Xeviat
Dungeon Mistress, she/her
The basic assumptions of encounter/day design spelled out in the DMG suggest 6-8 encounters a day, with 2 short rests a day. It even specifically spells out that if you have more than a 3rd of the recommended daily XP in challenges between short rests, you're pushing the difficulty of the encounters up.
If you take the monk's Ki and triple it, then compare it to the Spell Point progression in the DMG, you'll see that a Monk's Ki is very similar to a half-caster's (Paladin/Ranger) spell points:
[TABLE="width: 500"]
[TR]
[TD]Level[/TD]
[TD]Monk Ki x3[/TD]
[TD]Paladin SP[/TD]
[/TR]
[TR]
[TD]1[/TD]
[TD]-[/TD]
[TD]-[/TD]
[/TR]
[TR]
[TD]2[/TD]
[TD]6[/TD]
[TD]4[/TD]
[/TR]
[TR]
[TD]3[/TD]
[TD]9[/TD]
[TD]6[/TD]
[/TR]
[TR]
[TD]4[/TD]
[TD]12[/TD]
[TD]6[/TD]
[/TR]
[TR]
[TD]5[/TD]
[TD]15[/TD]
[TD]14[/TD]
[/TR]
[TR]
[TD]6[/TD]
[TD]18[/TD]
[TD]14[/TD]
[/TR]
[TR]
[TD]7[/TD]
[TD]21[/TD]
[TD]17[/TD]
[/TR]
[TR]
[TD]8[/TD]
[TD]24[/TD]
[TD]17[/TD]
[/TR]
[TR]
[TD]9[/TD]
[TD]27[/TD]
[TD]27[/TD]
[/TR]
[TR]
[TD]10[/TD]
[TD]30[/TD]
[TD]27[/TD]
[/TR]
[TR]
[TD]11[/TD]
[TD]33[/TD]
[TD]32[/TD]
[/TR]
[TR]
[TD]12[/TD]
[TD]36[/TD]
[TD]32[/TD]
[/TR]
[TR]
[TD]13[/TD]
[TD]39[/TD]
[TD]38[/TD]
[/TR]
[TR]
[TD]14[/TD]
[TD]42[/TD]
[TD]38[/TD]
[/TR]
[TR]
[TD]15[/TD]
[TD]45[/TD]
[TD]44[/TD]
[/TR]
[TR]
[TD]16[/TD]
[TD]48[/TD]
[TD]44[/TD]
[/TR]
[TR]
[TD]17[/TD]
[TD]51[/TD]
[TD]57[/TD]
[/TR]
[TR]
[TD]18[/TD]
[TD]54[/TD]
[TD]57[/TD]
[/TR]
[TR]
[TD]19[/TD]
[TD]57[/TD]
[TD]64[/TD]
[/TR]
[TR]
[TD]20[/TD]
[TD]60+[/TD]
[TD]64[/TD]
[/TR]
[/TABLE]
So, I consider a monk's ki to make them akin to a half-caster. At lower levels, 1 ki for +1 attack (flurry of blows over martial art's bonus action) is really comparable to the damage from a 2 sp smite (level 1) when misses are taken into account.
As for comparing the Monk to the Battle Master Fighter at low levels ... if you compare the Battle Master Fighter's potential damage across a day to a Champion or a Eldritch Knight, you may end up agreeing with me that the battle master has too many dice at low levels. I'd give them a progression 2/3rds of the monk's ki progression (since the Eldritch Knight is 1/3rd a Wizard, and the Monk is 1/2 a wizard, the Battle Master could be 2/3rds a Monk).
With 2 encounters per short rest, and an average of 3 rounds per encounter, a 6th level monk can easily start using 1 ki per round. Past that, they can easily start doubling up sometimes; stun and flurry some rounds, just stun and martial arts on other rounds.
If you take the monk's Ki and triple it, then compare it to the Spell Point progression in the DMG, you'll see that a Monk's Ki is very similar to a half-caster's (Paladin/Ranger) spell points:
[TABLE="width: 500"]
[TR]
[TD]Level[/TD]
[TD]Monk Ki x3[/TD]
[TD]Paladin SP[/TD]
[/TR]
[TR]
[TD]1[/TD]
[TD]-[/TD]
[TD]-[/TD]
[/TR]
[TR]
[TD]2[/TD]
[TD]6[/TD]
[TD]4[/TD]
[/TR]
[TR]
[TD]3[/TD]
[TD]9[/TD]
[TD]6[/TD]
[/TR]
[TR]
[TD]4[/TD]
[TD]12[/TD]
[TD]6[/TD]
[/TR]
[TR]
[TD]5[/TD]
[TD]15[/TD]
[TD]14[/TD]
[/TR]
[TR]
[TD]6[/TD]
[TD]18[/TD]
[TD]14[/TD]
[/TR]
[TR]
[TD]7[/TD]
[TD]21[/TD]
[TD]17[/TD]
[/TR]
[TR]
[TD]8[/TD]
[TD]24[/TD]
[TD]17[/TD]
[/TR]
[TR]
[TD]9[/TD]
[TD]27[/TD]
[TD]27[/TD]
[/TR]
[TR]
[TD]10[/TD]
[TD]30[/TD]
[TD]27[/TD]
[/TR]
[TR]
[TD]11[/TD]
[TD]33[/TD]
[TD]32[/TD]
[/TR]
[TR]
[TD]12[/TD]
[TD]36[/TD]
[TD]32[/TD]
[/TR]
[TR]
[TD]13[/TD]
[TD]39[/TD]
[TD]38[/TD]
[/TR]
[TR]
[TD]14[/TD]
[TD]42[/TD]
[TD]38[/TD]
[/TR]
[TR]
[TD]15[/TD]
[TD]45[/TD]
[TD]44[/TD]
[/TR]
[TR]
[TD]16[/TD]
[TD]48[/TD]
[TD]44[/TD]
[/TR]
[TR]
[TD]17[/TD]
[TD]51[/TD]
[TD]57[/TD]
[/TR]
[TR]
[TD]18[/TD]
[TD]54[/TD]
[TD]57[/TD]
[/TR]
[TR]
[TD]19[/TD]
[TD]57[/TD]
[TD]64[/TD]
[/TR]
[TR]
[TD]20[/TD]
[TD]60+[/TD]
[TD]64[/TD]
[/TR]
[/TABLE]
So, I consider a monk's ki to make them akin to a half-caster. At lower levels, 1 ki for +1 attack (flurry of blows over martial art's bonus action) is really comparable to the damage from a 2 sp smite (level 1) when misses are taken into account.
As for comparing the Monk to the Battle Master Fighter at low levels ... if you compare the Battle Master Fighter's potential damage across a day to a Champion or a Eldritch Knight, you may end up agreeing with me that the battle master has too many dice at low levels. I'd give them a progression 2/3rds of the monk's ki progression (since the Eldritch Knight is 1/3rd a Wizard, and the Monk is 1/2 a wizard, the Battle Master could be 2/3rds a Monk).
With 2 encounters per short rest, and an average of 3 rounds per encounter, a 6th level monk can easily start using 1 ki per round. Past that, they can easily start doubling up sometimes; stun and flurry some rounds, just stun and martial arts on other rounds.