What's that more successful person doing right now?
Chances are, they are going to go to bed long after you've reached the happy land of dreams.
Although site after site encourages you to rest and take care of yourself--usually advocating tons of sleep to make sure your brain isn't overworked--training yourself to fight exhaustion is a very powerful statement. The beauty of staying up late is that you have uninterrupted time to focus on yourself--and your work. If you choose, over and over again, to fight against what your body naturally wants to do, then that shows a great combination of self-discipline and control.
Staying up later will encourage you to get more done and to complete every assignment that you meant to finish from the day before. Holding ourselves accountable for deadlines and due dates to which we have already committed long ago is just half the battle. Showing up to finish the work on time is the rest of the struggle.
In a self-paced world, it makes sense that one's self-control and willpower would be greatly tested. There's nobody else around to push you--so if you can't keep up, it's nobody's fault but your own.
Conditioning our bodies to stay awake even if they're a little tired, training ourselves that--in some instances--it really is mind over matter, teaches us how to overcome small hurdles on our path to productivity.
Discovering over time that it is more important to push through and finish your work than to give in, give up, and fall asleep is a small part of the entire learning process that has ideals which can be applied to all aspects of life.
If you're bothered by something very small, don't let it get to you. Find a way to shake off the negative emotion, reaching around annoyance or irritation to maintain a positive attitude throughout the interaction. And, if you think you're too tired to continue, show yourself that often, this kind of discouraging attitude is all in the mind. Being present and deciding to conquer the beast shows that it's always, to some extent, mind over matter.
So start putting your mind over matter, and put your productivity first.