Heres a scenario: You have a room 25x25x25 (xyz, not that it matters much) and you'd like to put spawners in it to make a fast spawn farm. You wouldn't just fill it to the brim with spawners because the spawners in the middle would have nowhere to spawn their mobs.
I know, the best way to arrange spawners vertically is to just stack them on top of each other in pillars, but here I'd like to focus on horizontally. Minecraft spawners start off spawning 4 mobs on 4 randomly chosen spots in the radius of the spawner, but if the space randomly chosen is taken that mob will not spawn, and if all 4 mobs have nowhere to spawn it will try every tick to spawn just one mob anywhere it can around itself until it's successful. The wiki says that it's also a higher chance for mobs to spawn closer to the spawner, but the spawn chance percentages per ring around the spawner (x,z) are not given. So if the inside rings are "worth" more than the outside rings the inside rings should have less spawners in them. How close can two spawners get before the space the second spawner occupies is worth more than its spawn rate?
Taking all of this into account, and not taking into account resources or player radius to the spawners or too many mobs in the chunk, what is the optimal placement of spawners for the best use of space and fastest spawn rates?
Update: Keep posting answers, there's yet to be a mathematically based answer so keep thinkin' boys.