The Poisson Process

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The Poisson Process
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Definition

• What is A Poisson Process?
• The Poisson Process is a counting that counts
  the number of occurrences of some specific event
  through time

Examples - Number of customers arriving to a
counter - Number of calls received at a
telephone exchange - Number of packets entering
a queue
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The Poisson Process
3rd Event Occurs
1st Event Occurs
2nd Event Occurs
4th Event Occurs
X4
X1
X3
X2
time
t0

• X1, X2, represent a sequence of ve independent
  random variables with identical distribution
• Xn depicts the time elapsed between the (n-1)th
  event and nth event occurrences
• Sn depicts a random variable for the time at
  which the nth event occurs
• Define N(t) as the number of events that have
  occurred up to some arbitrary time t.

The counting process N(t), tgt0 is called a
Poisson process if the inter-occurrence times X1,
X2, follow the exponential distribution
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The Poisson Process Example
For some reason, you decide everyday at 300 PM
to go to the bus stop and count the number of
buses that arrive. You record the number of buses
that have passed after 10 minutes
Sunday
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The Poisson Process Example
For some reason, you decide everyday at 300 PM
to go to the bus stop and count the number of
buses that arrive. You record the number of buses
that have passed after 10 minutes
Monday
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The Poisson Process Example
For some reason, you decide everyday at 300 PM
to go to the bus stop and count the number of
buses that arrive. You record the number of buses
that have passed after 10 minutes
Tuesday
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The Poisson Process Example

• Given that Xi follow an exponential distribution
  then N(t10) follows a Poisson Distribution

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The Exponential Distribution

• The exponential distribution describes a
  continuous random variable

Cumulative Distribution Function (CDF)
1
0
Probability Density Function (PDF)
?
0
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Mean of the Exponential Distribution
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Variance of the Exponential Distribution
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Laplace Transform of Exponential PDF

• The Laplace transform of any PDF for a
  continuous random variable may be used to deduce
  different parameters and characteristics of the
  distribution

What could F(S) be used for
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Probability Density Function for Sk

• The probability density function for the sum of
  k independent random variables (X1, X2, , Xk)
  could be deduced from the Laplace transform of
  fXn(x) as follows

From Laplace Transform Tables
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Cumulative Distribution Function for Sk
From Laplace Transform Tables
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The Poisson Distribution
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Mean of the Poisson Distribution

• On average the time between two consecutive
  events is 1/?
• This means that the event occurrence rate is ?
• Consequently in time t, the expected number of
  events is ?t

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Variance of the Poisson Distribution
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Moment Generating Function of Poisson Distribution

• The Moment Generating Function of any PMF for a
  discrete random variable may be used to deduce
  different parameters and characteristics of the
  distribution

What could G(Z) be used for
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Remaining Time of Exponential Distributions
Xk1 is the time interval between the kth and
k1th arrivals Condition T units have passed and
the k1th event has not occurred
yet Question Given that Xk1 is the remaining
time until the k1th event occurs What is
PrXk1x
kth Event Occurs
k1th Event Occurs
T
Xk1
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Remaining Time of Exponential Distributions
Xk1 follows an exponential Distribution, i.e.,
PrXk1t1-e-?t
The remaining time Xk1 follows an exponential
distribution with the same mean 1/? as that of
the inter-arrival time Xk1
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The Memoryless Property of Exponential
Distributions

• The Memoryless Property
• The waiting time until the next arrival has the
  same exponential distribution as the original
  inter-arrival time regardless of long ago the
  last arrival occurred
• Memoryless Property of Exponential Distribution
  and the Poisson Process

In the Poisson process, the number of arrivals
within any time interval s follows a Poisson
distribution with mean ?s
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Merging of Poisson Processes

• N1(t), t 0 and N2(t), t 0 are two
  independent Poisson processes with respective
  rates ?1 and ?2,
• Ni (t) corresponds to type i arrivals.
• The merged process N(t) N1(t) N2(t), t 0.
  Then N(t), t 0 is a Poisson process with rate
  ? ?1 ?2.
• Zk is the inter-arrival time between the (k -
  1)th and kth arrival in the merged process
• Ik i if the kth arrival in the merged process is
  a type i arrival,
• For any k 1, 2, . . . ,
• PIk i Zk t ?i/(?1?2) , i 1, 2,
  independently of t .

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Splitting of Poisson Processes

• N(t), t 0 is a Poisson process with rate ?.
• Each arrival of the process is classified as
  being a type 1 arrival or type 2 arrival with
  respective probabilities p1 and p2, independently
  of all other arrivals.
• Ni (t) is the number of type i arrivals up to
  time t .
• N1(t) and N2(t) are two independent Poisson
  processes having respective rates ?p1 and ?p2.