Hello. I have a question regarding poisson processes.

Suppose we have a room with x people, and occasionally people leave from the room. Can we model the number of people that leave from the room at a given small time period S as a poisson process, so P(one person leaves) = lamda*S?

I think that the number of people that leave during S' depends on the number of people that are still in the room, so it depends on the number of people that left during S. As a result there is no memorylessness.

However I've told that if the number of people in the room is much larger than the number of people leaving at each time period then the process can be approximated as poisson. Is this correct?

If it is correct, suppose that initially the number of people in the room is much larger than the number of people leaving the room. However, as the process continues and people leave, at some time the difference will stop being much larger. Does it affect the correctness of our model?

Thank you!