Alienware X51 R2 screen "freezing"

So here is the rundown of the issue I'm currently having:

- System boots without issue

- Between 1 to approx 10 minutes after windows started, the screen might suddenly "freeze" in place, so the pc becomes unusable.

IMPORTANT:

* If the issue doesn't happen shortly after starting, it won't happen later on. I went on using the pc for hours and nothing happened

* The pc itself hasn't crashed. If you keep it going, it will even enter sleep mode.

* Performance monitor does NOT report a crash. Just an incorrect Windows shutdown (since I am forced to shut it)

* The issue happens regardless of what the pc is running. It can even happen without doing anything, so it's not related to heating up during games.

Additional data:

- I'm running Windows 7 64-bit

- BIOS is updated to A12

- I'm using a 330W AC Adapter (default one) This one

- I performed both a RAM and Hard Disk checks, and got no errors

- I just performed a clean installation and the issue still presents itself

- The Dell driver check says everything is installed

- I'm using AVG as anti-virus system

Additional question:

- I could update to win10. Could that help?