Write the successive gaps of the rate-$\lambda$ <Poisson process> as independent <exponential random variables> $E_j$. The first arrivals in the three streams occur at $E_1$, $E_1+E_2$, and $E_1+E_2+E_3$. They have <Erlang distributions> of shapes one, two and three:
$$
\boxed{f_i(t)=\frac{\lambda^i t^{i-1}e^{-\lambda t}}{(i-1)!},\quad t\ge0,\quad i=1,2,3.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 26J / a / Solution

 Solution (source code)