[1] It results from a system of one or more inter-related Poisson processes occurring in sequence, or phases.

The sequence in which each of the phases occurs may itself be a stochastic process.

The distribution can be represented by a random variable describing the time until absorption of a Markov process with one absorbing state.

It has a discrete-time equivalent – the discrete phase-type distribution.

It is usually assumed the probability of process starting in the absorbing state is zero (i.e. α0= 0).

The Erlang distribution has two parameters, the shape an integer k > 0 and the rate λ > 0.

The Erlang distribution can be written in the form of a phase-type distribution by making S a k×k matrix with diagonal elements -λ and super-diagonal elements λ, with the probability of starting in state 1 equal to 1.

For example, E(5,λ), and For a given number of phases, the Erlang distribution is the phase type distribution with smallest coefficient of variation.

The generalised Coxian distribution relaxes the condition that requires starting in the first phase.

BuTools includes methods for generating samples from phase-type distributed random variables.

Approximating a deterministic distribution of time 1 with 10 phases, each of average length 0.1 will have variance 0.1 (because the Erlang distribution has smallest variance[2]).