I'm having trouble with the following question.
A common car service between cities in Israel is a sheroot. A sheroot is a seven-seat cab that leaves from its stand as soon as it has collected seven passengers. Suppose that potential passengers arrive at the stand according to a Poisson process with rate $λ$. An arriving person who sees no cab at the stand goes elsewhere and is lost for the particular car service. Empty cabs pass the stand according to a Poisson process with rate $μ$. An empty cab stops only at the stand when there is no other cab.
(a) Let $X(t)$ be the number of passengers waiting at the stand time $t$. Determine times {$S_{n}$}$_{n}$ such that ${X(t)}_{t}$ is a (delayed) regenerative process.
(b) Determine the fraction of passengers that are being transported from the stand (by using appropriately picked renewal times).
(c) Determine the fraction of busses that doesn't stop at the stand (by using appropriately picked renewal times).
(d) Compare the answers from (b) en (c), give an explanation.
My answer to (a) is to define $S_{n}$ as the $n_{th}$ moment that $X(t) = 7$ (not completely sure if this is correct).
However I got stuck at (b). Intuitively the fraction of passengers that are being transported should be the same as 1 - the fraction of passengers that arrive at the stand while there is no bus waiting. Now since the poisson proces is also a renewal proces, the amount of passengers that arrive within a certain time frame is $\lambda$, while the amount of busses that arrive in that timeframe is $\mu$. I have no idea how to continue however..