Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 26I / a / Solution

Assume the first-fork choices are independent Bernoulli routings, as in Poisson thinning. Put $r=2\lambda /3$. Cars assigned to A form a Poisson process of rate $\lambda /3$, also an ordinary renewal process with independent exponential holding times of rate $\lambda /3$. The stream to the second fork is an independent Poisson process of rate $r$.

B receives its odd-numbered arrivals. Its initial delay is exponential of rate $r$, followed by independent Erlang distributions of shape two and rate $r$ between successive B arrivals. Thus B is a delayed renewal process, not Poisson. C receives the even-numbered arrivals: its first and all subsequent holding times are independent Erlang$(2,r)$, so C is an ordinary renewal process, not Poisson. Their holding-time density and survivor are $r^{2}t{e}^{-rt}$ and $e^{-rt}(1+rt)$. Both have asymptotic rate $r/2=\lambda /3$; equality of these mean rates with A does not make the processes Poisson.