GLSL fragment processing order for full screen quad

Question

I'm using GLSL for some image processing stuff, so drawing a full screen quad and doing processing in the fragment shader. I'm wondering if we can expect fragments to be processed in any particular priority order?

I know the fragments are being processed in parallel and we can't make any guarantees on the finish time for any particular fragment, so how is this handled? Is it just a big queue? And what would the pattern look like ie. scanline, blocks etc.

Will this be driver dependent? As you can tell I have several questions in this area, and I'm having trouble finding documentation on it.