Mob spawners spawn up to 4 mobs once every 10-40 seconds and they only count mobs of the type within a 9x9x9 volume (there must be less than 6 mobs, and less than 2 to ensure that up to 4 more spawn. This also means that spawnproofing the area is not needed and two or more spawners next to each other spawning different mobs would not interfere, other than existing mobs blocking attempts; the best way to maximize the spawn rate is to ensure that mobs are cleared out within 10 seconds):
Is there any way I can increase spawn rates for this farm as is? It's on a realm, and I've mob-proofed a large area around the farm and inside it from pigmen, etc.
From the picture, that looks like a version of MumboJumbo's piston funnel design (a design noted as being more 'cool' than intended for maximum efficiency).
Moving the 'funnel' down so that newly spawned blaze sink directly out of the 9-block cube checked by the spawner would maximize rates for that design (assuming no blazed aggro-ed in the kill box were able to rise back into the spawn check cube), but the need to move back and forth between the spawn enabling station and the kill station places a limit on efficiency for piston funnel designs. (This may not be a consideration if you've shortened the funnel to the kill station remains within 16 euclidean of the spawner.)
[Mumbo seems to have built his without remembering that blaze only drop rods for player kills. ]
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WARNING: I have an extemely "grindy" playstyle; YMMV — if this doesn't seem fun to you, mine what you can from it & bin the rest.
Blazes fall really slowly when they are not aggro. If you want them to exit more quickly, you will need to take control of their pathfinding AI to make them run to a spot of your choosing. Gnembon has 2 Fun Farms videos about blaze farms that goes into detail about mob pathfinding AI (the second video is a refinement that makes the blaze farm design redstoneless).