Like any other bright light source (such as the Sun), the light coming from the monitor breaks down some of the melatonin that your body naturally produces. You may still become tired when using a computer and subsequently fall asleep, but the quality of that sleep will be adversely affected unless you account for reduced melatonin levels somehow (either via a program like f.lux, or with supplemental pills).
Melatonin is prescription only in the UK. So, go careful if you're travelling from US to UK with off the shelf sleep meds.
I read some article that investigated melatonin supplements in the US. It said that because supplements aren't as heavily regulated you couldn't be certain which pills had melatonin or how much.
