Chapter 6 Fundamentals of traffic flow

Reference: Chapter 6 [Traffic and Highway Engineering (Nicholas J. Garber Lester A. Hoel)]

1. 15/12/2019
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Fundamental Principles of
Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
1
1
Fayaz Rashid, MSc
Taxila Institute of Transportation Engineering
(TITE)
2
•Traffic Flow Elements
•Flow-density Relationship
•Shock Waves In Traffic Stream
•Gap And Gap Acceptance
•Queuing Theory
•Numerical Problems
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
1
2

2. 15/12/2019
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Outlines
Importance of Traffic flow theory
•Traffic flow theory involves the development of
mathematical relationships among the primary elements of
a traffic stream: flow, density, and speed.
•These relation-ships help the traffic engineer in planning,
designing, and evaluating the effectiveness of
implementing traffic engineering measures on a highway
system.
•This chapter, however, will introduce only those aspects of
traffic flow theory that can be used in the planning,
design, and operation of highway systems
3Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Importance
of Traffic
flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
Outlines
TRAFFIC FLOW ELEMENTS
4Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Time-Space Diagram
•The time-space diagram is a graph that describes the
relationship between the location of vehicles in a traffic
stream and the time as the vehicles progress along the
highway
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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•Time-Space Diagram
•Figure 6.1 shows a time-space diagram for six vehicles
with distance plotted on the vertical axis and time on the
horizontal axis. At time zero, vehicles 1, 2, 3, and 4 are at
respective distances d1, d2, d3, and d4 from a reference
point whereas vehicles 5 and 6 cross the reference point
later at times t5 and t6, respectively.
5
TRAFFIC FLOW ELEMENTS
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
524-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
6
TRAFFIC FLOW ELEMENTS
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Primary Elements of Traffic Flow
The primary elements of traffic flow are flow, density, and
speed. Another element,
associated with density, is the gap or headway between two
vehicles in a traffic stream.
Flow (q):
It is the equivalent hourly rate at which vehicles pass a point on
a highway
during a time period less than 1 hour. It can be determined by
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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7
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Density(k):
•The number of vehicles (n) occupying a given length (l) of a
lane or roadway at a particular instant
•Unit of density is vehicles per mile (vpm).
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
8
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Speed(u):
It is the distance traveled by a vehicle during a unit of time. It can be
expressed in miles per hour (mi/h), kilometers per hour (km/h), or
feet per second (ft/sec).
There are two types of mean speeds:
1)time mean speed
2)space mean speed
1) Time mean speed (spot speed)
It is the arithmetic mean of the speeds of vehicles passing a point on
a highway during an interval of time. The time mean speed is found
by
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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9
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Space mean speed (spot speed)
It is the harmonic mean of the speeds of vehicles passing a point on
a highway during an interval of time. It is obtained by dividing the
total distance traveled by two or more vehicles on a section of
highway by the total time required by these vehicles to travel that
distance. This is the speed that is involved in flow-density
relationships. The space mean speed is found by
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
10
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Relationship between time mean speedand space mean speed
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
Direct relationship developed by Garber and Sankar
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6. 15/12/2019
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11
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Headway:
It is the separation between a given point on adjacent vehicles i.e.,
from front bumper to front bumper. It Can be measured in distance
or time. There are two types of headway i.e. time head way(h) and
space headway(s).
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
Outlines
12Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Primary Elements of Traffic Flow
Time Headway (h):
It is the difference between the time the front of a vehicle arrives at
a point on the highway and the time the front of the next vehicle
arrives at that same point. Time headway is usually expressed in
seconds. For example, in the time space diagram (Figure 6.1), the
time headway between vehicles 3 and 4 at d1 is h3 –4.
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
11
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7. 15/12/2019
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13
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Space Headway (d):
It is the distance between the front of a vehicle and the front of the
following vehicle and is usually expressed in feet. The space
headway between vehicles 3 and 4 at time t5 is d3 –4 (see Figure
6.1).
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
14
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Example 6.1
Determining Flow, Density, Time Mean Speed, and
Space Mean Speed
Figure 6.3 shows vehicles traveling at constant speeds on a two-
lane highway between sections X and Y with their positions and
speeds obtained at an instant of time by photography. An observer
located at point X observes the four vehicles passing point X during
a period of T sec. The velocities of the vehicles are measured
as 45, 45, 40, and 30 mi/h, respectively. Calculate the flow, density,
time mean speed, and space mean speed.
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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8. 15/12/2019
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15
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
16
Primary Elements of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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17
FLOW-DENSITY RELATIONSHIPS
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
The general equation relating flow, density, and space mean
speed is given as
Flow = density ×space mean speed
Each of the variables in Eq. 6.7 also depends on several other
factors including the
characteristics of the roadway, characteristics of the vehicle,
characteristics of the
driver, and environmental factors such as the weather.
Other relationships that exist among the traffic flow variables
are given here.
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
18
FLOW-DENSITY RELATIONSHIPS
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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19
Fundamental Diagram of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Therelationshipbetweenthedensity(veh/mi)andthe
correspondingflowoftrafficonahighwayisgenerallyreferredto
asthefundamentaldiagramoftrafficflow.Thefollowingtheory
hasbeenpostulatedwithrespecttotheshapeofthecurve
depictingthisrelationship:
1)Whenthedensityonthehighwayis0,theflowisalso0
because there are no
vehiclesonthehighway.
2)Asthedensityincreases,theflowalsoincreases.
3)However,whenthedensityreachesitsmaximum,generally
referred to as the
jamdensity(kj),theflowmustbe0becausevehicleswilltend
to line up end
toend.
4)Itfollowsthatasdensityincreasesfrom0,theflowwillalso
initiallyincreasefrom0toamaximumvalue.Further
continuousincreaseindensitywillthenresultincontinuous
reductionoftheflow,whichwilleventuallybe0whenthe
densityisequaltothejamdensity.
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of
Traffic Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
20
Fundamental Diagram of Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of
Traffic Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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21
6.2.2 Mathematical Relationships
Describing Traffic Flow
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Mathematical relationships describing traffic flow can be
classified into two general categories—macroscopic and
microscopic—depending on the approach used in the
development of these relationships. The macroscopic
approach considers flow density relationships whereas the
microscopic approach considers spacing's between vehicles
and speeds of individual vehicles.
Macroscopic Approach
The macroscopic approach considers traffic streams and
develops algorithms that relate the flow to the density and
space mean speeds. The two most commonly used
macroscopic models are the Green shields and Greenberg
models.
Outlines
Importance of
Traffic flow
theory
TRAFFIC FLOW
ELEMENTS
Primary
Elements of
Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of
Traffic Flow
6.2.2
Mathematical
Relationships
Describing
Traffic Flow
Greenshields
Model
Model
Application
Calibration of
Macroscopic
Traffic Flow
Models
22
Greenshields Model
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields
Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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23Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Greenshields Model.
Greenshields carried out one of the earliest recorded works
in which he studied the relationship between speed and
density. He hypothesized that a linear relationship existed
between speed and density which he expressed as
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields
Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
24
Greenshields Model
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Eqs. 6.14 and 6.15 indicate that if a linear relationship in the form
of Eq. 6.13 is assumed for speed and density, then parabolic
relationships are obtained between flow and density and between
flow and speed. The shape of the curve shown in Figure 6.4a will
therefore be a parabola. Also, Eqs. 6.14 and 6.15 can be used to
determine the corresponding speed and the corresponding density
for maximum flow.
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields
Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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13. 15/12/2019
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25
Greenshields Model
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields
Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
26
Greenshields Model
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields
Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
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27
Greenberg Model.
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Several researchers have used the analogy of fluid flow to develop
macroscopic relationships for traffic flow. One of the major
contributions using the fluid-flow analogy was developed by
Greenberg in the form
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenberg
Model
Model Application
Calibration of
Macroscopic Traffic
Flow Models
28
Model Application
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Use of these macroscopic models depends on whether they
satisfy the boundary criteria of the fundamental diagram of
traffic flow at the region that describes the traffic conditions.
For example, the Greenshields model satisfies the boundary
conditions.
•when the density k is approaching zero as well as when the
density is approaching the jam density kj. The Greenshields
model can therefore be used for light or dense traffic. The
Greenberg model, on the other hand, satisfies the boundary
conditions when the density is approaching the jam density
but it does not satisfy the boundary conditions when k is
approaching zero. The Greenberg model is therefore useful
only for dense traffic conditions.
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model
Application
Calibration of
Macroscopic Traffic
Flow Models
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29
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•The traffic models discussed thus far can be used to
determine specific characteristics, such as the speed and
density at which maximum flow occurs, and the jam density
of a facility.
•This usually involves collecting appropriate data on the
particular facility of interest and fitting the data points
obtained to a suitable model. The most common method of
approach is regression analysis. This is done by minimizing
the squares of the differences between the observed and
expected values of a dependent variable.
•When the dependent variable is linearly related to the
independent variable, the process is known as linear
regression analysis. When the relationship is with two or
more independent variables, the process is known as multiple
linear regression analysis
.
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
30
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
29
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31
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Example 6.2 Fitting Speed and Density Data to the Greenshields
Model .Let us now use the data shown in Table 6.1 (columns
1 and 2) to demonstrate the use of the method of regression
analysis in fitting speed and density data to the macroscopic
models discussed earlier
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
32
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
31
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33
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
34
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Example 6.3
Fitting Speed and Density Data to the Greenberg Model
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
33
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35
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
36
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
35
36

19. 15/12/2019
19
37
Calibration of Macroscopic
Traffic Flow Models
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Importance of
Traffic flow theory
TRAFFIC FLOW
ELEMENTS
Primary Elements
of Traffic Flow
FLOW-DENSITY
RELATIONSHIPS
Fundamental
Diagram of Traffic
Flow
6.2.2 Mathematical
Relationships
Describing Traffic
Flow
Greenshields Model
Model Application
Calibration of
Macroscopic
Traffic Flow
Models
38
Contents
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
1)Microscopic Approach of Traffic flow
2)Shock waves in traffic stream
3)Velocity of shock waves
4)Numerical Problems
5)Special case of Shock wave propagation
6)Gap and Gap acceptance
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39
Microscopic Approach
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Also referred to as car-following theory and follow-the-
leader theory.
•Considers space between vehicles and speed of individual
vehicles.
•Consider two vehicles A and B on a highway, if A is n
th
then
B is (n+1)
th
vehicle and their distance from a fixed cross-
section at any time t is x
n
and x
n+1
respectively.
Microscopic
Approach
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
Outlines
40
Microscopic Approach
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Microscopic
Approach
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
39
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41
Microscopic Approach
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•If the driver of vehicle B maintains an additional separation
distance P above the separation distance at rest S such that P is
proportional to the speed of vehicle B, then
•Where
•We can write
•Where S is the distance between front bumpers of vehicles at
rest
Outlines
Microscopic
Approach
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
42
Microscopic Approach
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Differentiating this equation we get
•Which is the basic equation of microscopic models and it
describes the stimulus response of the models.
•Researches show that a time lag exists for a driver to respond to
any stimulus that is induced by the vehicle just ahead and the
equation can therefore be written as.
Outlines
Microscopic
Approach
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
41
42

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43
Microscopic Approach
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Where
T = Time lapse of response to the stimulus
λ = (1/р) sometimes referred to as sensitivity
•A general expression for λ is given in the form of
Outlines
Microscopic
Approach
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
44
Microscopic Approach
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•The general expression for microscopic model thus takes the
form
•
•
•Where a, ℓand mare constants.
•This equation can be used to determine the velocity, flow, and
densityof a traffic stream when the traffic stream is moving in
a steady state. Macroscopic models can also be derived from
this equation.
Outlines
Microscopic
Approach
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
43
44

23. 15/12/2019
23
45
Shockwaves in Traffic
Streams
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•When density on a traffic stream changes from a higher value C
1
to a lower value C
2
, the speeds of the vehicles will have to be
reduced while passing the bottleneck. The point at which the
speed reduction takes place can be approximately noted by the
turning on of the brake lights of the vehicles. An observer will see
that this point moves upstream as traffic continues to approach the
vicinity of the bottleneck indicating an upstream movement of the
point at which flow and density change.
•This phenomenon is referred to as a shock wavein the traffic
stream.
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
Traffic
Streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
1
46
Shockwaves in Traffic
Streams
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
Traffic
Streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
45
46

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47
Types of Shockwaves
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
Traffic
Streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
•Frontal stationary shock waves
•Backward forming shock waves
•Backward recovery shock waves
•Rear stationary and forward recovery shock waves
48
Velocity of Shock Waves
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Consider two different traffic densities, k
1
and k
2
, along a
highway where k
1
> k
2
. Assume that these densities are
separated by the line w representing the shock wave moving at a
speed uw. If the line w moves in the direction of the arrow (that
is, in the direction of the traffic flow), uwis positive.
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
47
48

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49
Velocity of Shock Waves
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•the speed of the vehicle in area P relative to line w is
•The number of vehicles crossing line w from area P during a
time period t is
•Similarly in Q area,
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
50
Velocity of Shock Waves
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Since net change is zero i.e. N
1
= N
2
, we get
•As q
1
= k
1
* u
1
and q
2
= k
2
* u
2
•That is
•This indicates that the velocity of the shock wave created by a
sudden change of density from k
1
to k
2
and is the slope of the
chord joining the points associated with k
1
and k
2
on the volume
density curve for that traffic stream.
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
49
50

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51
Numerical Problem
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
Solution
52
Special Cases of Shock
Wave Propagation
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•The shock wave phenomenon can be explained by considering a
continuous change of flow and density in the traffic stream. If the
change in flow and the change in density are very small, we can
write;
(q2 –q1) = ∆q (k2 –k1) = ∆k
•Wave velocity can be written as;
dk
dus
kusuw 
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
51
52

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53
Special Cases of Shock
Wave Propagation
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Flow = density x space mean speed
Asdensityincreases,spacemeanspeeddecreases,givinganegative
valuefordus/dk.
Whenboththeflowanddensityareverylow: thedeferentialof
usw.r.tk(dus/dk)thentendstozero,andwavevelocityequalsto
spacemeanspeed.
Whenflowishigherthanzerobutlowerthancapacityof
restrictedarea:WavevelocityislessthanSMS(Spacemean
speed),andwavemovesforwardrelativetotheroad.
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
1
54
Special Cases of Shock
Wave Propagation
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Whenflowisgreaterthanthecapacityofrestrictedarea:speedof
waveislessthantheSMSandwavemovesbackwardrelativetothe
road.
Themovementofwavetowardstheu/ssectionofthetrafficcreates
shockwaveinthetrafficstream,resultinginbackupswhichmovesu/s
ofthetrafficstream.
TheGreenshieldmodelfitsflowdensityrelationshipforatrafficflow,
thenthespeedofshockwaveisgivenas;
uw= uf[ 1 –(
η
1 +
η
2) ] … (6.46)
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
53
54

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55
Special Cases of Shock
Wave Propagation
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Density Nearly Equal:
When
η
1 =
η
2;
uw= uf[ 1 –(2
η
1) ]
StoppingWaves:
Eq6.46canbeusedtodeterminethevelocityoftheshockwavedueto
thechangefromgreentoredofasignalatanintersection.
Normaliseddensityatgreenphase=
η
1
Valueof
η
2duringjamdensity=1
Speedofshockwavewillbethen;
uw=-uf
η
1…….(6.47)
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Gap and
Gap
acceptance
Special
Cases of
Shock Wave
Propagation
1
56
Special Cases of Shock
Wave Propagation
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
StartingWaves:
Whensignalchangesfromredtogreen,
Valueof
η
1=1
Speedofshockwavewillbethestartingwaveandrepresentedas;
uw=-uf
η
2
but
Velocityofshockwavewillbe;
uw=-uf+us2
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Numerical
Problems
55
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57
Given Problem
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
PROBLEM 6.6:
Studieshaveshownthatthetrafficflowonasingle-lane
approachtoasignalizedintersectioncanbedescribedbythe
Greenshieldsmodel.Ifthejamdensityontheapproachis130veh/mi,
determinethevelocityofthestoppingwavewhentheapproachsignal
changestoredifthedensityontheapproachis45veh/miandthe
spacemeanspeedis40mi/h.Attheendoftheredinterval,what
lengthoftheapproachupstreamfromthestoplinewillvehiclesbe
affectediftheredintervalis35sec?
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Given
Problems
1
58
Given Problem
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Given
Problems
57
58

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59
Gap and Gap Acceptance
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Importantmeasuresthatinvolvetheconceptofgapacceptance;
Mergingistheprocessbywhichavehicleinonetrafficstreamjoins
anothertrafficstreammovinginthesamedirection,suchasaramp
vehiclejoiningafreewaystream.
Divergingistheprocessbywhichavehicleinatrafficstreamleaves
thattrafficstream,suchasavehicleleavingtheoutsidelaneofan
expressway.
Weavingistheprocessbywhichavehiclefirstmergesintoastream
oftraffic,obliquelycrossesthatstream,andthenmergesintoasecond
streammovinginthesamedirection;forexample,themaneuver
requiredforarampvehicletojointhefarsidestreamofflowonan
expressway.
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
1
60
Gap and Gap Acceptance
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Gapistheheadwayinamajorstream,whichisevaluatedbyavehicle
driverinaminorstreamwhowishestomergeintothemajorstream.It
isexpressedeitherinunitsoftime(timegap)orinunitsofdistance
(spacegap).
Timelagisthedifferencebetweenthetimeavehiclethatmergesinto
amaintrafficstreamreachesapointonthehighwayintheareaof
mergeandthetimeavehicleinthemainstreamreachesthesame
point.
Spacelagisthedifference,ataninstantoftime,betweenthedistance
amergingvehicleisawayfromareferencepointintheareaofmerge
andthedistanceavehicleinthemainstreamisawayfromthesame
point.
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
59
60

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61
Gap and Gap Acceptance
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
CriticalGapAcceptance:
Greenshieldsreferredtoitasthe“acceptableaverageminimumtime
gap”anddefineditasthegapacceptedby50%ofthedrivers.
Raffdefineditasthegapforwhichthenumberofacceptedgaps
shorterthanitisequaltothenumberofrejectedgapslongerthanit.
Time-Space Diagram for Vehicles in the
Vicinity of a Stop Sign
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
1
62
Gap and Gap Acceptance
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Determination of Critical gap using Raff’s
Method:
Algebraic Method:
Outlines
Microscopic
Approach
of Traffic
flow
Shock
waves in
traffic
streams
Velocity of
shock waves
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
61
62

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63
Gap and Gap Acceptance
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Determination of Critical gap using Raff’s
Method:
Graphical Method:
Cumulative Distribution Curves for Accepted and Rejected Gaps
Outlines
Microscopic
Approach
of Traffic
flow
Shock waves
in traffic
streams
Velocity of
shock waves
Numerical
Problems
Numerical
Problems
Special
Cases of
Shock Wave
Propagation
Gap and
Gap
acceptance
64
Contents
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
1)Stochastic approach to Gap and Gap
acceptance
2)Queuing Theory and analysis
3)Deterministic analysis of queues
4)Stochastic analysis of Queues
5)Queues classification
6)Examples
63
64

33. 15/12/2019
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65
Stochastic Approach to Gap
and Gap Acceptance
Problems
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Itisusuallyassumedthatforlight-to-mediumtrafficthedistributionis
Poisson,althoughassumptionsofgammaandexponentialdistributions
havealsobeenmade.Assumingthatthedistributionofmainstream
arrivalisPoisson,thentheprobabilityofxarrivalsinanyintervalof
timetseccanbeobtainedfromtheexpression:
Substituting zero for x in above equation yields;
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
66
Stochastic Approach to Gap
and Gap Acceptance
Problems
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
LetusassumethatTisequalto1hrandthatVisthevolumeinveh/h
onthemainstreamflow.Since(V1)gapsoccurbetweenVsuccessive
vehiclesinastreamofvehicles,thentheexpectednumberofgaps
greater
orequaltotisgivenas:
and the expected number of gaps less than t is given as
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
65
66

34. 15/12/2019
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67
Given Problem
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
PROBLEM 6.7:
Thepeakhourvolumeonanexpresswayatthevicinityofthemerging
areaofanonrampwasdeterminedtobe1800veh/h.Ifitisassumed
thatthearrivalofexpresswayvehiclescanbedescribedbyaPoisson
distribution,andthecriticalgapformergingvehiclesis3.5sec,
determinetheexpectednumberofacceptablegapsforrampvehicles
thatwilloccurontheexpresswayduringthepeakhour.
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
68
Given Problem
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
67
68

35. 15/12/2019
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69
Given Problem
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
70Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
QUEUING THEORY AND
QUEUING ANALYSIS
•Introduction to Queuing theory
•Basics and importance of Queuing analysis
•Deterministic Analysis of Queues
•Example 6.9
•Stochastic Analyses of Queues
•Examples 6.10 & 6.11
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
69
70

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71
QUEUING THOERY AND
ANALYSIS
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Background
•Traffic engineers try their best to provide economical and
efficient transport network.
•IncreaseinpopulationandUrbanization.
•Peopledemandmoreofaservicethanthatservicecould
provide.
•Seriouscongestionsontraffichighways(Peakhours).
•Ultimatelyresultinginwaitinglines(Queues)onroadsand
delays.
•Delays is the difference between the actual travel time on a
given segment and some ideal (under free flow) travel time
of that segment.
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
72Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
QUEUING THOERY AND
ANALYSIS
•Aprimarygoalisfindingthebestlevelofservice
•Bestcyclelengthandphaselengthsfortrafficsignals.
•Optimizingthefrequencyatwhichbusesortrucksshouldbe
dispatchedalongaroute.
•Formoreaccuratedesignofanintersection.
•Fundamentally,queueinganalysisisusedtodeterminethe
differencebetweenhowlongittakestocompleteatrip,andhow
longitwouldhavetakeniftherewerenoqueueingor
congestion.
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
71
72

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73Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
QUEUING THOERY
•Uses mathematical algorithms to describe the processes that
result in the formation of queues, so that a detailed analysis of
the effects of queues can be undertaken.
•Basically, queuing theory involves analysis of a queuing system
which consists of
•Servers:resource that provides the service to the
customer
•Segment of roadway, bus, gate in an airport, tollbooth,
etc.
•Customers: persons or things that await service
•Travelers, vehicles, bikes, freight container, etc.
•Queues: group of customers waiting to be served
•Congested flow of vehicles, group of passengers
waiting at gates, etc.
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
74Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
QUEUING THOERY
•Application of queuing theory in Transportation
Engineering
•Two approaches
•Deterministic analysis of queues
•Stationary (steady-state) situation
•Stochastic analysis of queues
•Non-stationary situations
•More realistic in transportation systems
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
73
74

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75
Deterministic analysis of queues
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Assumptions
•Demands capacities and volumes are known
•All traffic characteristics of the queue are deterministic
•Application of this approach
•There are two conditions where this approach has been
used,
•varying service rate and constant demand condition
•Significant reduction in highway capacity (due
to an incident)
•varying demand and constant service rate condition
•Demand exceeding capacity
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
76
Deterministic analysis of queues
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Varying service rate and constant demand
condition
•A three lane Highways having capacity C veh/hr.
•An incident occurs
•Results in closure of one lane out of three lanes of the
highway.
•Reducing its capacity to Cr veh/hrfor time t
(clearance time required).
•Demand volume being constant is,
•Less than the original capacity (before the incident)
•Higher than the reduced capacity (after the incident),
queue formation.
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
75
76

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77
Deterministic analysis of queues
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Important Parameters for Queue analysis
•Maximum queue length
•Duration of the queue
•Average queue length
•Maximum individual delay
•Time a driver spends in the queue
•average queue length while the queue exists
•Maximum individual delay
•The total delay
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
78Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Maximum queue length (q
max
)
•The excess demand rate multiplied by the duration of the
incident.
•Time duration of the queue (tq)
•The queue length divided by the difference between the
capacity and the demand rate.
•
Deterministic analysis of queues
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
77
78

40. 15/12/2019
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1
79Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Deterministic analysis of queues
Average queue length (q
av
)
Total delay (d
T
)
The time duration of the queue multiplied by the average queue
length.
Note: Ensure that same units are used for all the variables
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
80Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
79
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Stochastic Analyses of Queues
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Probabilistic approach
•Assumption
•Traffic characteristics are not always determined e.g. Arrival
rates.
•A queue is formed when arrivals wait for a service such as,
•The collection of tolls at a tollbooth , parking fees at a parking
garage etc.
•This approach determines
•The probability that an arrival will be delayed.
•The expected waiting time for all arrivals.
•Various Models based on traffic conditions
•Merging of ramp traffic to freeway traffic
•Intersection at pedestrian crossings
•Sudden reduction of capacity on freeways
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
82
Pre-requisites for Stochastic
Analyses of Queues
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•ForproperanalysisofQueue,fullspecificationofQueue
characteristicsisveryimportant.
•FollowingcharacteristicsofQueuemustbegiven;
•Characteristicdistributionofarrivals(uniform,Poissonetc.)
•Methodofservice(firstcome–firstserved,random,priority
based)
•Characteristicofthequeuelength(finiteorinfinite,depends
onspace)
•Distributionofservicetimes
•Thechannellayout(single,multiplechannelsandinthecase
ofmultiplechannels,whethertheyareinseriesorparallel).
•Note:basedontheabovecharacteristicsqueuesarefurther
classifiedforeverycasei.e.singlechannelqueue,finitequeue
etc.
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
81
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Queues Classification
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Oversaturated Queues
•Arrival rate is greater than the service rate
•Its length is never steady
•Its length increases with time.
•Under saturated Queues
•The arrival rate is less than the service rate.
•Its length varies, and become steady upon arrival of
units.
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
84
Analysis of Single-Channel, Under
saturated, Infinite Queues
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Rate of service = Q veh/hr
•Rate of arrival = q veh/hr
•Queue is
•Single channel
•Unsaturated i.e. Q>q
•Arrival and service rates are random
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
83
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Analysis of Single-Channel, Under
saturated, Infinite Queues
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Parameters to be determined for Queue analysis
•Probability of n units in the system, P(n):
•
• where n is the number of units in the
system, including the unit being serviced.
•Expected number of units in the system, E(n):
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
86Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Analysis of Single-Channel, Under
saturated, Infinite Queues
•Mean queue length, E(m):
•Expected number of units waiting to be served
•Average waiting time in the queue, E(w):
•Average waiting time of an arrival, including queue and
service, E(v):
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
85
86

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87Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Analysis of Single-Channel, Under
saturated, Infinite Queues
•Probability of spending time t or less in the system:
•Probability of waiting for time t or less in the queue:
•Probability of more than N vehicles being in the system, that
is, P (n > N):
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
Outlines
1
88
The Bottom Line
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Graphs summarizes the whole concept.
•Traffic intensity
•P= q/Q
•Infinite units at P=1
•Saturation
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
87
88

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89Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
90
Example 6.10 Cont...
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•(b) the average number of vehicles in the system
•(c) the average waiting time for the vehicles that wait
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
89
90

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Single-Channel, Under saturated,
Finite Queues
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
•Finite Queue
•Maximum No (N) of units in the system is specified.
•Important relations
•Probability of n units in the system:
•The expected number of units in the system:
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
92Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
91
92

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Example 6.11 Cont…..
Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
(b) The percent of time the ramp is full
•Probability of 10 vehicles on the ramp
•The ramp is full only 2.3% of the time.
(c) The expected number of vehicles on the ramp during the peak hour
•The expected number of vehicles on the ramp is 3.
Outlines
Stochastic
approach to
Gap and Gap
acceptance
Queuing
Theory and
analysis
Deterministic
analysis of
queues
Stochastic
analysis of
Queues
Queues
classification
Examples
1
94Department of Civil Engineering, UET Taxila
(MSC Transportation Engineering)
24-Oct-19
93
94