Suppose buses arrive at the bus stop according Poisson process with rate $\lambda$. You get on bus if it is not full. The probability that a bus is not full is $p$ and is independent of arrival time. What is the expected time to get on bus?
The question is motivated by the recent question. I don't know what is asked in the question and its author disappeared, so I decided to ask my question. I know that $k$-th bus arrival time is a random variable with Erlang distribution, but I don't know does it lead to compact solution.
The expected time of $n$-th arrival in Poisson process with rate $\lambda$ is $$ E[T_n] = \frac{n}{\lambda}$$. Not full buses arrive according Poisson process with rate $\lambda p$. The question asks for expected time of the first arrival of not full bus, it is $$E=\frac{1}{\lambda p}$$