Download Poisson process

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Poisson process is one of the most important
models used in queueing theory.
Often the arrival process of customers can be
described by a Poisson process.
In teletraffic theory the “customers” may be
calls or packets. Poisson process is a viable
model when the calls or packets originate
from a large population of independent
users.
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Mathematically the process is described by
the counter process Nt or N (t). The counter
tells the number of arrivals that have
occurred in the interval (0, t) or, more
generally, in the interval (t1, t2)
N (t) = number of arrivals in the interval (0, t)
N (t1 , t2 ) = number of arrival in the interval
(t1 , t2)
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The Poisson process can be defined in three
different (but equivalent) ways:
1. Poisson process is a pure birth process:
In an infinite time interval dt there may occur
λ only one arrival. This happens with the
probability
λ dt independent of arrivals outside the
interval.
2.The number of arrivals N (t) in a finite
interval of length t obeys the Poisson(λt)
distribution,
(λt)n/( −λt )n!
P{N (t) = n} =
e
3.The interarrival times are independent and
obey the Exp(λ) distribution:
P{interarrival time > t} = e−λt
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The distribution was first introduced
by Siméon Denis Poisson (1781–1840) and
published, together with his probability theory.
In 1837 in his work Research on the Probability
of Judgments in Criminal and Civil Matters.
The work focused on certain random
variables N that count, among other things, the
number of discrete occurrences (sometimes
called "events" or “arrivals”) that take place
during a time-interval of given length.
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The Poisson distribution can also be used for the
number of events in other specified intervals such as
distance, area or volume.
For Example 4 pieces of mail per day on average.
There will be, however, a certain spread: sometimes a
little more, sometimes a little less, once in a while
nothing at all.Given only the average rate, for a certain
period of observation (pieces of mail per day,
phonecalls per hour, etc.), and assuming that the
process, or mix of processes, that produces the event
flow is essentially random, the Poisson distribution
specifies how likely it is that the count will be 3, or 5,
or 10, or any other number, during one period of
observation. That is, it predicts the degree of spread
around a known average rate of occurrence.
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A practical application of this distribution was
made by Ladislaus Bortkiewicz in 1898 when
he was given the task of investigating the
number of soldiers in the russian army killed
accidentally by horse kick; this experiment
introduced the Poisson distribution to the
field of reliability engineering.
The Poisson distribution is defined by:
f ( x) 
x 
 e
x!
Where f(x) is the probability of x occurrences in
an interval
 is the expected value or mean value of
occurrences within an interval
e is the natural logarithm. e = 2.71828
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Telecommunication example: telephone calls arriving in a
system.
Astronomy example: photons arriving at a telescope.
Biology example: the number of mutations on a strand
of DNA per unit length.
Management example: customers arriving at a counter or
call centre.
Civil engineering example: cars arriving at a traffic light.
Finance and insurance example: number of Losses/Claims
occurring in a given period of Time.
Earthquake seismology example: an asymptotic Poisson
model of seismic risk for large earthquakes. (Lomnitz,
1994).
Radioactivity Example: Decay of a radioactive nucleus.
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The Poisson process has several interesting (and useful)
properties:
1) Conditioning on the number of arrivals. Given that in
the interval (0, t) the number of arrivals is N (t) = n, these
n arrivals are independently and uniformly distributed in
the interval.
One way to generate a Poisson process in the interval (0, t) is as
follows:
– draw the total number of arrivals n from the Poisson(λt)
distribution
– for each arrival draw its position in the interval (0, t)
from the uniform distribution, independently of the
others
2) Superposition. The superposition of two Poisson
processes with intensities λ1 and λ2 is a Poisson process
with intensity λ = λ1 + λ2.
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3)Random selection. If a random selection is made
from a Poisson process with intensity λ such that
each arrival is selected with probability p,
independently of the others, the resulting process
is a Poisson process with intensity pλ.
4)Random split. If a Poisson process with intensity
λ is randomly split into two sub processes with
probabilities p1 and p2, where p1 + p2 = 1, then
the resulting processes are independent Poisson
processes with intensities p1λ & p2λ.
(This result allows an straight forward
generalization to a split into more than two sub
processes.)
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5) PASTA. The Poisson process has the so
called PASTA property (Poisson Arrivals See
Time Averages): for instance, customers with
Poisson arrivals see the system as if they
came into the system at a random instant of
time (despite they induce the evolution of the
system).