Poisson distribution (PDF):
Erlang distribution (PDF):
where is the arrival rate of a Poisson process, and is the number of arrival events. The Poisson distribution tells the probability of the number of arrivals over a given amount of time . The Erlang distribution tells the probability of the time interval over a given number of arrivals . The difference is that, in Erlang distribution, it is assumed to have an arrival at time (which is not counted toward ) and (which is counted toward ), respectively, whereas Poisson does not.
Poisson is derived using the law of rare events: Assume denote the binomial distribution of Bernoulli events with “success” probability . Assume the rate of occurrence of “success” is , then where is the time to finish Bernoulli events. Define to be the number of “success” events occurred. So
The formula for Erlang distribution is derived by induction. The case of is trivial because this is the exponential distribution, i.e. the time for the next arrival is .
Consider , then is defined by the convolution: