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 capital ships. 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 TIEs 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?