Long-tail Traffic - The Poisson Distribution and Traffic

The Poisson Distribution and Traffic

Before the heavy-tail distribution is introduced mathematically, the memoryless Poisson distribution, used to model traditional telephony networks, is briefly reviewed below. For more details, see the article on the Poisson distribution.

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 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 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.:

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

where h is the Mean Holding Time (MHT) .

Information on the fundamentals of statistics and probability theory can be found in the external links section.

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