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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
1
Distributions for Traffic Analysis Poisson Distribution: light traffic conditions
e.g.
Several poisson distributions: m1, m2, m3, …
Then
2
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
Binomial Distribution
For congested traffic flow ---
P is the probability that one car arrives
Mean value:
Variance:
3
Distributions for Traffic Analysis
1mean
variance
xnxn
x pPcxP )1()(
npm
)1(2 pnps
x = 0, 1, 2, …, n
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
4
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, …
5
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)(
6
Negative exponential frequency curve
Bar indicate observed data taken on sample size of 609
7
Statistical distributions of traffic characteristics
8
Dashed curve applies only to probability scale
Shifted Exponential Distribution
9
)/()()( TtethP
)/()(1)( TtethP
,0)( tP
)]/()(exp[1
)(
TtT
tP
at t<
and
10
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)
11
Example of fhifted exponential fitted to freeway data
Erlang Distribution
12
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
Lognormal Distribution
especially for traffic in platoons
13
Composite Headway Model
Constrained flows
Unconstrained , free flows
14
)exp(1)exp(1)1()(
21
1
T
t
T
tthP
Selection of Headway Distribution
Generalized Poisson distribution (Dense Traffic)
15
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,
16
Distribution Models for Speeds
Normal distributions of speeds
Lognormal model of speeds
Gap acceptance distribution model
17
18
Cumulative (normal) distributions of speeds of four locations
19
Same data as above figure but with each distribution normalized
20
Lognormal plot of freeway spot speeds
21
Comparison of observed and theoretical distributions of rejected gaps
22
Lag and gap distribution for through movements
23
Distribution of accepted and rejected lags and gaps at intersection left turns