Why a busy machine queues
Capacity divided by hours tells you how busy a machine is on average. It does not tell you how long work waits in front of it, and the two do not move together: the wait grows much faster than the load.
The standard single-queue result — one machine, work arriving unevenly (Poisson), processing times
that vary as an exponential does, the M/M/1 queue — gives the mean number of jobs
waiting as
jobs waiting = busy² / (1 − busy)
| how busy the machine is | jobs waiting in front of it (M/M/1) | with a perfectly steady process time (M/D/1) |
|---|---|---|
| 70% | 1.63 | 0.82 |
| 75% | 2.25 | 1.13 |
| 80% | 3.20 | 1.60 |
| 85% | 4.82 | 2.41 |
| 90% | 8.10 | 4.05 |
Going from 70% to 90% busy is 29% more work through the same machine; the queue in front of it goes from 1.63 to 8.10 jobs, five times as long. Check the last row by hand: 0.9 squared is 0.81, divided by 0.1 is 8.1.
A machine whose process time does not vary at all queues exactly half as much (the M/D/1 column,
busy² / (2 (1 − busy))), and the curve keeps its shape.
What the formula cannot do
A real plant is not one queue. It is dozens feeding each other, with breakdowns, changeovers and shared crews; some of that steepens the curve and some flattens it, and no closed form says which way yours goes. The engine itself is checked against both formulas, among its other checks against exact mathematics, in Validation.