USE OF INTERSECTION DELAY FUNCTIONS TO
IMPROVE RELIABILITY OF TRAFFIC ASSIGNMENT
MODEL
Prepared for:
Presentation at the 14th Annual International EMME/2 Conference
Chicago, Illinois
October 22, 1999
Prepared by:
Hedayat Z. Aashtiani
Sharif University of Technology
Tehran, Iran
Hamid Iravani
Tehran Comprehensive Transportation and Traffic Studies (TCTTS)
No. 171/1, Mirdamad Ave.
Tehran 19116, Iran
Currently with:
WILBUR SMITH ASSOCIATES
9800 Richmond Avenue, Suite 400
Houston, Texas 77042-4524
(713) 785-0080
Fax (713) 785-8797
E-mail: hiravani@wilbursmith.com
USE OF INTERSECTION DELAY FUNCTIONS TO IMPROVE
RELIABILITY OF TRAFFIC ASSIGNMENT MODEL
Abstract
This study develops a simple method for estimating delay at intersections with and
without signals. A knowledge of the magnitude of intersection delay can assist planners to
optimize traffic flow at intersections. The technique described is applied by using EMME/2
transportation planning software. Calculation of intersection delay is also deemed necessary
when modeling traffic flow. An accurate estimation of intersection delay will lead to more
precise traffic assignment results.
The area of analysis comprised of the various
transportation analysis zones in the city of Tehran. The model output was shown to be
statistically significant. Also; our calculations were found to be valid when compared with
observed data.
USE OF INTERSECTION DELAY FUNCTIONS TO IMPROVE
RELIABILITY OF TRAFFIC ASSIGNMENT MODEL
Introduction
Physical limitations and constraints have hindered further expansion of the road
network systems.
Hence, there has been a tendency to optimize the use of existing
intersections. However, such optimization demands an accurate estimation of peak hour
delay. Moreover, use of reliable intersection delay in traffic models could bring estimated
volumes closer to observed data and improve the overall result. Estimation of intersection
delay however has received little attention in transportation modeling. Through a case study
in Tehran, the capital city of Iran, a new approach was developed to simulate delays at
intersections. As a result, the precision of the model was raised significantly. The method
used is applicable in other studies, in particular, when input data for intersections for
calculating delays can not be readily available, which is the case in most large cities, especially
in developing countries where this problem is even more acute. In addition, the proposed
methodology allows forecasting delays at any intersection in the auto network.
Project Description
The City of Tehran has recently completed a comprehensive study to develop a
transportation plan. In this effort the first multi-modal transportation model was developed
and it included many sub-models. The major challenge in this regard has been the fact that
most elements of the model had to be developed from scratch, coupled with the large scale of
the study area and lack of suitable data. Also, the model had to be compatible for local needs.
Tehran has currently a population of 7 million span over an area of 600 squared kilometer.
The Model area includes peripheral cities as well. In developing the model, emme/2
transportation planning computer software was used. The study area was divided into 583
transportation analysis zones, and the network system was coded into emme/2 as links (street
segments) and nodes (intersections). The existing condition includes 4295 nodes and 12768
links.
When model calibration was completed, various alternatives were developed, a
sophisticated alternative evaluation technique applied, and finally a preferred alternative was
selected. Our model proposed many innovative approaches. This paper intends to explain
how this model obtained a more reliable result when simulating delays at intersections.
Study Approach
In transportation models, link delay or volume delay functions are the essential
elements that need to be computed when traffic assignment on network is conducted. The
purpose of this function is to determine the time on the network links. Based on the
magnitude of time, shortest path could then be found when traffic volume is being assigned on
different routes. In an inter city network the cost associated with the nodes are negligible and
can be ignored. However, in an urban area where intersections cause significant delays for
travelers, their impacts are high on travel time and, sometimes, improving intersections could
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be even more cost advantageous than widening the streets. Consequently, modelers should
also consider this factor in their designs. However, treatment of intersection delay for real size
network could be a complex method to simulate. For a small size network, traffic engineers
have undertaken the delay at intersections, at case by case level, and they have obtained
reasonably accurate results. This approach is not as straightforward in modeling practices
when using large networks. In most computer packages for transportation planning, delay at
intersections are treated as turn penalty functions. Usually, these turn functions are assumed
to be constants and used only at selected nodes to force the model to fit the observed data. In
the City of Tehran, intersection delay functions were modeled for both intersections, with and
without traffic signals.
Methodology
In most of the existing traffic assignment algorithms, delays are considered only at
links. Furthermore, to guarantee the convergence of the algorithms, it is assumed that the
delay function for each link is only a function of the traffic flow of that link without
consideration of the opposing links. One approach is to consider delay at intersections, is to
use the current algorithms and expand the nodes and representing each turn as a dummy
link. Obviously, delay at intersecting links depends not only on the physical characteristics
and control policies of intersections, but also on the traffic flows of other links. There are two
major problems associated with converting every turn to a link. First, the size of the network
becomes too large. Secondly, the delay function for dummy links depend on the traffic flow of
other links. To overcome these difficulties two assumptions are made. First, it is assumed
that all turns corresponding to each link entering an intersection have the same delay. By this
assumption there is no need to expand the nodes, and the delay at intersection could be
included as a part of the delay function for the entering link. Second, the delay associated
with intersection for each entering link depends on the physical characteristics and control
policy of that intersection and the volume of traffic on that link. Although, at operation level
these assumptions, especially for unsignalized intersections, seem to be unrealistic, but, at
planning stage when the objectives narrow down to strategic decisions, they are justified and,
in comparison to exclusion of delay at intersection, is more realistic.
Delay Functions for Signalized Intersections
The function used for calculation of signalized intersection is based on Webster’s
formula. However, our model involved additional steps for calculation of red and total cycle
used in the expression. Based on Webster’s model the general equation to replicate the time
involved for delay is:
d  x 
r2

x 
2 c  1 

 w 

where,
x
= Traffic volume on entering link, in vehicle (passenger car equivalent) per hour.

= Exiting rate of traffic volume, in vehicle (passenger car equivalent) per one meter width
per hour.
2
r
c
d
w
= Red time of the traffic light, in minutes.
= Cycle length of the traffic light, in minutes.
= Average delay at intersection, in minutes.
= Width of the link, in meter.
The magnitude of the  , based on survey was, 533 vehicles per one meter width per
hour. In order to avoid possible occurrence of error, max{[1-x/(*w)] and 0.01} was used
instead of [1-x/(*w)].
As explained, at each link the corresponding volume delay function was also used.
This function is same as the BPR equation and the general format is
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
 x  
t ( x )  l t0  1  015
. 
 
c w   





In this equation, t(x) and t0 are travel time on link and the free flow travel time in
minutes, l is the length of the link and c is the capacity of each link per meter of the street
width which varies depending on the functional class of street. In our model street width is
used instead of number of lanes, this is due to driving behavior in Tehran, where number of
lanes do not necessarily dictate the number of cars being accommodated across the width of
the street. Considering both equations, the general function used in estimating link delay is
expressed as follows:
t  t  x  d  x
Estimation of Red Time and Total Cycle Length
The data corresponding to cycle length and red times might not be available for some
intersections, especially at planning level. Thus, a simple method has been developed to
estimate these data.
Assuming node j is an intersection which is equipped with traffic signal, sj is the set of links
ending to node j. The following weights, for each link entering node j , could be defined, based
upon their functional class, as follows:
wij
2,
= 3,
4,
5,
if
if
if
if
the
the
the
the
link
link
link
link
is
is
is
is
local or collector
minor arterial
major arterial
express way
The following formula used the above weights to calculate the total cycle length:
  wij

 is j
sj 
cj  1 


4 
 8



where cj is total cycle time in minutes at node j.
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To calculate the red cycle, the following equation was developed:



s j  wij 

rij  1.2  c j  1 
 2   wij 
i s j


The value of
r2
2c
is assumed to be greater than 0.2 and less than one minute. The functional
form of these two equations have the following desirable characteristics:
 Each link of any intersection takes into account the effect of all other links belonging to the
same intersection.
 Intersections with more legs will cause more delay time, assuming the rest of explanatory
variables remain unchanged.
 Intersections with higher functional classification will cause more delay time, assuming the
rest of explanatory variables being the same.
 At any given intersection, legs with higher functional classifications will take less red cycle
time comparing to other legs of the same intersection.
 Delay on every signalized intersection, on any size of network, could be calculated using the
equation.
Delay Functions for Unsignalized Intersections
The following equation was developed to estimate delay for intersections without any
traffic control.
 x 

d2  x  d2  Min
 . and .1
 c w 

where,
 n *  k  1
d2  0.05 

2


p

where,
n
= Number of links ending at the intersection
k
= Number of links exiting from the intersection
p
= Number of turns prohibited at each intersection
n*(k-1) shows the number of possible turns. The functional form on this equation has the
following desirable characteristics:
 Each link of any intersection, takes into account, the effect of all other links belonging to
the same intersection.
 Intersections with more legs will cause more delay time, assuming the rest of explanatory
variables remain the same.
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 Intersections with more turn prohibition will end up with less delay.
 Delay on intersections without traffic signal, on any size of network, could be calculated
using the equation.
This function was added to the link delay function on the last link terminating at the last
node. The equation could include the effect of stop signs, wherever applicable. Links
intersecting with centroid connectors were not considered
as an intersection in our
computation.
Results
As a consequence of the applied method, the accuracy of the city of Tehran model was
raised significantly; estimated volumes on auto network were compared against observed
traffic count along screen lines. Table 1 shows the result of regression analysis, conducted on
auto assignment result as estimated volumes were compared against observed data. The
results indicate that, the effect of using both types of intersection delay functions on the
assigned auto volume is fairly significant and is recommended to be used in other studies.
Table 1: Regression results; estimated traffic volumes against observed data
Scenario Description
r2
Without using any intersection delay functions
0.69
Using just unsignalized intersection delay functions
0.72
Using just signalized intersection delay functions
0.75
Using both signalized and unsignalized delay functions
0.77
Summary
The paper presented a simple method for calculating delay at different types of
intersections in the city of Tehran for existing and future conditions. The main feature of our
methodology includes the use of readily available data, and therefore lower cost. The findings
showed our estimation to be valid when compared with observed data. Despite the limitations
related to exclusion of opposing traffic volume, our methodology proved to be statistically valid
and improved the calibration process. In addition, by knowing the magnitude of delay at
different intersections, planners can use the model to prioritize intersections according to their
delay time and, hence, alleviate congestion. The equations used, could also be applied as a
guidance to lead engineers to optimize traffic signal timing. An area to consider for further
research is the inclusion of opposing traffic volume in intersection delay.
Note
This paper is based on study undertaken by Tehran Comprehensive Transportation
and Traffic Studies (TCTTS), a division of the Municipality of Tehran, to develop the city’s
regional transportation plan.
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References
1. EMME/2 User’s Manual, Software Release 7. INRO, Montreal, Quebec, Canada,
1994.
2. Final Structure of Tehran Transportation Model in EMME/2 Software, Traffic
Assignment Report; Tehran Comprehensive Transportation and Traffic Studies.
Tehran, Iran. 1996.
3. Highway Capacity Manual, Special Report 209; Transportation Research Board.
Washington, D.C., 1980.
4. Webster, F.V., Traffic Signal Setting, Report 39, Road Research Laboratory
Crowthorne. Berkshire, England, 1958.
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