LECTURE 14
Bernoulli review
• Discrete time; success probability p
The Poisson process
• Number of arrivals in n time slots:
16
BernoulliLECTURE
review
binomial pmf
• Readings: LECTURE
Start Section
166.2.
• Discrete time; success
probability
p
The Poisson
process
• Interarrival
times: geometric
pmf
• D
• Readings: Start Section 5.2.
• Review of Bernoulli process
• Number
of arrivals in
n time
slots:5.2.
• Readings:
Start
Section
• Time to k arrivals: Pascal pmf
binomial pmf
• N
b
• Definition of
Poisson
process
Lecture
outline
• Memorylessness
• Interarrival
time pmf:
geometric
pmf
Lecture
outline
• I
•• Distribution
of number
of arrivals
Review of Bernoulli
process
• Time•toReview
k arrivals:
Pascal pmf
of Bernoulli
process
• T
•• Distribution
interarrival
times
Definition ofof
Poisson
process
• Memorylessness
• Definition of Poisson process
• M
The Poisson process
Lecture outline
•• Other
properties
of the of
Poisson
Distribution
of number
arrivalsprocess
• Distribution of number of arrivals
• Distribution of interarrival times
• Distribution of interarrival times
• Other properties of the Poisson process
• Other properties of the Poisson process
Definition of the Poisson process
Definition of the Poisson process
t2
t1
0
x x
x x
t3
x x
xx
PMF of NumberPoisson
of Arrivals N
PMF Definition
of Numberofofthe
Arrivals N process
x
t2
t1
!
x xk −λτ
x x
0 (λτ ) e
x x
Time
P (k, τ ) =
k!
,
t3
x x
xx
!
x
k = 0, 1, . . .
x x
Time
• E[N ] •= λτ
P (k, τ ) = Prob. of k arrivals in interval
• Finely
discretize [0, t]: approximately Bernoulli
2 = λτof duration τ
• σN
•• Time
P (k, τ )homogeneity:
= Prob. of k arrivals in interval
Pof(k,duration
τ ) = Prob.
of k arrivals in interval
τ
of duration τ
• Assumptions:
• Numbers of arrivals in disjoint time
– Numbers of arrivals in disjoint time inintervals
independent
tervals are
are independent
• Assumptions:
λt(esdiscrete
−1)
Nte(of
• =
approximation): binomial
• MN (s)
– Numbers of arrivals in disjoint time intervals are independent
• Taking δ → 0 (or n → ∞) gives:
– For VERY small δ:
−λτ
(λτ )k
eaccording
Example: You
to a
P (k, get
τ ) =email 
 1 −,λδ ifk k==0,01, . . .
k!
Poisson process at
a
rate
of
λ
=
0.4
mesP (k, δ) ≈ λδ
if k = 1


sages per hour. You check your
0 email ifevery
k>1
• E[Nt] = λt,
var(Nt) = λt
thirty minutes.
– λ = “arrival rate”
– Prob(no new messages)=
For VERY
small
δ:
• –Small
interval
probabilities:

For VERY small 
δ:
1
 − λδ if k = 0
P (k, δ) ≈
=1

1 λδ
− λδ, ifif kk=
0;



0
if k > 1
P (k, δ ) ≈ λδ,
if k = 1;



– λ = “arrival rate”
0,
if k > 1.
– λ: “arrival rate”
– Prob(one new message)=
1
• E
• σ
• M
Exa
Poi
sag
thir
–
–
Example
Interarrival Times
Interarrival Times
• Yk time of kth arrival
• You get email according to a Poisson
process at a rate of λ = 5 messages per
hour. You check your email every thirty
minutes.
• Yk time of kth arrival
• Erlang distribution:
k k−1
e−λy
λ y
fYk (y) = k k−1 −λy ,
λ y(k −e1)!
fYk (y) =
• Prob(no new messages) =
to the
past
(k − 1)!
flr (l)
fY (y)
k
• Prob(one new message) =
y≥0
y≥0
,
r=1
r=2
r=3
k=1
k=2
k=3
0
y
l
Image by MIT OpenCourseWare.
• First-order interarrival
times (k = 1):
exponential
• Time of first arrival (k = 1):
(y) = λe−λy , f y
≥0
fY1exponential:
(y) = λe−λy , y ≥ 0
Y1
– –Memoryless
the
Memorylessproperty:
property: The
The time
time to
to the
next
arrival
is
independent
of
the
past
next arrival is independent of the past
Adding
Poisson
Processes
Merging
PoissonProcesses
Bernoulli/Poisson Relation
Bernoulli/Poisson Relation
! ! ! ! ! ! ! !
0
x
x
n = t /!
x
Arrivals
• •Sum
of of
independent
Poisson
Sum
independent
Poissonrandom
randomvariables
is Poisson
variables
is Poisson
Time
p ="!
np =" t
Merging
of independent
Poisson
processes
• •Sum
of independent
Poisson
processes
Poisson
is is
Poisson
0
1):
!
• Erlang distribution:
POISSON
Times of Arrival
Times Arrival
of Arrival
Rate
POISSON
Continuous
Continuous
λ/unit time
BERNOULLI
Discrete
Poisson
λ/uni
t time
Interarrival Time Distr.
Exponential
Geometric
Erlang
Pascal
Time to k-th arrival
Poisson
"1
Discrete
p/per
trial
PMF of Rate
# of Arrivals
Arrival
PMF of # of Arrivals
Red bulb flashes
(Poisson)
BERNOULLI
"2
Binomial
p/per trial
Green bulb flashes
(Poisson)
Binomial
Interarrival Time Distr.
Exponential
Geometric
Time to k-th arrival
Erlang
Pascal
All flashes
(Poisson)
– What is the probability that the next
– What is the probability that the next
arrival comes from the first process?
arrival comes from the first process?
m vari-
2
0
MIT OpenCourseWare
http://ocw.mit.edu
6.041 / 6.431 Probabilistic Systems Analysis and Applied Probability
Fall 2010
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