Motorway problem: how to minimize the number of encounters? I was working on the motorway problem: At time t=0, cars are launched from the same entrance of highway following a Poisson process with parameter $\lambda$, and the speed of cars ($v$) follows a known distribution of $S$. The observer is on one of the cars with a constant speed of $v_0$, and the amount of observation time is $T$. Since the speed of each car might be different, observer's car can catch up with some cars ahead of it, or be caught up by some cars from behind. The question is: how to choose $v_0$ such that the expected number of encounters as seen from the observation car is minimized, given only 3 parameters ($\lambda$, $v_0$, observation time $T$) and 2 distributions (Poisson distribution of launch time, and the speed distribution for cars $S$)?

EDIT: check Sheldon M. Ross, introduction to probability model, edition 9, example 5.19, or this link.