INTRODUCTION

Statistical Properties

Dr. Gang-Len Chang

Professor and Director of

Traffic Safety and Operations Lab.

University of Maryland-College Park

1

Definition of Time Headway

Poisson Arrival and Exponential

Distribution

General Poisson Properties

Poisson Example Applications

Other Types of Distributions

2

Time Headway Distribution

Given a time horizon: T n headways, the distribution of

such headways depends on traffic condition

Given a fixed time interval t ( ), the number of

arrivals during each t is a distribution

3

Time T

Distance

1st 3rd2nd

Gap

Occupancy

tkT

Poisson Arrival

In order to be able to describe traffic as a poisson process,

the following assumptions are required:

The traffic stream must be stationary

(mean arrival rate) = constant

The probability that m vehicles appear in the interval (t0, t0+t) is

independent of t0:

The traffic stream has no memory:

is independent of the details of the process up to

time t0

The simultaneous appearance of several vehicles at a location can

be neglected; i.e.,

4

][][000 , mMPmMP tttt

][00 , mMP ttt

0]1),,([

lim

t

ttxMP i

t

Poisson Arrival

The number of poisson arrivals occurring in a time

interval of is:

k = 0, 1, 2, …

The probability that there are at least k number of vehicles

arriving during interval t is:

Poisson is only applicable in light traffic conditions

5

)( tnt

!

)(])([

k

etktMP

tk

kk

tk

k

etktMP

' !

)(])([

Poisson Arrival Interarrival Times = headway

Let Lk = time for occurrence of the kth arrival, k = 1, 2, 3,…

The pdf fLk(x)dx

P[kth arrival occurs in the interval x to x+dx]

= P[exactly k-1 arrivals in the interval [0,x] and exactly one arrival in

[x,x+dx]]

6

Lk

Time

kth1

!1

)(

)!1(

)( 1 dxxk edx

k

ex

)!1()!1(

)( 11

k

exedx

k

ex xkkdx

xk

dxxfkL )(

Poisson Arrival

, x 0; k = 1, 2, 3,…

the kth - order interarrival time distribution for a poisson

process is a kth - order Erlang pdf

set k = 1 (headway)

x 0 (negative exponential distribution)

The probability

(C.D.F.)

7

)!1()(

1

k

exxf

xkk

Lk

x

L exf )(1

)( xhP

x

x

x edxe

Poisson Arrival

From a Poisson perspective:

If “No vehicle arrives during the time length x”

a time headway x

(same as the previous case)

Note:

Headway is a continuous distribution:

Arrival rate is a discrete distribution:

8

xx

x eex

MP !0

)(]0[

0

xexhP )(

!

)()(

m

exmMP

xm

x

Poisson Arrival

Congested Traffic Conditions

Two types of headways between and within platoons during

the same period T

T = T1+T2 , each period has a different mean headway 1 and 2

9

Time T

Distance

1st

platoon

Multiple Independent Poisson Processes Two poisson processes: 1 and 2

The combined process: N(t) = N1(t) + N2(t) is also a poisson

process

pdf for 1 x1 0 (time-period)

pdf for 2 x2 0 (time-period)

The two are independent:

What is the probability that an arrival form process 1 (type 1

arrival) occurs before an arrival from process 2 (type 2

arrival)?

10

11

1

xe

22

2

xe

Multiple Independent Poisson Processes x1 and x2 are both random variables

x1 0, x2 0

Similarly,

11

0 212121

121

),(][x xx dxdxxxfxxP

2211

2121 212121, )()(),(xx

xxxx edxfxfxxf

][ 21 xxP

02121

1

2211

x

xxeedxdx

0

11 )( 2211 xxeedx

021

1

21

1

due u

21

2

12 ][

xxP

Multiple Independent Poisson Processes For the entire process: T = T1 + T2 (1 and 2)

The probability of a time-headway X > x is?

Total number of arrivals during T period

= T11 + T22

P(X > x) during T1 period and T2 period

= and

= (weighted average)

Generalization, : arrival rate

12

xe 1 x

e 2

arrivals of number Total

x headways their having arrivals Total

2211

221121

TT

eTeTxx

k

iii

k

i

x

ii

T

eT

xXP

i

1

1][

Constrained Flow-Platoon

Headway within a platoon are exponentially distributed with a mean

arrival rate and minimum headway z0

(shifted exponential distribution)

The relation between and ’

The expected value of the shifted distribution must be equal to the

actual mean headway

13

0

zz

0

zz for e

zx for xXP

,

,1][

)(' 0

Constrained Flow-Platoon The arrival rate for such a shifted distribution ’

where

’ cannot be observed

actually observed

14

0

1'

zz /1z

01'

z

))(1

( 0

0][zz

zexXP

Some Travel Free, Some Are in Platoon Combination of two poisson processes:

P[X > x] = P[X >x | occurs in travel free traffic]

+ P[X > x | in platoon traffic]

= P1 + P2

T1: total observed period during which traffic is not moved in

platoon

T2: total observed period during which vehicles are moved in

platoon

15

)221

11

1

1

TT ( arrivals of number Total

eTP

1

x

2211

))(1

(

222

020

2

TT

eTP

zzz

Problem Pedestrians approach from the size of the crossing in a Poisson manner

with average arrival rate arrivals per minute (Figure). Each pedestrian

then waits until a light is flashed, at which time all waiting pedestrians must

cross. We refer to each time the light is flashed as a “dump” and assume that

a dump takes zero time (i.e., pedestrians cross instantly). Assume that the left

and right arrival processes are independent

16

right

left

R

L

Pedestrian Traffic

Automobile Traffic

Pedestrian Traffic

Pedestrian Crossing Light

RL

We wish to analyze three possible decision rules for operating the

light:

Rule A: Dump every T minutes

Rule B: Dump whenever the total number of waiting pedestrians

equals N0

Rule C: Dump whenever the first pedestrian to arrive after the

precious dump has waited T0 minutes

Presumably, implementation of each rule requires a particular type of

technology with its attendant costs, and thus it is important to

determine the operating characteristics of each in order to understand

tradeoffs between performance and cost

17

For each decision rule, determine:

The expected number of pedestrians crossing left to right on any

dump

The probability that zero pedestrians crossing left to right on any

particular dump

The pdf for the time between dumps

The expected time that a randomly arriving pedestrian must wait

until crossing

The expected time that a randomly arriving observer, who is not a

pedestrian, will wait until the next dump

18