Traffic Flow Characteristics. Dr. Attaullah Shah

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Transportation Engineering - I
Traffic Flow Characteristics. Dr. Attaullah Shah

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Traffic Flow

• Traffic flow No of vehicle passing per unit time
  at a road section
• q n/t n No of vehicles t duration of
  time, usually an hour
• Time head way The time between passing of
  successive vehicles i.e. bumpers
• t S hi t duration of time interval hi
  time head way of ith vehicle.
• q n/ S hi Average mean time headway
• The average speed is defined in two ways
• Arithmetic mean of total speeds at a particular
  point also called time mean speed Mean Ut
  Sui/n
• Space Mean Speed Assumed that the average speed
  for all vehicles is measured at the same length
  of roadway.
• Space mean speed in unit distance per unit time
  length of strip/mean time

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Example

• The speed of five vehicles is measured with radar
  at the midpoint of 0.8Km strip (0.5 mi) The
  speeds are measured as 44,42,51,49 and 46 mi/h.
  Assuming the vehicles were travelling constant
  speed, determine the time mean speed.
• Tim Mean speed Ut Sui/n (4442514946)/5
  46.4 mi/h
• The space mean
  5/ 1/441/421/511/491/46
• 46.17 mi/h
• Space mean speed is always smaller than time mean
  speed.

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Traffic density

• K n/l The No of vehicles per unit length.
• The density is also related to individual
  spacing of the vehicles
• The total length of roadway l S si
• k 1/( Mean space)
• Traffic flow q uk
• q flow of traffic No of vehicles/hour
• Speed ( Space mean speed)
• k density units of vehicle/mile.
• Example 5.2

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Traffic Flow

• Complex between vehicles and drivers, among
  vehicles
• Stochastic variability in outcome, cannot
  predict with certainty
• Theories and models
• Macroscopic aggregate, steady state
• Microscopic disaggregate, dynamics
• Human factor driver behavior

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Speed (v)

• Rate of motion
• Individual speed
• Average speed
• Time mean speed
• Arithmetic mean
• Space mean speed
• Harmonic mean

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Individual Speed
(1)
Spot Speed
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Time Mean Speed
Mile post
Observation Period
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Space Mean Speed
Observation Distance
Observation Period
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Volume (q)

• Number of vehicles passing a point during a given
  time interval
• Typically quantified by Rate of Flow (vehicles
  per hour)

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Volume (q)
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Density (k)

• Number of vehicles occupying a given length of
  roadway
• Typically measured as vehicles per mile (vpm),
• or vehicles per mile per lane (vpmpl)

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Density (k)
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Density (k)
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Spacing (s)

• Front bumper to front bumper distance between
  successive vehicles

S1-2
S2-3
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Headway (h)

• Time between successive vehicles passing a fixed
  point

T3sec
T0 sec
h1-23sec
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Spacing and Headway
spacing
headway
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Spacing and Headway
What are the individual headways and the average
headway measured at location A during the 25 sec
period?
A
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Spacing and Headway
What are the individual headways and the average
headway measured at location A during the 25 sec
period?
A
h1-2
h2-3
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Lane Occupancy

• Ratio of roadway occupied by vehicles

L1
L2
L3
D
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Clearance (c) and Gap (g)

• Front bumper to back bumper distance and time

Clearance (ft) or Gap (sec)
Spacing (ft) or headway (sec)
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Basic Traffic Stream models

• 1. Speed density Model Relationship between
  speed and density
• Initially when there is only one vehicle, the
  density is very low and driver will move freely
  at the speed closely to the design speed of the
  highway. This is called free flow speed.
• When more and more vehicles would come, the
  density of the road will increase and the speed
  of vehicles would reduce. The speed of the
  vehicle may ultimately reduce to u0
• This high traffic density is referred as Jam
  Density
• This model can be expressed as U Uf ( 1- k/kj)
  
• U Space mean speed in mi/h (km/h)
• Uf Free flow speed in mi/h
• k density in veh/mi ( veh/km)
• This gives relationship between traffic flow
  speed and density
• The speed density relationship tends to be non
  linear at low density and very high density.
• Non linear relationship at low densities that has
  speed slowly declining from free flow to value
• Linear relationship over the medium density
  region ( speed declining linearly with the
  density).
• Non linear relationship near the traffic jam
  density relationship

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Speed vs. Density
ufFree Flow Speed
Speed (mph)
kj Jam Density
Density (veh/mile)
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Basic Traffic Stream models

• Flow Density Model
• As we know that q uk and U Uf ( 1- k/kj)
• Parabolic density model q Uf ( k k2/Kj )
• The max flow rate qcap The highest flow rate
• The traffic density that corresponds to this
  capacity flow rate is kcap and the corresponding
  values qcap ,Kcap ,Ucap
• dq/dk Uf (1-2k/kf ) 0 Since the free flow
  speed cannot be zero.
• Kcap Kj / 2
• Substituting the value Ucap ( k Kj /2kj )
  Uf /2

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Flow vs. Density
Congested Flow
Optimal flow, capacity, qm
FLow (veh/hr)
kj Jam Density
kmOptimal density
Uncongested Flow
Density (veh/mile)
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Speed Flow Model

• This is parabolic relationship
• Two speeds are possible as it is quadratic
  equation, up to the highway capacity
• Similarly two densities are possible for given
  flow

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Speed vs. Flow
ufFree Flow Speed
Uncongested Flow
um
Speed (mph)
Optimal flow, capacity, qm
Congested Flow
Flow (veh/hr)
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Example
At a section of highway in Islamabad the free
flow speed is 90km/h and capacity of 5000 veh/h.
During traffic census at a particular section
2500 vehicles were counted. Determine the space
mean speed of the vehicle. Solution The
traffic flow is related to the space mean speed
and jam density as The jam To determine the
jam density From above equation of traffic flow
we can rearrange the equation as
Both the speeds are feasible as
shown on previous slide
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Traffic Flow Models-Poisson Model

• The Poisson distribution
• Is a discrete (as opposed to continuous)
  distribution
• Is commonly referred to as a counting
  distribution
• Represents the count distribution of random events

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Poisson Distribution
P(n) probability of having n vehicles arrive in
time t ? average vehicle arrival rate in
vehicles per unit time t duration of time
interval over which vehicles are counted e base
of the natural logarithm (e2.718)
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Example

• An observer counts 360veh/h at a specific highway
  location. Assuming the arrival of vehicles at
  this point follows Poissons distribution.
  Estimate the probabilities of 0,1,2,3,4 and 5
  vehicles arriving in 20 sec time interval.
• Solution
• The average arrival rate ? 360 veh/h
  360/60x600.1 veh/sec
• t20 sec
• For probability of no vehicle
• P(1) 0.271 P(2) 0.271 P(3) 0.180 P(4)
  0.090
• For 5 or more vehicles
• P(ngt 5) 1-P(nlt5) 1-(0.1350.2710.2710.180.090
  0.053
• Draw the histogram of the distribution

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Poisson Example

• Example
• Consider a 1-hour traffic volume of 120 vehicles,
  during which the analyst is interested in
  obtaining the distribution of 1-minute volume
  counts

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Poisson Example

• What is the probability of more than 6 cars
  arriving (in 1-min interval)?

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Poisson Example

• What is the probability of between 1 and 3 cars
  arriving (in 1-min interval)?

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Poisson distribution

• The assumption of Poisson distributed vehicle
  arrivals also implies a distribution of the time
  intervals between the arrivals of successive
  vehicles (i.e., time headway)

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Negative Exponential

• To demonstrate this, let the average arrival
  rate, ?, be in units of vehicles per second, so
  that

• Substituting into Poisson equation yields

(Eq. 5.25)
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Negative Exponential

• Note that the probability of having no vehicles
  arrive in a time interval of length t i.e., P
  (0) is equivalent to the probability of a
  vehicle headway, h, being greater than or equal
  to the time interval t.

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Negative Exponential

• So from Eq. 5.25,

(Eq. 5.26)
Note
This distribution of vehicle headways is known as
the negative exponential distribution.
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Negative Exponential Example

• Assume vehicle arrivals are Poisson distributed
  with an hourly traffic flow of 360
  veh/h.Determine the probability that the
  headway between successive vehicles will be less
  than 8 seconds.Determine the probability that
  the headway between successive vehicles will be
  between 8 and 11 seconds.

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Negative Exponential Example

• By definition,

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Negative Exponential Example
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Negative Exponential
For q 360 veh/hr
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Negative Exponential