```INTRODUCTION
Statistical Properties
Dr. Gang-Len Chang
Professor and Director of
Traffic Safety and Operations Lab.
University of Maryland-College Park
1





Poisson Arrival and Exponential
Distribution
General Poisson Properties
Poisson Example Applications
Other Types of Distributions
2
Distance
1st
2nd
3rd
Gap
Occupancy


Time T
Given a time horizon: T ⇒ n headways, the distribution of
such headways depends on traffic conditions
Given a fixed time interval ∆t (T = k ⋅ ∆t ), the number of
arrivals during each ∆t is a distribution
3
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
⇒ λ (mean arrival rate) = constant
The probability that m vehicles appear in the interval (t0, t0+∆t) is
independent of t0: Pt ,t + ∆t [ M = m] = Pt [ M = m]
The traffic stream has no memory:
Pt ,t + ∆t [ M = m] is independent of the details of the process up to
time t0
The simultaneous appearance of several vehicles at a location can
P[ M ( xi , t , ∆t ) > 1]
be neglected; i.e.,
=0
0

0

0
0
0
lim
∆t →∞
∆t
4
Poisson Arrival

The number of Poisson arrivals occurring in a time
interval of t (= n ⋅ ∆t ) is:
(λt ) k ⋅ e − λt
k = 0, 1, 2, …
P[ M (t ) = k ] =
k!

The probability that there are at least k number of vehicles
arriving during interval t is:
(λt ) k ⋅ e − λt
P[ M (t ) ≥ k ] = ∑
k!
k '= k
∞
∴ Poisson is only applicable in light traffic conditions
5
Poisson Arrival
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]]
1
Lk
kth
Time
k
 (λx) k −1 ⋅ e −λx   (λ ⋅ dx) ⋅ e −λ ⋅dx 
(λx k −1 ) ⋅ e − λx
⋅ x k −1 ⋅ e − λx
λ
− λ ⋅dx
=
⋅
]=
⋅ [λ ⋅ dx ⋅ e

 =
(
k
−
1
)!
1
!
(
k
−
1
)!
(k − 1)!

 

= f L ( x)dx
k
6
Poisson Arrival
∴
f L ( x) =
k
λk ⋅ x k −1 ⋅ e − λx
(k − 1)!
, x ≥ 0; k = 1, 2, 3,…
⇒ the kth - order interarrival time distribution for a poisson
process is a kth - order Erlang pdf
f L ( x) = λ ⋅ e − λx x ≥ 0 (negative exponential distribution)
The probability = P(h ≥ x)
∞
= ∫x λ ⋅ e −λx ⋅ dx = e −λx (C.D.F.)
1
7
Poisson Arrival
From a Poisson perspective:
If “No vehicle arrives during the time length x”
≡ a time headway ≥ x
(λx) 0 −λx
P [ M = 0] =
⋅ e = e −λx (same as the previous case)
x
0!
Note:
= λx
 Headway is a continuous distribution: P ( h ≥ x ) = e
(λx) m ⋅ e − x
 Arrival rate is a discrete distribution: Px ( M = m) =
m!
8
Poisson Arrival

Congested Traffic Conditions
platoon
Distance
1st
Time T


Two types of headways ⇒ between and within platoons during
the same period T
T = T1+T2 , each period has a different mean headway λ1 and λ2
9
Multiple Independent Poisson Processes
Two Poisson processes: λ1 and λ2
The combined process: N(t) = N1(t) + N2(t) is also a poisson
process
pdf for λ1 ⇒ λ1 ⋅ e − λ x x1 ≥ 0 (time-period)
pdf for λ2 ⇒ λ2 ⋅ e − λ x x2 ≥ 0 (time-period)
1 1
2 2
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
Multiple Independent Poisson Processes
x1 and x2 are both random variables
∞
P[ x1 < x2 ] = ∫0
∞
∫x f x x ( x1 , x2 ) ⋅ dx1 ⋅ dx2
x1 ≥ 0, x2 ≥ 0
1
1 2
f x , x ( x1 , x2 ) = f x ( x1 ) ⋅ f x ( x2 ) = λ1λ2 ⋅ d − λ x e − λ
1 1
1
∴
2
1
2
∞
∞
P[ x1 < x2 ] = ∫0 dx1 ∫x dx2 ⋅ λ1λ2 ⋅ e
− λ1 x1
⋅e
1
=
Similarly,
2 x2
λ1
λ1 + λ2
∫
∞
0
e −u du =
P[ x2 < x1 ] =
− λ2 x 2
∞
= ∫ dx1 ⋅ λ1 ⋅ e −λ1x1 (e −λ2 x2 )
0
λ1
λ1 + λ2
λ2
λ1 + λ2
11
Multiple Independent Poisson Processes
For the entire process: T = T1 + T2 (λ1 and λ2)
The probability of a time-headway X > x is?
 Total number of arrivals during T period
= T1λ1 + T2λ2
 P(X > x) during T1 period and T2 period
= e − λ x and e − λ x
1

2
Total arrivals having their headways > x
Total number of arrivals
T1λ1 ⋅ e − λ x + T2λ2 ⋅ e − λ
=
T1λ1 + T2λ2
1
2x
(weighted average)
k
∑ Ti λi ⋅ e− λ x
i
Generalization, P[ X > x] =
i =1
k
∑ Ti λi
i =1
λ: arrival rate
12
Constrained Flow-Platoon

Headway within a platoon are exponentially distributed with a mean
arrival rate λ and minimum headway z0
for x < z0
1,
(shifted exponential distribution)
P[ X > x] =
e −λ '( z − z ) , for z ≥ z

0
0


The relation between λ and λ’
The expected value of the shifted distribution must be equal to the
13
Constrained Flow-Platoon

The arrival rate for such a shifted distribution λ’
1
where z = 1 / λ
λ'=
z − z0
∴ λ'=
λ
1 − z0 λ
∴ λ’ ⇒ cannot be observed
λ ⇒ actually observed
∴ P[ X > x] = e
−(
λ
1− z0 λ
)( z − z0 )
14
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
T1λ1 ⋅ e − λ x
P1 =
Total number of arrivals ( = T1λ1 + T2 λ2 )
1
−(
P2 =


λ2
)( z − z 0 )
T2λ2 ⋅ e
T1λ1 + T2λ2
1− z 0 λ 2
T1: total observed period during which traffic is not moved in
platoon
T2: total observed period during which vehicles are moved in
platoon
15
Problem

 left
the right
Pedestrians approach from
size of the crossing in a Poisson manner
λ
with average arrival rate λ 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
Automobile Traffic
L
R
λL
Pedestrian Traffic
λR
Pedestrian Traffic
Pedestrian Crossing Light
16


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
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
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