# To calculate the time needed for a vacuum robot to cover whole area

A vacuum robot runs on randomized direction when it is operating in a room.

I want to know how long it takes to fully cover the whole area (or at least most part of the area) with given information:

1. the area in the room is 10 square meters.
2. the dimension of the round shape robot is 40 centimeter.
3. the robot travels at a speed of 20 meters per minute.
4. the robot only travels straightly.
5. the robot forwards to a random direction when hit the wall.

Below the left is a vacuum robot. The right is the imagination of a working robot.

I'm not sure if it relates to routing problem or other Operations Research method. (neither sure if it is a problem that can be optimized by Operation Research but just allow me to put it here.)

Can you please advise how to solve this problem? Thank you

• "or at least most part of the area" needs to be defined properly. There might be an area on the plain (in the extreme case a point) that is never touched with a positive probability which implies that the expected time of covering the entire area is infinity. – r.beigi Jan 3 at 0:35
• I think this is a very interesting mathematics question. I suggest that you post this question to either math.stackexchange or mathoverflow and use the keywords expectation, expected-value and geometric-probability. This question or its variants may have already been solved. Otherwise, you will know if the question is relatively new or still open. – r.beigi Jan 3 at 1:34
• @Mark K, would you see this or this links. They would be helpful. – A.Omidi Jan 3 at 7:22
• @r.beigi, A.Omidi, thank you both for the comments! – Mark K Jan 4 at 0:56