"My" water change analysis is confusing when briefly stated until one thinks about it.
and old thread here explains in a hopefully light hearted humorous way:
https://forums.saltwaterfish.com/index.php?threads/beaslbobs-water-change-analysis.369985/#post-3183747
My analysis is simplified to where does the tank wind up as opposed to all the intermediate values.
To Beth's point sure if nothing is increasing or decreasing due to the biological actions in the system, then the tank winds up at whatever the replacement water had. 30ppm nitrate or 400ppm calcium water means the tank will wind up at 30ppm nitrates and 400ppm calcium.
If the tank starts out at say 0 nitrates and 400ppm calcium and nitrates are increasing at 1ppm/day and calcium is decreasing a 1 ppm/day and the replacement water has 0 nitrates and 400ppm calcium, then where does the tank wind up when the values just after a water change are always the same?
In order for that to happen the total changes between water changes must be removed by the water change. So those changes will build up until the water change completely removes the total change.
If you change 1/10 of the water the tank will build up to 10 times the changes. So the 1/10 water change completely removes the buildup.
1/20 the tank builds up to 20 times and so on.
So for a 1/10 water change every 10 days, the tank will build up to 10 times the changes. The total change is 10*1ppm=10ppm. The tank builds up to 10ppm*10=100ppm. So nitrates will be 100ppm and calcium will be 400ppm-100ppm=300ppm. the nitrates removed by the water change is 100ppm/10=10ppm. Then after the water changes nitrates will be 100ppm-10=90ppm. The calcium added is 400ppm/10=+40ppm. the calcium removed is -300ppm/10=-30ppm/ So the calcium after the water change is 300ppm+40ppm-30ppm=310ppm. The just before the next water change the values are 90+10ppm=100ppm nitrates and 310-10=300ppm calcium.
Sure that's a lot of math hence my simplification that the tank winds up to a pint where the water changes completely remove the changes.
By contrast if you consume all nitrates all the time with algae or bacteria nitrates will be unmeasureable regardless of the water change schedule.
And if you dose calcium exactly as it is being consumed, calcium will be 400ppm regardless of the water change schedule.
Hence my point what is important is to balance out and stabilize the system so the nitrates and calcium are not changing. At that point the frequency and size of water changes are irrelevant.
But all that's it probably too long winded also.
Bottom line is creating an environment which does not change is more effective than water changes convenient to hobbiests.
my .02