Macroscopic model

Om Namo Narayanaya Namah
TFT 6: Macroscopic Models and
Empirical Data

Learning Objectives
• Define occupancy and express relationship
between occupancy and density
• Compare flow density and flow speed
relationships from models and empirical data
• Illustrate difficulties in measuring flow density
curves with empirical data
• Describe premise and applications of car
following models

Occupancy and Density
• Concentration: General term indicating measure
of intensity of vehicles over space
• Concentration classified as: density or
occupancy
• Density: No of vehicles per unit length (e.g. got
from aerial photography)
• Occupancy: fraction of time detector is occupied
by vehicles

Why occupancy?
• Density is an area based measurement, and can’t be
obtained at a point
• Occupancy is a point measurement and hence easier.
• Occupancy: defined because most detectors take up
space on the road
• They also give continuous reading at 50-60 Hz.
• Depends on detector type, size and nature, the
readings may vary for identical traffic.

Occupancy Definition
• Occupancy: Si ti
occupied / T
• Where ti
occupied is the time duration over which
vehicle i occupies (or is present) on the detector
• And T is the total interval/duration of observation.

Relationship between occ and K
• ti = time period when detector is occupied by
vehicle i = (li + d)/vi
• li = length of vehicle i
• d = width of detector
• vi = speed of vehicle i
• Assume all vehicles are of same dimension L.

Relationship between occ and K
• Occ = Si ti
occupied / T
• = Si (L + d) /(vi T)
• = [(L+d)/T] Si(1/vi)
• = c Si(1/vi)
• Let N vehicles pass through in time T
• Occ = c N / N Si(1/vi)

• Occ = cN [Si(1/vi)/ N]
• = CN/ Vs2 (Why?)
• = (L+d) (N/T) / Vs2
• = (L+d) Q/ Vs2 (Why?)
• = (L+d) K
• Occupancy is proportional to density
• So a point measurement can provide good indication of an area
measurement.
• Note occupancy is dimensionless (%), whereas density

Greenshield’s Model
• 1930’s
• Relationship between speed u and density k
• Empirical data showed that the relationship was
approximately linear
• U = U0(1 – k/kj)
• FFS = U0
• Jam density = Kj

Greenshield’s Model
• Traffic Flow Eqn
• Q = KU
= kU0(1 – k/kj)
At Qmax
dQ/dk = 0
U0 – 2kU0/Kj = 0
Or Kcr = Kj/2
Qmax = KjU0(1 – kj/2kj)/2
= (kj/2)(U0/2)
So Ucr = U0/2
K
U
Sf
Kj

Flow (Q)-density (k) Relationship
Qm
Kcr Kj
Sf Ucr
Q
K

Speed(U) vs Flow (Q) relationship
U
Q Qmax
Sf
Ucr
Uncongested
Flow
Congested Flow

Empirically Observed Data Q vs K
Qm
Q
K

Empirically Observed Data U vs Q
U
Q
Sf
X X X X
X
LOS A LOS C
LOS E

Observations
• Empirical data has lot of gaps.
• So to predict congestion we need theoretical
models
• However, the theoretical models deviate
significantly from field data once capacity is
reached.
• For expressways and highways, density is more
sensitive measure of LOS than speed

Comparison of Theoretical Models
and Field Data
• Theoretical Models
• Restrictive Assumptions
• Do not predict very well
as v/c -> 1
• Problem often due to
extrapolation of model
beyond original domain
of data
• Empirical Data
• Difficult to get complete Q-K
curve
• Empirical measurements are
location dependent
• As v/c -> 1 gaps are there in
data
• Generalization may not be
true. Ability to generalize to
other or changed condition
may be limited.

Example to illustrate location
dependence of measurements
A B C D
Sections A, B, D have 3 lanes, and C has 2 lanes – Capacity Drop
A is far from capacity reduction
B is immediately upstream of reduction
D is after capacity reduction
Let Qm = capacity of one lane

U-K and Q-K curves – Secn A
Say volume keeps increasing in the morning peak period
Gradually at location A, from Qm, 2Qm, 2.5Qm etc.
Location A has 3 lanes so with increasing flow:
Speed will decrease
But flow is less than capacity
So flow, speed, and density will be uncongested
U
Sf
K
K
Q

U-K and Q-K curves at B
U
Sf
K
K
Q
Upto 2Qm, Section B will be uncongested
Beyond that it will become congested because capacity
Of section C is 2Qm and will start queuing for larger vols

U-K and Q-K curves at C
U
Sf
2Kj/3
K
2Qm
Note sharp drop in speed
Jam density is smaller
Peak flow is 2Qm
Flow will be in uncongested state as there is no queue

U-K and Q-K curves at D
U
Sf
K
K
Q
Inflow will always be <= 2Qm
Capacity = 3Qm
So flow is always uncongested

Empirical Measurements
Note that at no location a complete u-k or Q-k curve is obtained
Problem is at C, but effect is felt at B
Cause and effect are separated over space and time
Section B transitions from 2.5Qm to 2Qm without passing through
Capacity
Bottleneck at C leads to uncongested flow at D, but
Efficiency is low
Q-K and K-U plots can be used to identify bottlenceks and problem
spots

Macroscopic model

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