) 1 1

 fb (6.19)
v a (1 q/Q) 1 q/s

Wardrop (1968) directly incorporated average street width and

average signal spacing into a relation between average speed and Using an expression for running speed found for central London
flow, where the average speed includes the stopped time. In (Smeed and Wardrop 1964; RRL 1965),
order to obtain average speeds, the delay at signalized
0.70q  430
intersections must be considered along with the running speed vr
31 (6.20)
between the controlled intersections, where running speed is 3w
defined as the average speed while moving. Since speed is the
inverse of travel time, this relation can be expressed as:

Figure 6.10
Speed-Flow Relations in Inner and Outer Zones of Central Area
(Thomson 1967a, Figure 5).

  
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where w is the average roadway width in feet, and an average For the delay term, five controlled intersections per mile and a
street width of 42 feet (in central London), Equation 6.20 g/c of 0.45 were found for central London. Additionally, the
becomes vr = 28 - 0.0056 q, or 24 mi/h, whichever is less. The intersection capacity was assumed to be proportional to the
coefficient of q was modified to 0.0058 to better fit the observed average stop line width, given that it is more than 5 meters wide
running speed. (RRL 1965), which was assumed to be proportional to the
roadway width. The general form for the delay equation (second
Using observed values of 0.038 hours/mile stopped time, 2180 term of Equation 6.21) is
pcu/hr flow, and 2610 pcu/hr capacity, the numerator of the
second term of Equation 6.19 (fb) was found to be 0.0057.
fb
Substituting the observed values into Equation 6.19, fd
(6.23)
1 q /k  w
1 1 0.0057

 .
v 28 0.0058 q q
1
2610 where k is a constant. For central London, w = 42,  = 0.45,
and kw = Q = 2770, thus k = 147, yielding

Simplifying,
fb
fd
(6.24)
1 q /147  w
1 1 1

 (6.21)
v 28 0.0058 q 197 0.0775 q
Given that f = 5 signals/mile and fb = 0.00507 for central
London, b = 0.00101, yielding
Revising the capacity to 2770 pcu/hour (to reflect 1966 data),
f
thus changing the coefficient of q in the second term of Equation fd
. (6.25)
6.21 to 0.071, this equation provided a better fit than Thomson's 1000 6.8 q/  w
linear relation (Thomson 1967b) and recognizes the known
information on the ultimate capacity of the intersections.
Combining, then, for the general equation for average speed:
Generalizing this equation for urban areas other than London,
and knowing that the average street width in central London was
1 1 f
12.6 meters, the running speed can be written
 .
v 140 q
31 0.0244 1000 6.8 q (6.26)
w w w
430
vr
31 aq
3w w
140 aq

31 . The sensitivity of Equation 6.26 to flow, average street width,
w w number of signalized intersections per mile, and the fraction of
green time are shown in Figures 6.11, 6.12, and 6.13. By
calibrating this relation on geometric and traffic control features
Since a/w = 0.0058 when w = 42 by Equation 6.21, a=0.0244, in the network, Wardrop extended the usefulness of earlier speed
then flow relations. While fitting nicely for central London, the
applicability of this relation to other cities in its generalized
140
vr
31 0.0244 q . (6.22) format (Equation 6.26) is not shown, due to a lack of available
w w data.

  
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Figure 6.11
Effect of Roadway Width on Relation Between Average (Journey)
Speed and Flow in Typical Case (Wardrop 1968, Figure 5).

Figure 6.12
Effect of Number of Intersections Per Mile on Relation Between
Average (Journey) Speed and Flow in Typical Case (Wardrop 1968, Figure 6).

  
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Figure 6.13
Effect of Capacity of Intersections on Relation Between
Average (Journey) Speed and Flow in Typical Case (Wardrop 1968, Figure 7).

Godfrey (1969) examined the relations between the average -relationship, below, and the two-fluid theory of town traffic.
speed and the concentration (defined as the number of vehicles The two-fluid theory has been developed and applied to a greater
in the network), shown in Figure 6.14, and between average extent than the other models discussed in this section, and is
speed and the vehicle miles traveled in the network in one hour, described in Section 6.3.
shown in Figure 6.15. Floating vehicles on circuits within the
network were used to estimate average speed and aerial Zahavi (1972a; 1972b) selected three principal variables, I, the
photographs were used to estimate concentration. traffic intensity (here defined as the distance traveled per unit
area), R, the road density (the length or area of roads per unit
There is a certain concentration that results in the maximum flow area), and v, the weighted space mean speed. Using data from
(or the maximum number of miles traveled, see Figure 6.15), England and the United States, values of I, v, and R were found
which occurs around 10 miles/hour. As traffic builds up past for different regions in different cities. In investigating various
this optimum, average speeds show little deterioration, but there relationships between I and v/R, a linear fit was found between
is excessive queuing to get into the network (either from car the logarithms of the variables:
parking lots within the network or on streets leading into the
designated network). Godfrey also notes that expanding an
intersection to accommodate more traffic will move the queue to I
 ( v/R)m , (6.27)
another location within the network, unless the bottlenecks
downstream are cleared. where  and m are parameters. Trends for London and
Pittsburgh are shown in Figure 6.16. The slope (m) was found
to be close to -1 for all six cities examined, reducing Equation
6.2.3 General Network Models 6.27 to
Incorporating Network Parameters I
 R/v , (6.28)

Some models have defined specific parameters which intend to

quantify the quality of traffic service provided to the users in the where  is different for each city. Relative values of the
network. Two principal models are discussed in this chapter, the variables were calculated by finding the ratio between observed

  
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Figure 6.14
Relationship Between Average (Journey)
Speed and Number of Vehicles on Town
Center Network (Godfrey 1969, Figure 1).

Figure 6.15
Relationship Between Average (Journey) Speed of Vehicles
and Total Vehicle Mileage on Network (Godfrey 1969, Figure 2).

  
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Figure 6.16
The -Relationship for the Arterial Networks of London and Pittsburgh,
in Absolute Values (Zahavi 1972a, Figure 1).

values of I and v/R for each sector and the average value for the The two-fluid model also uses parameters to evaluate the level
entire city. The relationship between the relative values is of service in a network and is described in Section 6.3.
shown in Figure 6.17, where the observations for London and
Pittsburgh fall along the same line.
6.2.4 Continuum Models
The physical characteristics of the road network, such as street
widths and intersection density, were found to have a strong Models have been developed which assume an arbitrarily fine
effect on the value of  for each zone in a city. Thus,  may grid of streets, i.e., infinitely many streets, to circumvent the
serve as a measure of the combined effects of the network errors created on the relatively sparse networks typically used
characteristics and traffic performance, and can possibly be used during the trip or network assignment phase in transportation
as an indicator for the level of service. The  map of London is planning (Newell 1980). A basic street pattern is superimposed
shown in Figure 6.18. Zones are shown by the dashed lines, over this continuum of streets to restrict travel to appropriate
with dotted circles indicating zone centroids. Values of  were directions. Thus, if a square grid were used, travel on the street
calculated for each zone and contour lines of equal  were network would be limited to the two available directions (the x
drawn, showing areas of (relatively) good and poor traffic flow and y directions in a Cartesian plot), but origins and destinations
conditions. (The quality of traffic service improves with could be located anywhere in the network.
increasing .)
Individual street characteristics do not have to be specifically
Unfortunately, Buckley and Wardrop (1980) have shown that  modeled, but network-wide travel time averages and capacities
is strongly related to the space mean speed, and Ardekani (per unit area) must be used for traffic on the local streets. Other
(1984), through the use of aerial photographs, has shown that  street patterns include radial-ring and other grids (triangular, for
has a high positive correlation with the network concentration. example).

  
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Figure 6.17
The -Relationship for the Arterial Networks of London and Pittsburgh,
in Relative Values (Zahavi 1972a, Figure 2).

Figure 6.18
The -Map for London, in Relative Values (Zahavi 1972b, Figure 1).

  
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While the continuum comprises the local streets, the major (within the constraints provided by the superimposed grid) to the
streets (such as arterials and freeways) are modeled directly. network of major streets.
Thus, the continuum of local streets provides direct access

6.3 Two-Fluid Theory

An important result from Prigogine and Herman's (1971) kinetic
theory of traffic flow is that two distinct flow regimes can be Vr
Vm f r n , (6.29)
shown. These are individual and collective flows and are a
where Vm and n are parameters. Vm is the average maximum
function of the vehicle concentration. When the concentration
running speed, and n is an indicator of the quality of traffic
rises so that the traffic is in the collective flow regime, the flow
service in the network; both are discussed below. The average
pattern becomes largely independent of the will of individual
speed, V, can be defined as Vr fr , and combining with Equation
drivers.
6.29,
Because the kinetic theory deals with multi-lane traffic, the two- V
Vm f rn1 . (6.30)
fluid theory of town traffic was proposed by Herman and
Prigogine (Herman and Prigogine 1979; Herman and Ardekani Since f r + fs = 1, where fs is the fraction of vehicles stopped,
1984) as a description of traffic in the collective flow regime in Equation 6.30 can be rewritten
an urban street network. Vehicles in the traffic stream are
divided into two classes (thus, two fluid): moving and stopped V
Vm (1 f s) n 1 . (6.31)
vehicles. Those in the latter class include vehicles stopped in the Boundary conditions are satisfied with this relation: when fs=0,
traffic stream, i.e., stopped for traffic signals and stop signs,
V=Vm , and when fs=1, V=0.
stopped for vehicles loading and unloading which are blocking
a moving lane, stopped for normal congestion, etc., but excludes
This relation can also be expressed in average travel times rather
those out of the traffic stream (e.g., parked cars).
than average speeds. Note that T represents the average travel
time, Tr the running (moving) time, and Ts the stop time, all per
The two-fluid model provides a macroscopic measure of the
unit distance, and that T=1/V, Tr=1/Vr , and Tm=1/Vm , where Tm
quality of traffic service in a street network which is independent
is the average minimum trip time per unit distance.
of concentration. The model is based on two assumptions:
The second assumption of the two-fluid model relates the
(1) The average running speed in a street network is
fraction of time a test vehicle circulating in a network is stopped
proportional to the fraction of vehicles that are moving,
to the average fraction of vehicles stopped during the same
and
period, or
(2) The fractional stop time of a test vehicle circulating in
a network is equal to the average fraction of the Ts
vehicles stopped during the same period. fs
. (6.32)
T
The variables used in the two-fluid model represent network- This relation has been proven analytically (Ardekani and
wide averages taken over a given period of time. Herman 1987), and represents the ergodic principle embedded
in the model, i.e., that the network conditions can be represented
The first assumption of the two-fluid theory relates the average by a single vehicle appropriately sampling the network.
speed of the moving (running) vehicles, Vr , to the fraction of
moving vehicles, fr , in the following manner: Restating Equation 6.31 in terms of travel time,

  
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observations of stopped and moving times gathered in each

T
Tm ( 1 f s) (n1) . (6.33) network. The log transform of Equation 6.35,
Incorporating Equation 6.32, 1 n
ln Tr
ln Tm  ln T (6.37)
(n1) n1 n1
T
Tm 1 ( Ts /T ) , (6.34)
provides a linear expression for the use of least squares analysis.
realizing that T = Tr + Ts , and solving for Tr ,
1 n Empirical information has been collected with chase cars
Tr
Tm n1 T n1 . (6.35) following randomly selected cars in designated networks. Runs
have been broken into one- or two-mile trips, and the running
The formal two-fluid model formulation, then, is time (Tr) and total trip time (T) for each one- or two-mile trip
1 n from the observations for the parameter estimation. Results tend
Ts
T Tm n1 T n1 . (6.36) to form a nearly linear relationship when trip time is plotted
against stop time (Equation 6.36) as shown in Figure 6.19 for
A number of field studies have borne out the two-fluid model data collected in Austin, Texas. The value of Tm is reflected by
(Herman and Ardekani 1984; Ardekani and Herman 1987; the y-intercept (i.e., T at Ts=0), and n by the slope of the curve.
Ardekani et al. 1985); and have indicated that urban street Data points representing higher concentration levels lie higher
networks can be characterized by the two model parameters, n along the curve.
and Tm . These parameters have been estimated using

Note: Each point represents one test run approximately 1 or 2 miles long.

Figure 6.19
Trip Time vs. Stop Time for the Non-Freeway Street Network of the Austin CBD
(Herman and Ardekani 1984, Figure 3).

  
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6.3.1 Two-Fluid Parameters congestion, and when congestion is high, vehicles when moving,
travel at a lower speed (or higher running time per unit distance)
The parameter Tm is the average minimum trip time per unit than they do when congestion is low. In fact, field studies have
distance, and it represents the trip time that might be shown that n varies from 0.8 to 3.0, with a smaller value
experienced by an individual vehicle alone in the network with typically indicating better operating conditions in the network.
no stops. This parameter is unlikely to be measured directly, In other words, n is a measure of the resistance of the network to
since a lone vehicle driving though the network very late at night degraded operation with increased demand. Higher values of n
is likely to have to stop at a red traffic signal or a stop sign. indicate networks that degrade faster as demand increases.
Tm , then, is a measure of the uncongested speed, and a higher Because the two-fluid parameters reflect how the network
value would indicate a lower speed, typically resulting in poorer responds to changes in demand, they must be measured and
operation. Tm has been found to range from 1.5 to 3.0 evaluated in a network over the entire range of demand
minutes/mile, with smaller values typically representing better conditions.
operating conditions in the network.
While lower n and Tm values represent, in general, better traffic
As stop time per unit distance ( Ts ) increases for a single value operations in a network, often there is a tradeoff. For example,
of n, the total trip time also increases. Because T=Tr+Ts , the two-fluid trends for four cities are shown in Figure 6.20. In
total trip time must increase at least as fast as the stop time. If comparing Houston (Tm=2.70 min/mile, n=0.80) and Austin
n=0, Tr is constant (by Equation 6.35), and trip time would (Tm=1.78 min/mile, n=1.65), one finds that traffic in Austin
increase at the same rate as the stop time. If n>0, trip time moves at significantly higher average speeds during off-peak
increases at a faster rate than the stop time, meaning that running conditions (lower concentration); at higher concentrations, the
time is also increasing. Intuitively, n must be greater than zero, curves essentially overlap, indicating similar operating
since the usual cause for increased stop time is increased conditions. Thus, despite a higher value of n, traffic conditions

Note: Trip Time vs. Stop Time Two-Fluid Model Trends for CBD Data From the Cities of Austin, Houston, and San Antonio,
Texas, and Matamoros, Mexico.
Figure 6.20
Trip Time vs. Stop Time Two-Fluid Model Trends
(Herman and Ardekani 1984, Figure 6).

  
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are better in Austin than Houston, at least at lower being chased imitating the other driver's actions so as to reflect,
concentrations. Different values of the two-fluid parameters are as closely as possible, the fraction of time the other driver spends
found for different city street networks, as was shown above and stopped. The objective is to sample the behavior of the drivers
in Figure 6.21. The identification of specific features which have in the network as well as the commonly used routes in the street
the greatest effect on these parameters has been approached network. The chase car's trip history is then broken into one-
through extensive field studies and computer simulation. mile (typically) segments, and Tr and T calculated for each mile.
The (Tr ,T) observations are then used in the estimation of the
two-fluid parameters.
6.3.2 Two-Fluid Parameters: Influence
of Driver Behavior One important aspect of the chase car study is driver behavior,
both that of the test car driver and the drivers sampled in the
Data for the estimation of the two-fluid parameters is collected network. One study addressed the question of extreme driver
through chase car studies, where the driver is instructed to follow behaviors, and found that a test car driver instructed to drive
a randomly selected vehicle until it either parks or leaves the aggressively established a significantly different two-fluid trend
designated network, after which a nearby vehicle is selected and than one instructed to drive conservatively in the same network
followed. The chase car driver is instructed to follow the vehicle at the same time (Herman et al. 1988).

Note: Trip Time vs. Stop Time Two-Fluid Model Trends for Dallas and Houston, Texas, compared to the trends in Milwaukee,
Wisconsin, and in London and Brussels.

Figure 6.21
Trip Time vs. Stop Time Two-Fluid Model Trends Comparison
(Herman and Ardekani 1984, Figure 7).

  
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The two-fluid trends resulting from the these studies in two cities at lower network concentrations, the aggressive driver can take
are shown in Figure 6.22. In both cases, the normal trend was advantage of the less crowded streets and significantly lower his
found through a standard chase car study, conducted at the same trip times.
time as the aggressive and conservative test drivers were in the
network. In both cases, the two-fluid trends established by the As shown in Figure 6.22b, aggressive driving behavior more
aggressive and conservative driver are significantly different. In closely reflects normal driving habits in Austin, suggesting more
Roanoke (Figure 6.22a), the normal trend lies between the aggressive driving overall. Also, all three trends converge at
aggressive and conservative trends, as expected. However, the high demand (concentration) levels, indicating that, perhaps, the
aggressive trend approaches the normal trend at high demand Austin network would suffer congestion to a greater extent than
levels, reflecting the inability of the aggressive driver to reduce Roanoke, reducing all drivers to conservative behavior (at least
his trip and stop times during peak periods. On the other hand, as represented in the two-fluid parameters).

Note: The two-fluid trends for aggressive, normal, and conservative drivers in (a) Roanoke, Virginia, and (b) Austin, Texas

Figure 6.22
Two-Fluid Trends for Aggressive, Normal, and Conservative Drivers
(Herman et al. 1988, Figures 5 and 8).

  
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The results of this study reveal the importance of the behavior of approaches with signal progression. Of these, only two features
the chase car driver in standard two-fluid studies. While the (average block length and intersection density) can be
effects on the two-fluid parameters of using two different chase considered fixed, and, as such, not useful in formulating network
car drivers in the same network at the same time has not been improvements. In addition, one feature (average number of
investigated, there is thought to be little difference between two lanes per street), also used in the previous study (Ayadh 1986),
well-trained drivers. To the extent possible, however, the same can typically be increased only by eliminating parking (if
driver has been used in different studies that are directly present), yielding only limited opportunities for improvement of
compared. traffic flow. Data was collected in ten cities; in seven of the
cities, more than one study was conducted as major geometric
changes or revised signal timings were implemented, yielding
6.3.3 Two-Fluid Parameters: Influence nineteen networks for this study. As before, the two-fluid
parameters in each network were estimated from chase car data
of Network Features (Field Studies)
and the network features were determined from maps, field
studies, and local traffic engineers. Regression analysis yielded
Geometric and traffic control features of a street network also
the following models:
play an important role in the quality of service provided by a
network. If relationships between specific features and the two-
fluid parameters can be established, the information could be Tm
3.93  0.0035 X5 0.047 X6 0.433 X10 (6.39)
used to identify specific measures to improve traffic flow and and n
1.73  1.124 X2 0.180 X3 0.0042 X5 0.271 X9
provide a means to compare the relative improvements.

Ayadh (1986) selected seven network features: lane miles per

where X2 is the fraction of one-way streets, X3 the average
square mile, number of intersections per square mile, fraction of
number of lanes per street, X5 the signal density, X6 the average
one-way streets, average signal cycle length, average block
speed limit, X9 the fraction of actuated signals, and X10 the
length, average number of lanes per street, and average block
fraction of approaches with good progression. The R2 for these
length to block width ratio. The area of the street network under
equations, 0.72 and 0.75 (respectively), are lower than those for
consideration is used with the first two variables to allow a direct
Equation 6.38 (both very close to 1), reflecting the larger data
comparison between cities. Data for the seven variables were
size. The only feature in common with the previous model
collected for four cities from maps and in the field. Through a
(Equation 6.38) is the appearance of the fraction of one-way
regression analysis, the following models were selected:
streets in the model for n. Since all features selected can be
Tm
3.59 0.54 C6 and changed through operational practices (signal density can be
n
0.21  2.97 C3  0.22 C7 (6.38) changed by placing signals on flash), the models have potential
practical application. Computer simulation has also been used
to investigate these relationships, and is discussed in Section
6.3.4.
where C3 is the fraction of one-way streets, C6 the average
number of lanes per street, and C7 the average block length to
block width ratio. Of these network features, only one (the
fraction of one-way streets) is relatively inexpensive to 6.3.4 Two-Fluid Parameters: Estimation
implement. One feature, the block length to block width ratio, by Computer Simulation
is a topological feature which would be considered fixed for any
established street network. Computer simulation has many advantages over field data in the
study of network models. Conditions not found in the field can
Ardekani et al. (1992), selected ten network features: average be evaluated and new control strategies can be easily tested. In
block length, fraction of one-way streets, average number of the case of the two-fluid model, the entire vehicle population in
lanes per street, intersection density, signal density, average the network can be used in the estimation of the model
speed limit, average cycle length, fraction of curb miles with parameters, rather than the small sample used in the chase car
parking allowed, fraction of signals actuated, and fraction of studies. TRAF-NETSIM (Mahmassani et al. 1984), a

  
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microscopic traffic simulation model, has been used successfully can be simulated by recording the trip history of a single vehicle
with the two-fluid model. for one mile, then randomly selecting another vehicle in the
network. Because the two-fluid model is non-linear
Most of the simulation work to-date has used a generic grid (specifically, Equation 6.35, the log transform of which is used
network in order to isolate the effects of specific network to estimate the parameters), estimations performed at the
features on the two-fluid parameters (FHWA 1993). Typically, network level and at the individual vehicle level result in
the simulated network has been a 5 x 5 intersection grid made up different values of the parameters, and are not directly
entire of two-way streets. Traffic signals are at each intersection comparable. The sampling strategy, which was found to provide
and uniform turning movements are applied throughout. The the best parameter estimates, required a single vehicle
network is closed, i.e., vehicles are not allowed to leave the circulating in the network for at least 15 minutes. However, due
network, thus maintaining constant concentration during the to the wide variance of the estimate (due to the possibility of a
simulation run. The trip histories of all the vehicles circulating relatively small number of "chased" cars dominating the sample
in the network are aggregated to form a single (Tr , T) estimation), the estimate using a single vehicle was often far
observation for use in the two-fluid parameter estimation. A from the parameter estimated at the network level. On the other
series of five to ten runs over a range of network concentrations hand, using 20 vehicles to sample the network resulted in
(nearly zero to 60 or 80 vehicles/lane-mile) are required to estimates much closer to those at the network level. The much
estimate the two-fluid parameters. smaller variance of the estimates made with twenty vehicles,
however, resulted in the estimate being significantly different
Initial simulation runs in the test network showed both T and Ts from the network-level estimate. The implication of this study
increasing with concentration, but Tr remaining nearly constant, is that, while estimates at the network and individual vehicle
indicating a very low value of n (Mahmassani et al. 1984). In levels can not be directly compared, as long as the same
its default condition, NETSIM generates few of the vehicle sampling strategy is used, the resulting two-fluid parameters,
interaction of the type found in most urban street networks, although biased from the "true" value, can be used in making
resulting in flow which is much more idealized than in the field. direct comparisons.
The short-term event feature of NETSIM was used to increase
the inter-vehicular interaction (Williams et al. 1985). With this
feature, NETSIM blocks the right lane of the specified link at 6.3.5 Two-Fluid Parameters:
mid-block; the user specifies the average time for each blockage Influence of Network Features
and the number of blockages per hour, which are stochastically
applied by NETSIM. In effect, this represents a vehicle stopping (Simulation Studies)
for a short time (e.g., a commercial vehicle unloading goods),
blocking the right lane, and requiring vehicles to change lanes to The question in Section 6.3.3, above, regarding the influence of
go around it. The two-fluid parameters (and n in particular) geometric and control features of a network on the two-fluid
were very sensitive to the duration and frequency of the short- parameters was revisited with an extensive simulation study
term events. For example, using an average 45-second event (Bhat 1994). The network features selected were: average
every two minutes, n rose from 0.076 to 0.845 and Tm fell from block length, fraction of one-way streets, average number of
2.238 to 2.135. With the use of the short-term events, the values lanes per street, signals per intersection, average speed limit,
of both parameters were within the ranges found in the field average signal cycle length, fraction of curb miles with parking,
studies. Further simulation studies found both block length and fraction of signalized approaches in progression. A
(here, distance between signalized intersections) and the use of uniform-precision central composite design was selected as the
progression to have significant effects on the two-fluid experimental design, resulting in 164 combination of the eight
parameters (Williams et al. 1985). network variables. The simulated network was increased to 11
by 11 intersections; again, vehicles were not allowed to leave the
Simulation has also provided the means to investigate the use of network, but traffic data was collected only on the interior 9 by
the chase car technique in estimating the two-fluid parameters 9 intersection grid, thus eliminating the edge effects caused by
(Williams et al. 1995). The network-wide averages in a the necessarily different turning movements at the boundaries.
simulation model can be directly computed; and chase car data Ten simulation runs were made for each combination of

  
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variables over a range of concentrations from near zero to about NETSIM reflected traffic conditions in San Antonio, NETSIM
35 vehicles/lane-mile. was calibrated with the two-fluid model.

Regression analysis yielded the following models: Turning movement counts used in the development of the new
signal timing plans were available for coding NETSIM.
Tm
1.049  1.453 X2  0.684 X3 0.024 X6 and
(6.40) Simulation runs were made for 31 periods throughout the day,
n
4.468 1.391 X3 0.048 X5  0.042 X6
and the two-fluid parameters were estimated and compared with
those found in the field. By a trial and error process, NETSIM
was calibrated by
where X2 is the fraction of one-way streets, X 3the number of
lanes per street, X5 average speed limit, and X 6 average cycle  Increasing the sluggishness of drivers, by increasing
length. The R2 (0.26 and 0.16 for Equation 6.40) was headways during queue discharge at traffic signals
considerably lower than that for the models estimated with data and reducing maximum acceleration,
from field studies (Equation 6.39). Additionally, the only  Adding vehicle/driver types to increase the range of
variable in common between Equations 6.39 and 6.40 is the sluggishness represented in the network, and
number of lanes per street in the equation for n. Additional work  Reducing the desired speed on all links to 32.2 km/h
is required to clarify these relationships. during peaks and 40.25 km/h otherwise (Denney
1993).

Three measures of effectiveness (MOEs) were used in the

6.3.6 Two-Fluid Model: evaluation: total delay, number of stops, and fuel consumption.
A Practical Application The changes noted for all three MOEs were greater between
calibrated and uncalibrated NETSIM results than between
When the traffic signals in downtown San Antonio were retimed, before and after results. Reported relative improvements were
TRAF-NETSIM was selected to quantify the improvements in also affected. The errors in the reported improvements without
the network. In order to assure that the results reported by calibration ranged from 16 percent to 132 percent (Denney
1994).

6.4 Two-Fluid Model and Traffic Network Flow Models

Computer simulation provides an opportunity to investigate fluid model, to the concentration. In addition, using values of
network-level relationships between the three fundamental flow, speed, and concentration independently computed from the
variables of traffic flow, speed (V), flow (Q), and concentration simulations, the network-level version of the fundamental
(K), defined as average quantities taken over all vehicles in the relation Q=KV was numerically verified (Mahmassani et al.
network over some observation period (Mahmassani et al. 1984; Williams et al. 1987).
1984). While the existence of "nice" relations between these
variables could not be expected, given the complexity of network Three model systems were derived and tested against simulation
interconnections, simulation results indicate relationships similar results (Williams et al. 1987; Mahmassani et al. 1987); each
to those developed for arterials may be appropriate (Mahmassani model system assumed Q=KV and the two-fluid model, and
et al. 1984; Williams et al. 1985). A series of simulation runs, consisted of three relations:
as described in Section 6.3.4, above, was made at concentration
levels between 10 and 100 vehicles/lane-mile. The results are
shown in Figure 6.23, and bear a close resemblance to their
V
f (K) , (6.41)
counterparts for individual road sections. The fourth plot shows
the relation of fs , the fraction of vehicles stopped from the two-

  
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Figure 6.23
Simulation Results in a Closed CBD-Type Street Network.
(Williams et al. 1987, Figures 1-4).

A model system is defined by specifying one of the above relationships; the other two can then be analytically derived. (A
relation between Q and V could also be derived.)

Q
g (K) , and (6.42) Model System 1 is based on a postulated relationship between
the average fraction of vehicles stopped and the network
f s
h (K) . (6.43) concentration from the two-fluid theory (Herman and Prigogine

  
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1979), later modified to reflect that the minimum fs > 0 then by using Q=KV,
(Ardekani and Herman 1987):

f s
fs,min  ( 1 fs,min ) ( K/Kj )% , Q
Vf ( K K 2 /Kj ) . (6.49)
(6.44)

Equations 6.47 through 6.49 were fitted to the simulation data

where fs,min is the minimum fraction of vehicles stopped in a and are shown in Figure 6.25. The difference between the
network, Kj is the jam concentration (at which the network is Method 1 and Method 2 curves in the fs-K plot (Figure 6.25) is
effectively saturated), and % is a parameter which reflects the described above. Model System 3 uses a non-linear bell-shaped
quality of service in a network. The other two relations can be function for the V-K model, originally proposed by Drake, et al.,
readily found, first by substituting fs from Equation 6.44 into for arterials (Gerlough and Huber 1975):
Equation 6.31:
V
Vf exp[  ( K/Km )d ] , (6.50)
n1
V
Vm (1 fs,min ) [1 (K/Kj )%]n1 , (6.45)

where Km is the concentration at maximum flow, and  and d are

then by using Q=KV, parameters. The fs-K and Q-K relations can be derived as shown
for Model System 2:
Q
K Vm (1 fs,min )n1 [1 (K/Kj )%]n1 . (6.46)
f s
1 { (Vf /Vm ) exp [  (K/Km )d ] }1/(n1) and (6.51)

Equations 6.44 through 6.46 were fitted to the simulated data Q

K Vf exp[  ( K/Km )d ] . (6.52)
and are shown in Figure 6.24. Because the point representing
the highest concentration (about 100 vehicles/lane-mile) did not
lie in the same linear lnTr - lnT trend as the other points, the two-
fluid parameters n and Tm were estimated with and without the
highest concentration point, resulting in the Method 1 and Equations 6.50 through 6.52 were fitted to the simulation data
Method 2 curves, respectively, in the V-K and Q-K curves in and are shown in Figure 6.26.
Figure 6.24.
Two important conclusions can be drawn from this work. First,
Model System 2 adopts Greenshields' linear speed-concentration that relatively simple macroscopic relations between network-
relationship (Gerlough and Huber 1975), level variables appear to work. Further, two of the models
shown are similar to those established at the individual facility
level. Second, the two-fluid model serves well as the theoretical
V
Vf (1 K/Kj ) , (6.47) link between the postulated and derived functions, providing
another demonstration of the model's validity. In the second and
third model systems particularly, the derived fs-K function
where Vf is the free flow speed (and is distinct from Vm ; performed remarkable well against the simulated data, even
Vf  Vm always, and typically Vf < Vm ). The fs-K relation can be though it was not directly calibrated using that data.
found by substituting Equation 6.47 into Equation 6.31 and
solving for fs :
f s
1 [(Vf /Vm) (1 K/Kj ) ]1/(n1) , (6.48)

  
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<

Figure 6.24
Comparison of Model System 1 with Observed Simulation Results
(Williams et al. 1987, Figure 5, 7, and 8).

  
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Figure 6.25
Comparison of Model System 2 with Observed Simulation Results
(Williams et al. 1987, Figures 9-11).

  
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Figure 6.26
Comparison of Model System 3 with Observed Simulation Results
(Williams et al. 1987, Figures 12-14).

  
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6.5 Concluding Remarks

As the scope of traffic control possibilities widens with the optimization of the control system) becomes clear. While the
development of ITS (Intelligent Transportation Systems) models discussed in this chapter are not ready for easy
applications, the need for a comprehensive, network-wide implementation, they do have promise, as in the application of
evaluation tool (as well as one that would assist in the the two-fluid model in San Antonio (Denney et al. 1993; 1994).

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