In the below picture, imagine that the balls are freely moving like Brownian motion. Since the box is empty and there are sufficient balls outside the box, the box will be filled eventually. Therefore, we have the boundary conditions of
$$c = 0\ (0\%), t=0\\
c = 9\ (100\%), t=\infty
$$
The motion of the balls can be simulated by random walk. However, in the regular random walk, the probabilities for all directions are the same. Here, we know that the probability towards the right (into the box) is more.
How can we simulate the process?