A New Hope: Why doesn't the Death Star scramble more TIE fighters?

In A New Hope, the Rebels assault the Death Star with X-Wing fighters launched from a moon around the planet Yavin. However, the Death Star only sends out a few squadrons of of TIE Fighters to counter them. It did send out Darth Vader himself, along with a few elite pilots, but given that the Death Star is a massive space station, shouldn't it be able to send out TIE's by the hundreds (if not the thousands) and completely wipe out the opposition before any of them got a chance to line up a shot on that exhaust vent?