Alice sends messages to Bob, with rate described by $\mathrm{Poisson}(1)$. However, Bob receives them only after a delay of $X$, for some non-constant r.v. $X$ (i.e. the messages may not arrive in the sent order).
How should $X$ be distributed so that Bob can claim that also he receives messages with rate described by $\mathrm{Poisson}(1)$?