Queueing Theory / Poisson process


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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