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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