= Poisson migration stream without a return path
{c}
{title2=$j\to k\text{ stream has rate }\alpha_jp_{jk}$}
If no directed routing path returns from $k$ to $j$, the set of colonies reachable from $k$ has no outgoing edge to its complement. The complement is an autonomous <open migration process>, and the marked $j\to k$ transition is an exit from it. In <time reversal of an open migration process>, that exit is a state-independent Poisson immigration channel of rate $\alpha_jp_{jk}$. Reversing a stationary two-sided <Poisson process> preserves its law, proving the forward stream is Poisson. This argument does not assert Poisson internal streams for networks with feedback.
Back to article page
