Transportation and
Traffic Flow
Anil V. Kantak
Traffic Flow
Anil Kantak
1
•Vehicular Flow:
•When fixed facilities are used simultaneously
by streams of vehicles, a vehicular flow is
constructed.
•Resulting traffic conditions may be almost
free flow when only a few unconstrained
vehicles are present on the roadway.
•Resulting traffic condition may be a highly
congested flow condition when a lot of
streams are combined together.
•Traffic rules and regulations try to maximize
their speeds while maintaining an acceptable
level of safety. This is usually achieved by
Traffic Flow
Anil Kantak
2
Adjusting the distance between vehicles by
adjusting the vehicular speed.
• Basic Variables of Traffic Flow:
• Flow
• Concentration
• Mean Speed
Fundamental relationship between these
variables is postulated and applied to several
traffic flow conditions.
•Vehicular Following:
Distance between subsequent vehicles is
computed that is safe if the leading vehicle
needs to decelerate suddenly.
Traffic Flow
Anil Kantak
3
•Assume
that two vehicles are moving on a
long stretch of a road without signals and other
restrictions (such as a freeway). With
  Initial speed of the tw o cars
dl  Decelerati on rate of the leading vehicle
d f  Decelerati on rate of the follow ing vehcle
  Perception - reaction time for the follow ing vehicle
x 0  Safety margin after stop
L  Length of the follow ing car
N  Number of vehicles if there are more than2 2 vehicles
υ
x l  Leading vehicle breaking distance 
2 dl
x f  Follow ing vehicle breaking distance  υδ 
υ2
2 df
x f  s  x l - NL - x 0
s  υ 
Traffic Flow
υ2
2 df
υ2
2 dl
 NL  x 0
Anil Kantak
4
•There are three levels of decelerations:
•Normal or comfortable deceleration: This type
of deceleration is subjective because it is
related to passenger comfort.
•Emergency deceleration: This situation arises
when an emergency occurs and is then
recognized by the driver of the vehicle.
•Instantaneous stop or stonewall stop: This
situation occurs when an accident or a stalled
vehicle or obstruction suddenly comes within
the perception field of the subject vehicle.
•The safest level of operation occurs when
spacing between vehicles is such that following
Traffic Flow
Anil Kantak
5
vehicle can safely stop by applying normal
deceleration even when leading vehicle comes
to a stonewall stop.
• In general, the higher the level of safety,
higher is required spacing just to avoid a
collision. However, by increasing the level of
safety, capacity of system, i.e., the maximum
number of vehicles or passengers that can be
accommodated during a given period of time
suffers. Consequently, a trade-off between
safety and capacity must be done.
•Spacing and Concentration: Suppose cars are
uniformly spaced on a length of roadway and
they
are all going withAnilaKantak
uniform speed.
Traffic Flow
6
•The ratio of number of equally spaced vehicles
on the roadway to the length of the roadway
segment is called the concentration (symbol k)
of the vehicular stream. Because of the uniform
flow, i.e., constant separation and speed the
concentration remains constant on any length of
the roadway.
•In actual practice the vehicles are neither
separated by a constant length nor they all go
with the same speed making the concentration
time variable and different at different places on
the same roadway.
Traffic Flow
Anil Kantak
7
The dimension of concentration k are vehicles
per length of the roadway such as vehicles per
mile = veh/mi. Relationship between spacing
(average spacing if not constant) and
concentration is
l
Concentrat ion  s 
 Density
k
Note: concentration is also called the density.
• Headways:
Interval of time between
successive cars is called the headways
between vehicles and is described by the
symbol h. Note: concentration is also called
the density of the flow.
Traffic Flow
Anil Kantak
8
Headways can be constant or variable
depending upon speeds of the vehicles. In any
case, during a time period of T headways can
be counted each corresponding to an
individual vehicle in relation to its leader.
Number of vehicles counted at the point of
observation divided by the total observation
time is called the stream flow and is given a
symbol q. The flow is also called Volume that
is measured in vehicles per time = Veh/Hour.
Note:
h 
Traffic Flow
1
q
Anil Kantak
9
Average or Mean Speed: When all the vehicles

are moving with the same speed,
average
speed of all the vehicle is also  In actual
practice the vehicles move at different speeds
and consequently the are two different
methods of computing the average speed,
Time Mean Speed Ut and Space Mean Speed
Us. The time speed is the arithmetic average of
the spot speeds taken.
Ut
1 N

U
N 1
i
Ui is the speed measured for the ith car. The
space speed is computed using the following
equation
Traffic Flow
Anil Kantak
10
Us 
N x
N
 t i
1
Where Delta ti is the time taken by the ith
vehicle to cover a fixed distance delta x. There
are many different ways of computing but the
general analysis the space velocity is used.
•The Fundamental Equation of Vehicular
Stream: If two vehicles are traveling at spacing
s and 
with speed then the headway between
them then the fundamental equation of traffic is
q  k
Traffic Flow
Anil Kantak
11
•Highway
Traffic flow: In case of highway
(freeway) traffic the drivers make their own
decisions regarding speed and headway
tradeoff. Some drivers keep close to leading
car and keep their speed high and safety low
while others keep long distances between cars
keeping the speed low but safety high. In
addition, the freeway vehicles are not all the
same. All these differences results in a
statistical clustering of vehicles on the
roadways. Next slide demonstrates the u-k, u-k,
and q-k diagrams for traffic flow on freeways
(highways). u-k relationship is monotonically
decreasing that is depiction of the rule the
drivers
follow one another
Anil Kantak on the average.
Traffic Flow
12
One car spacing for every 10 mph speed is one
such rule. The q-u and q-k curves are convex
with respect to y and x axis respectively and the
maximum flow occurs at some intermediate
speed
shown in the diagrams.
Anil Kantak
Traffic Flow
13
•Stream Measurements
The method of least
squares can be used to determine the
relationship between two or more variables
based on a set of experimental observations.
Many vehicular stream measurements are
available in practice. Because flow, speed and
concentration
are
interrelated,
any
measurement method used must measure two
of the variables simultaneously the third may
be estimated by the equation given above. It
should be noted that measuring only one
variable does not serve the purpose.
•The moving observer method. This method
is developed to provide simultaneous
measurements while moving in relation to the
traffic stream being measured. To understand this
better, consider the following two cases first.
Case I: Observer is stationary and the traffic
stream is moving, If N0 vehicles overtake the
observer during the period of observation, T then
the observed flow q is given by
N0
q 
 N0  q T
T
Case II: Observer is moving and the traffic
stream is stationary, By traveling a distance L,
the observer will overtake a number of vehicles
N0 then the concentration of stream being
measured is given by
N0
k 
L
 N0  k L  k V T
Where V is the observer’s speed and T is the
time it takes the observer to traverse the
distance L.
Using these two cases the actual flow will be
measured using the following technique: Now
the observer is moving in the traffic stream
which is also moving. In this case M0 number of
vehicles will overtake the observer and the
observer will overtake Mp number of vehicles.
The numbers M0 and Mp will depend upon the
average speed of the traffic stream designated
as ‘u’. The difference M between M0 and Mp is
given by:
M  M0 - Mp  q T - k V T
and hence,
M
 q  ku
T
M, T, and V are known variable from the observer
while k and u are unknowns. The observer test
must be run two times to get two
sets of values of M, V, and T to obtain the values
of the two unknowns u and k. Using Ma, Va, and
Ta to be quantities when traveling against traffic
and Mw, Vw, and Tw to be the corresponding
values when moving with traffic, substituting in
the above equations, we get
Ma
Mw
 q - k Vw ;
 q  k Va
Tw
Ta
Solving for q
Mw  Ma
q 
Tw  Ta
Note that the signs in the first equation are due to
with the traffic and against directions
•Shock
Waves Traffic: Assume that roadway
has uniform traffic with uniform spaces and
velocities of the vehicles.
•Suppose a truck slows down to 10 mph all of
a sudden.
• Assuming
that vehicles are not allowed to
overtake the truck, the next vehicle will try to
slow down in a safe deceleration and come
close to the safe distance and follow the truck
with 10 mph speed.
•With
time, a moving platoon of vehicles
traveling at 10 mph will form behind the truck.
Traffic Flow
19
Anil Kantak
•In front of the truck road is clear and behind
the last vehicle in the platoon the vehicles are
going at the normal speed of the roadway. As
the time passes, other vehicles have caught up
with the platoon and it grows incessantly.
•Suppose
the truck either exists the roadway
or speed up to the usual speed of the roadway.
•Then next car will speed at a safe acceleration
and keep a safe distance between it and the car
ahead. Next car would do the same etc. If this
persisted for sufficient time then roadway will
return to its normal speed.
•This effect is called the shock wave.
Traffic Flow
Anil Kantak
20
•The Shock Wave Equation
It has been shown
that speed of a traffic shock wave is given by
the slope of the chord connecting two stream
conditions that define the shock wave on a q-k
diagram. Labeling them as ‘a’ and ‘b’ the
magnitude and direction of the speed of the
shock wave is given by:
USW
qb - qa

(mph)
kb - ka
If sign of shock wave speed due to above
equation is (+)ve then the shock wave is traveling
in direction of the stream flow, if it is zero then
the shock wave is stationary with respect
roadway and if it is negative, then the shock
Traffic Flow
Anil Kantak
21
wave moves in the upstream direction.
•Fleet
Size: Number of vehicles needed to
maintain a transit line flow of q vehicles per
hour for a time period T is affected by the fact
that some vehicles may be traversing the line
more than one time during T. A vehicle count
over the time period T will produce
N

qT
Vehicl es
Some of the vehicles will be counted more than
one time. If the round trip time of a vehicle Trt .
This vehicle on the average will traverse the line
approximately T/Trt .times. So F =N (T/Trt) = q Trt
Traffic Flow
Anil Kantak
22
•Some Definitions
•Capacity : Term Capacity refers to the flow on
the roadway corresponding to a specific safety
regime.
•Ideal Freeway Conditions:
• Lane width and lateral
clearance: Lanes
must be at least 12 ft wide and any
obstructions must be at least 6 ft from the
edge of the pavement.
• Trucks,
Busses and Grades: Level
roadways and vehicular stream that is
entirely made of passenger cars.
Traffic Flow
Anil Kantak
23
•Demand: Within statistically acceptable limits,
the flow should be uniform. Level roadways
and vehicular stream that is entirely made of
passenger cars (pc).
•Volume:
The number of vehicles passing a
point on a highway or highway lane during one
hour, expressed as vehicles/hour.
•Rate of Flow: The number of vehicles passing
a point on a highway or highway lane during
some period of time less than one hour.
•Conversion: Since the roadways do not have
only passenger cars, a formula is needed to
convert the pc traffic into the normal traffic
with the heavy vehicles and others. This is
done as follows:
q  q N fw fhv
*
Where, q is the prevailing flow in veh/h, q* is
the ideal flow with pc/h/lane, N is the number
of freeway lanes, fw is the adjustment for the
combined effect of lane widths other than 12 ft.
and lateral obstruction closer than 6 ft. Finally
fhv is the adjustment factor due to the
presence of heavy vehicles on the roadway.
•Pedestrain flow Models: Pedestrian flow have
been developed to bear a close resemblance to
the vehicle flow models. The speed of a
pedestrian regime is naturally measured in
units of distance divided by time such as feet
per second. Flow is given by pedestrians per
unit widht of walkway power unit time.
Concentration or density is measured per
number of pedestrians per unit area of the
walkway. The reciprocal of concentration is
called space and has units of surfae area per
pedestrian, such as square feet per pedestrian.
The fundamental relationship q = u k is valid
here too.