Onto the notation and interpretation of queueing theory-related markov chains

Question

If in a server with probability $p$ one job arrives and independently with probability $q$ one job departs, could you please explain to me what is the quantity and let me know if I have understood? The queue is infinite.

Note that during a time step, we might have both an arrival and transmission, or neither.

$1$. $r=p(1-q)$: This denotes the probability that there is a job at the server right? i.e in Markov chain notation $\mathbb P(X_n=1| X_{n-1}=0)$

$2.$ $s=q(1-p)$: This denotes the probability that there is no job at the server right? i.e in Markov chain notation $\mathbb P(X_n=0| X_{n-1}=1)$ (I am not sure)

$3.$ What is the quantity $r+s$ denotes?

$4.$ What about $1-(r+s)$?

Thanks for helping.

Answer 1

1. $r$ is the probability to have an arrival and no departure, i.e. $P(X_n=X_{n-1}+1)$.
2. $s$ is the probability to have a departure and no arrival, i.e. $P(X_n=X_{n-1}-1)$.
3. $r+s$ is the probability that the queue length changes, i.e. $P(X_n \neq X_{n-1})$.
4. $1-(r+s)$ is the probability that the queue length doesn't change, i.e. $P(X_n=X_{n-1})$.