Transcript lec2_final

Poisson Process
Review Session 2
EE384X
Winter 2006
EE384x
1
Point Processes

Supermarket model : customers arrive
(randomly), get served, leave the store
Arrival Process

Server
Queue
Departure Process
Need to model the arrival and departure
processes
Winter 2006
EE384x
2
What does Poisson Process
model?
Start time of Phone calls in Palo Alto
 Session initiation times (ftp/web servers)
 Number of radioactive emissions (or
photons)
 Fusing of light bulbs, number of accidents
 Historically, used to model packets
(massages) arriving at a network switch


(In Kleinrock’s PhD thesis, MIT 1964)
Winter 2006
EE384x
3
Properties
Say there has been 100 calls in an hour in
Palo Alto
 We expect that :





The start time of each call is independent of the others
The start time of each call is uniformly distributed over
the one hour
The probability of getting two calls at exactly the same
time is zero
Poisson Process has the above properties
Winter 2006
EE384x
4
Notation
0
Winter 2006
EE384x
5
Notation


A(t) : Number of points in (0,t]
A(s,t) : Number of points in (s,t]
 Arrival
points :
 Inter-arrival times:
Winter 2006
EE384x
6
Poisson Process- Definition
 A(0)=0 and each jump is of unit magnitude

Number of arrivals in disjoint intervals are
independent


For any
the random variables
are independent.
Number of arrivals in any interval of
length t is distributed Poisson(lt)
Winter 2006
EE384x
7
Basic Properties
Winter 2006
EE384x
8
Stationary Increments

The number of arrivals in (t,t+t] does not
depend on t
Winter 2006
EE384x
9
Orderliness


The probability of two or more arrivals in
an interval of length t gets small as
Arrivals occur “one at a time”
Winter 2006
EE384x
10
Poisson Rate


Probability of one arrival in a short interval
is (approx) proportional to interval length
Poisson process is like a continuous version
of Bernoulli IID
Winter 2006
EE384x
11
Additional Properties
Winter 2006
EE384x
12
Inter-arrival times

Inter-arrival times are Exponential(l) and
independent of each other
0
Winter 2006
EE384x
13
Merging two Poisson processes
merge

Merging two independent Poisson processes
with rates l1 and l2 creates a Poisson
process with rate l1+l2



A(0)=A1(0)+A2(0)=0
Number of arrivals in disjoint intervals are independent
Sum of two independent Poisson rv is Poisson
Winter 2006
EE384x
14
Sum of two Poisson rv

Characteristic function:

So
Winter 2006
EE384x
15
Splitting a Poisson process
:Poisson process with rate
l
Split

For each point toss a coin (with bias p):

With probability p

With probability 1-p the point goes to A2(t)
 A1(t) and
the point goes to A1(t)
A2(t) are two independent Poisson
processes with rates
Winter 2006
EE384x
16
proof

Define A1(t) and A2(t) such that:


A1(0)=0
A2(0)=0
Number of points in disjoint intervals are
independent for A1(t) and A2(t)

They depend on number of points in disjoint intervals of
A(t)

Need to show that number of points of A1
and A2 in an interval of size t are
independent Poisson(l1t) and Poisson(l2t)
Winter 2006
EE384x
17
Dividing a Poisson rv
Winter 2006
EE384x
18
Dividing a Poisson rv (cont)

So
Winter 2006
EE384x
19
Uniformity of arrival times

Given that there are n points in [0,t], the
unordered arrival times are uniformly
distributed and independent of each other.
Unordered variables
0
Ordered variables
Winter 2006
EE384x
20
Single arrival case
0
Winter 2006
EE384x
21
General case

It is the n order statistics of uniform
distribution.
Winter 2006
EE384x
22