Transportation and Traffic Flow

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Transportation andTraffic Flow

• Anil V. Kantak

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• 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

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• 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.

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• Assume that two vehicles are moving on a long
  stretch of a road without signals and other
  restrictions (such as a freeway). With

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• 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

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• 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 with a uniform speed.

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• 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.

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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

• 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.

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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
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• 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

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• 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 on the average.

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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.
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• 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

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• 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

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
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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
Note that the signs in the first equation are due
to with the traffic and against directions
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• 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.

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• 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.

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• 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

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
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• 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

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
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• 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.

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• 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

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convert the pc traffic into the normal traffic
with the heavy vehicles and others. This is done
as follows
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.
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• 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.