I created this problem while I was having my supper a few days back. So there maybe flaw in the formulation. Please point them out as you see one.
Suppose, there is a circular dining table with radius $R$ and $N$ persons are sitting around the table uniformly. There are $n$ pancakes at the middle of the table, distributed uniformly over a circular region of radius $r(r<R)$. Now, each of the $N$ persons need $m$ pancakes $(mN> n)$. As soon as they see the pancakes, they synchronously start to stretch their arm to snatch a pancake as soon as possible. It is assumed that a person can stretch its arm up to the center of the table from his/her side and each person can take a random velocity $v$ of stretching arm with $v\in (0,V]$. Each of them can take either $0$ or $1$ pancake at each attempt. Also it is assumed that no two person can take the same pancake (or pieces of it) simultaneously. Then they all go back to their position (synchronously) and again start doing this. This procedure is repeated until there is no pancake left at the center. Now there are several questions we can ask.
What is the expected amount of trials at the end of which no pancake is left at the center?
What is the expected number of persons who do not get $m$ pancakes?
Suggestions to model and solve this problem are welcome.