Why does the flow of ink increased when pens are about to run out of ink?

When the pen is full of ink, there is very little space for air inside the pen, so while writing the ink gradually oozes out through the cut in the nib. This leads to a very weak vacuum inside the pen. So the outside air tries to enter the pen and holds the ink from flowing out freely. This ensures a smooth flow of ink.

As the ink runs out and when there are only a few drops of ink the air column inside the pen is no longer isolated from outside the air. Hence the outside air can enter the pen freely and there is no difference in air pressure. Thus the force acting on the ink will be only the gravitational pull and so the ink begins to drain out.