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Transportation and Traffic Flow

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Title: Transportation and Traffic Flow


1
Transportation andTraffic Flow
  • Anil V. Kantak

2
  • 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.

8
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.

9
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

12
  • 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.

13
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.
14
  • 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

15
  • 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
19
  • 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.

20
  • 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.

21
  • 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
23
  • 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.

24
  • 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

25
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.
26
  • 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.
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