Imagine a room with a window through which flies fly into the room by poisson law. But they can also fly out with probability proportional to the number of flies already in the room. What is then probability that at moment t there are n flies in the room?
Although this problem seems quite fundamental and natural, the answer must be very complex. Because a much more simple case, when flies fly in and out with the same constant probability and we allow negative number of flies has a formula involving modified Bessel functions. But it's obvious that a fly cannot leave the room if this fly is not inside this room.
Yes, of course it is a M/M/∞ queue.
So flies fill a room by the law
$$ P_n(t)=\frac{a^n}{n!}e^{-a}, \ \ \ \ a=\frac{\lambda}{\mu}(1-e^{-\mu t}) $$ where $\lambda$ and $\mu$ are poisson parameters for a fly to enter or leave the room correspondingly. And $\mu$ is inversely proportional to the room volume.
So if, for example, usually there are ten flies in the room, then probability that the room is empty is close to exp(-10)
This formula is easy to derive because it is nothing more than diffusion out of infinite volume. All we should do is to find probability that a particle crossed the border odd number of times and then to find the limit of the corresponding Bernouilli distribution