Self-similar Process - The Poisson Distribution

The Poisson Distribution

Before the heavy-tailed distribution is introduced mathematically, the Poisson process with a memoryless waiting-time distribution, used to model (among many things) traditional telephony networks, is briefly reviewed below.

Assuming pure-chance arrivals and pure-chance terminations leads to the following:

The number of call arrivals in a given time has a Poisson distribution, i.e.:

$P(a)= \left ( \frac{\mu^a}{a!} \right )e^{-\mu},$

where a is the number of call arrivals in time T, and is the mean number of call arrivals in time T. For this reason, pure-chance traffic is also known as Poisson traffic.

The number of call departures in a given time, also has a Poisson distribution, i.e.:

$P(d)=\left(\frac{\lambda^d}{d!}\right)e^{-\lambda},$

where d is the number of call departures in time T and is the mean number of call departures in time T.

The intervals, T, between call arrivals and departures are intervals between independent, identically distributed random events. It can be shown that these intervals have a negative exponential distribution, i.e.:

$P=e^{-t/h},\,$

where h is the mean holding time (MHT).

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