At each time slot, a router's buffer receives a packet with probability $p$, or releases one with probability $q$, or stays the same with $r$. Initially empty, what is the distribution of the packets after $n$ slots? As $n\to\infty$, what is the typical buffer size?
I find that the above problem is similar to the birth-death process but have not been able to compute the asked results.
On first look and read, I felt that the distribution is the product of the mentioned probabilities and the number of slots $n$. But I feel that my idea is very wrong. I'd really like some help with this problem.