A pedestrian wishes to go across a single-lane road where the cars arrive according to a Poisson process with the rate λ. The time needed for him to cross the road safely is denoted as k. He will have to wait until he sees a gap of at least k between the coming cars. If the gap between the car that just arrived and the next one is at least k, he begins to cross the road. Let T denote the random time he needs to wait by the road. What is the expected time the pedestrian needs to cross this particular road?
I've encountered several similar problems on here, but none of them gives a detailed, step-by-step explanation, so I am really confused. I would appreciate any help! Thank you in advance.