Poisson process into a series of Bernouilli trial problem The ants of a colony arrives at a location with two food sources according to a poisson process $N(t),t\geq 0$ at the rate of $\lambda$. Once there, each ant will independently choose to eat form one of the sources $A$ or $B$ with respective probabilities $p, (1-p)$ respectively.
Let $\{T^{(a)}_i\}, i\in(0,\infty)$ be the interarrival sequence of ants that go to food source A. What is the distribution of $T^{(A)}_i$? Are these random variables independent?
I am not sure how to do this... Do I just plug in the expected amount of arrivals at each hour into the binomial equation to get the answer?

Clearly the long term arrival rate splits into $\lambda = p \lambda + (1-p)\lambda$.
If the number of ants at the origin is kept (supposed to be) constant then you can think to split the procession right at the origin and the two are clearly independent and Poisson distributed.
However the actual process looks to be that the ants return to the origin after a (constant ?) forth and back traveling time. If this is not negligible wrt $1/\lambda$, then the two processions are not longer independent.
In fact if at a certain moment there is e.g. a massive group of ants leaving for A, that will cause a massive decrease in the remaining number and thus a decrease in the number of ants leaving for B.
