08-16-2017, 10:53 PM
Queueing Theory / Poisson process
[attachment=336]
General
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.
In the following it is instructive to think that the Poisson process we consider represents
discrete arrivals (of e.g. calls or packets).
A Poisson process can be characterized in different ways:
Process of independent increments
Pure birth process
the arrival intensity (mean arrival rate; probability of arrival per time unit
The most random process with a given intensity
Properties of the Poisson process
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( t) distribution
for each arrival draw its position in the interval (0, t) from the uniform distribution,
independently of the others
[attachment=336]
General
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.
In the following it is instructive to think that the Poisson process we consider represents
discrete arrivals (of e.g. calls or packets).
A Poisson process can be characterized in different ways:
Process of independent increments
Pure birth process
the arrival intensity (mean arrival rate; probability of arrival per time unit
The most random process with a given intensity
Properties of the Poisson process
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( t) distribution
for each arrival draw its position in the interval (0, t) from the uniform distribution,
independently of the others