My question has a queue M/M/1/2, that is, a system with exponential interarrival and service times, one server and having a room only for 2 customers (including the one in service, and another that is waiting). Let the number of customers in the room at time $t$ be $X(t)$. I model this as a birth and death process with transition rate matrix being
$\begin{bmatrix} -\lambda & \lambda & 0 \\\\ \mu & -\lambda - \mu & \lambda \\\\ 0&\mu&-\mu \end{bmatrix}.$
I have trouble to understand and compute:
"the average number of customers in the room in the long run"
Is this to compute $\lim_{t \rightarrow \infty} X(t)/t$, or expectation of the limit distribution of $X(t)$ as $t \rightarrow \infty$?
"the proportion of potential customers that enter the room in the long run"
How is this represented in formula?
"If the server works twice faster, how much more business it will do"
How is "how much more business" represented in formula?
For each part, are there some theorems that can help me get started?
Thank you so much!