Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 26J (Applied Probability)

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:

Within each stream every subsequent gap is the sum of three consecutive original exponential random variables, hence has an Erlang distribution of shape three and rate $\lambda$. Successive gaps within that stream use disjoint original gaps, so they are independent and identically distributed, and independent of its first delay. Gaps from different streams can overlap in their constituent original gaps, so the three streams are not independent.

Streams $a$ and $b$ are delayed renewal processes; $c$ is an ordinary renewal process. Their first delays have Erlang shapes one and two, while subsequent gaps have shape three. Stream $c$ has shape three for its first gap as well.

For a renewal gap $T$ of mean $m$, the equilibrium first-delay density is $\mathbb{P}(T>t)/m$. Here $m=3/\lambda$ and the Erlang distribution survivor is $e^{-\lambda t}(1+\lambda t+{\lambda }^2{t}^2/2)$. The equilibrium delay of an Erlang renewal process is thereforeThis is an equal mixture of shapes one, two and three. For each equilibrium stream use this first delay, followed by independent shape-three gaps. A joint stationary version of the cyclic routing is obtained by choosing the initial position in the three-label cycle uniformly at random, independently of the Poisson arrivals. Its marginal streams have the stated equilibrium law, but remain dependent on one another.

Assign each arriving car independently to a uniformly chosen lot. The Poisson thinning theorem gives three independent Poisson processes, each of rate $\lambda /3$. For an interval of length $t$, their joint probability generating function isThe factorization proves independence of the counts in that interval; independence over disjoint intervals proves independence of the processes.