There is one result from queuing theory that changes how people look at a production line, and it fits on a single line. For a simple single-server queue, average waiting time is proportional to utilisation divided by one minus utilisation. As utilisation approaches one, waiting time goes to infinity — not gradually, but violently.

The number that surprises everyone

Compare two machines with identical processing rates. One runs at 80 per cent utilisation, the other at 95 per cent. The ratio of their average queue lengths is:

(0.95 / 0.05) / (0.80 / 0.20) = 19 / 4 = 4.75

Nearly five times the work-in-progress for a 15-point utilisation gain. Add variability — which real processes have in abundance — and the multiplier grows further. This is why "run every machine flat out" is a policy that manufactures WIP rather than product.

The practical form: Kingman's approximation

Real processes are not Markovian, so the textbook M/M/1 formula overstates the queue. Kingman's equation is the version worth memorising, because it separates the three levers you actually control:

Wq  ~=  ( (Ca^2 + Cs^2) / 2 )  x  ( rho / (1 - rho) )  x  te

where
  Ca  = coefficient of variation of arrivals
  Cs  = coefficient of variation of service time
  rho = utilisation
  te  = effective mean process time

The structure tells you the whole story. Waiting time is variability times utilisation pressure times process time. You can attack any of the three, and they multiply, so a 30 per cent reduction in variability and a 5-point drop in utilisation compound into something much larger than either alone.

Where each lever comes from

  • Ca is set by how work arrives. Batch transfers from an upstream process are the classic cause of a high Ca; moving from a batch of 50 to a batch of 10 often halves it.
  • Cs is set by process consistency: operator skill spread, machine breakdowns, changeovers. Breakdowns inflate Cs far more than people expect, because they turn a tight distribution into a bimodal one.
  • rho is set by capacity planning. This is the lever with the direct cost, and the one management prefers to push to the limit.

Sizing a buffer

Once you can estimate Wq, buffer size follows from Little's Law: L = lambda × W. If a station processes 120 pieces per hour and average wait is 4 minutes, expected queue is 120 × (4/60) = 8 pieces. Size the physical buffer for roughly three times the average to absorb normal fluctuation — around 24 pieces — and put a hard stop there. A buffer with no ceiling is not a buffer; it is a place where WIP goes to hide.

Where to spend buffer space

Buffers pay for themselves only in front of the constraint and immediately after high-variability operations. Everywhere else they add lead time and hide problems. The test is simple: if the station downstream of the buffer is never starved without it, the buffer is not earning anything.

The management conversation

The hardest part is not the arithmetic. It is explaining why deliberately planning for 85 per cent utilisation is a better business decision than planning for 98 and firefighting the difference. The Kingman form helps, because it is honest about the trade: you are buying short, predictable lead times with capacity. If lead time is what you sell, that is a good trade. If you sell on price and your product does not perish, it may not be.