Two clocks, and only the slower one counts
Work out the hands-on time first. Eight stalls a side at 0.8 minutes of routine per cow is 6.4 minutes of work in a row, and with one person that is 6.4 minutes of elapsed time. Now the machine: a unit is on for 5.5 minutes and it takes another minute to let the row out and get the next one in, so a stall is tied up for 6.5 minutes. In a double parlour one side is being worked while the other is milking, so that 6.5 minutes is spread over two sides and only obliges the row interval to be 3.25 minutes.
Six point four against three point two five. The person is slower, so the row interval is 6.4 minutes and the parlour does eight cows every 6.4 minutes — 75 cows an hour.
When the crew is the limit, stalls do nothing at all
This is the result worth having and it is exact rather than approximate. If the hands-on work is the binding constraint, the row interval is stalls times routine divided by people, and throughput is stalls times sixty divided by that. The stall count appears on the top and the bottom and cancels. What is left is sixty times the number of people, divided by the routine time. Nothing else.
On the defaults that is 60 divided by 0.8, which is 75 cows an hour — and the page prints 75 for eight stalls a side, 75 for ten, 75 for fourteen. Four extra stalls, four extra units, four more of everything, and not one more cow through the door in an hour. The number that does move it is people: a second person takes it to 147.7 cows an hour on the same eight stalls, at which point the machine becomes the limit and stalls start mattering again.
Where the changeover sits
The page prints the stall count at which the two clocks are equal. On the defaults it is 4.06 stalls a side. Below four, the crew finishes a row and stands waiting for units to come off, and every stall added is worth a straight increase in throughput. Above four, the units are waiting on the crew and stalls are inert.
Almost every parlour anyone actually builds sits well above that number, which is why the crew-limited case is the normal one and why the arithmetic that makes stalls irrelevant is the arithmetic that usually applies. It also means the routine time — the number everybody estimates and nobody times — is the single input that decides the answer.
The session
A hundred and twenty cows in eight-cow rows is 15 rows. Fifteen at 6.4 minutes is 96 minutes, the last row still needs its 6.5 minutes to finish, and setup and wash-up add 25, so the session runs 2.13 hours. Twice a day that is 4.25 hours, and across a year, with one person, 1,551 person-hours. That is most of a full-time job, before any other work on the place, and it is the number that makes routine time worth a stopwatch: shaving 0.1 minutes off the routine at the defaults drops the year to 1,405 person-hours, which is 146 hours saved for a change nobody in the parlour would feel.
What this model is not
It is a batch model. A row goes in, gets worked, comes out together. That is a herringbone or a parallel parlour and it is not a rotary, not a robot, and not a parlour where cows leave individually as they finish. It also treats every cow as identical, which no herd is — in a real row the slowest cow holds everybody, and the effect of that is worse in a batch parlour than an average suggests. Take the figures as a comparison between options rather than as a prediction of the clock on the wall.
Questions people ask
How many cows an hour can a parlour milk?
It is set by whichever is slower: the hands-on work in a row, or the time a stall is occupied divided by the number of sides. At the defaults — eight stalls a side, double, 0.8 minutes of routine, 5.5 minutes of unit-on and a minute to turn the row — the work is 6.4 minutes and the machine floor is 3.25, so the row interval is 6.4 and the parlour does 75 cows an hour. Both time figures are yours to measure.
Will adding stalls make my parlour faster?
Only if the machine is the limit. When the crew is the limit, throughput is sixty times the number of people divided by the routine time, and the stall count cancels out of the arithmetic entirely — the page prints 75 cows an hour at eight stalls a side, ten, and fourteen. The page also prints the stall count where it changes hands, which on the defaults is 4.06 a side.
How do I know whether people or the machine is my bottleneck?
Compare stalls times routine time divided by people, against unit-on plus turn time divided by the number of sides. Whichever is larger is the constraint, and the page says which one it is. On the defaults it is 6.4 minutes of work against a 3.25 minute machine floor, so the crew is holding it and the units spend time waiting.
How long does a milking session take?
At the defaults, 120 cows in rows of eight is 15 rows at 6.4 minutes, plus 6.5 minutes for the last row to finish and 25 minutes of setup and wash-up: 2.13 hours. Twice a day, every day, that is 1,551 person-hours a year with one person on it.
Does this work for a rotary or a robot?
No. It is a batch model — a row goes in together, gets worked, and comes out together, which is a herringbone or a parallel parlour. A rotary is a continuous flow and a robot is a queue with no operator in the loop, and neither is described by the arithmetic here.