Calls arrive at a telephone exchange as a Poisson process of rate $\lambda$, and the lengths of calls are independent exponential random variables of parameter $\mu$.
Assuming that infinitely many telephone lines are available, set up a Markov chain model for this process. What is this model called in the literature?
Show that for large $t$ the distribution of the number of lines in use at time $t$ is approximately Poisson with mean $\lambda/\mu$.
Find the mean length of the busy periods during which at least one line is in use.
Show that $m(t)$, the expected number of lines in use at time $t$, given that $n$ are in us at time $0$, is $$ m(t) = n \mathrm{e}^{-\mu t} + \frac{\lambda}{\mu} \left( 1 - \mathrm{e}^{-\mu t}\right). $$
The question is related to the calls arriving at a telephone exchange as a Poisson process of rate $\lambda$. I have been able to figure out the asymptotic distribution for it and also the mean length.
But I have been unable to understand as to how to go about solving the last part (4) of the question which asks: "about the expected number of lines in use at time $t$, given that $n$ are in use at time $0$"
Would be great if someone can explain to me how to solve this? (I was thinking that renewal theory could be used here, but I am not very sure if it is the best option).