# Why use poisson to model arrival of clients

1. Jun 5, 2012

### Mark J.

Hi
Almost every text use as example poisson/exponential distribution to model clients arrival.
What makes this distribution so good to fit in these cases?

Regards

2. Jun 5, 2012

### viraltux

Hi Mark,

Imagine you have a group of just 10 possible clients equally likely to arrive within a time gap with a probability p. That would follow a Binomial ~ B(10,p). Right? That's OK for 10 clients but how about 1000? or 1000000? You could still use a binomial, but it turns out that when the number of clients goes to infinity the distribution converges towards a Poisson ~ Poiss(λ) where λ is the average number of clients within the time gap.

So since in practice you can approximate the group of clients like if there were an infinite number of them, the Poisson distribution makes more sense in those situations than using a Binomial.

The Exponential distribution simply accounts for the time difference between events in a Poisson distribution, that is, once you use a Poisson to model the arrival of clients, the time length between one client and the next follows an Exponential distribution.

So this is why.

Last edited: Jun 6, 2012