TRAFFIC - City of Yarra · 2019-06-16 · AM Peak 8.1 AM-8:00 PM Peak SAO 1.-.00 INA
Traffic Flow Analysis - National Chiao Tung Universityocw.nctu.edu.tw/course/tco001/C1.pdf ·...
Transcript of Traffic Flow Analysis - National Chiao Tung Universityocw.nctu.edu.tw/course/tco001/C1.pdf ·...
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Traffic Flow Analysis Basic Properties
Dr. Gang-Len Chang
Professor and Director of
Traffic Safety and Operations Lab.
University of Maryland-College Park
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Distributions for Traffic Analysis Poisson Distribution: light traffic conditions
e.g.
Several poisson distributions: m1, m2, m3, …
Then
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time of nsobservatio Total
soccurrence Totalvalueavem ).(
1/2 m
!/)( xemxP mx
tm
x = 0, 1, 2,…
t: selected time interval meP )0(
x
m
mx
m
mx
m
xP
xPx
x
)exp()!1(
)exp(!
)1(
)(1
)1()( xPx
mxP
N
iimm
1
Limitations: only for discrete random events
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Binomial Distribution
For congested traffic flow ---
P is the probability that one car arrives
Mean value:
Variance:
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Distributions for Traffic Analysis
1mean
variance
xnxn
x pPcxP )1()(
npm
)1(2 pnps
x = 0, 1, 2, …, n
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Traffic counts with high variance – extend over both a peak period
and a n off-peak period
e.g. a short counting interval for traffic over a cycle, or downstream
from a traffic signal
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Distributions for Traffic Analysis
kkkx
k qPcxP 1
1)(
2ˆs
mp ms
mk
2
2
ˆ )ˆ1(ˆ pq
kpp )0(
)1(1
)(
xpqx
kxxp
x = 0, 1, 2, …
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Distributions for Traffic Analysis Interval Distribution Negative Exponential Distribution
Let V: hourly volume, = V/3600 (cars/sec)
If there is no vehicle arrive in a particular interval of length t, there will
be a headway of at least t sec.
P(0) = the probability of a headway t sec
Mean headway T = 3600/V
variance of headway = T2
!)
3600()(
3600/
x
etVxP
Vtx
3600/)0( VteP
3600/)( VtethP
TtethP /)(
TtethP /1)(
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Negative exponential frequency curve
Bar indicate observed data taken on sample size of 609
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Statistical distributions of traffic characteristics
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Dashed curve applies only to probability scale
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Shifted Exponential Distribution
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)/()()( TtethP
)/()(1)( TtethP
,0)( tP
)]/()(exp[1
)(
TtT
tP
at t<
and
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Shifted exponential distribution to represent the probability of
headways less then t with a prohibition of headways less than .
(Average of observed headways is T)
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Example of fhifted exponential fitted to freeway data
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Erlang Distribution
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1
0
/
!)()(
k
t
Tkti
i
e
T
ktthP
TkteT
ktthP /1)(
TkteT
kt
T
ktthP /2
!2
1)()(1)(
22 /~
STk
for k = 1
k: a parameter determining the shape of the distribution
for k = 2
for k = 3
Reduced to the exonential distribution
T: mean interval, S2 : variance
* k = 1, the data appear to be random
* k increase, the degree of nonrandomness appears to increase
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Lognormal Distribution
especially for traffic in platoons
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Composite Headway Model
Constrained flows
Unconstrained , free flows
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)exp(1)exp(1)1()(
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1
T
t
T
tthP
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Selection of Headway Distribution
Generalized Poisson distribution (Dense Traffic)
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eeP )0(
!3!2)1(
32
ee
P
k = 2,
1)(
!)(
ixk
xkj
j
j
exP
x = 0, 1, 2,…
k
i
ixk
ixk
exP
1
1
)!1(
)()(
)1(2/1 kkm
or x = 0, 1, 2,…
!2)0(
2
e
eeP
!5!4!3)1(
543
eee
P
k = 3,
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Distribution Models for Speeds
Normal distributions of speeds
Lognormal model of speeds
Gap acceptance distribution model
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Cumulative (normal) distributions of speeds of four locations
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Same data as above figure but with each distribution normalized
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Lognormal plot of freeway spot speeds
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Comparison of observed and theoretical distributions of rejected gaps
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Lag and gap distribution for through movements
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Distribution of accepted and rejected lags and gaps at intersection left turns