The arithmetic of being busy
There is a piece of mathematics most leadership teams have never seen, and it explains more about delivery speed than any process change they are considering.
Kingman's formula, published in 1961, approximates average waiting time in a queue. It has three components: utilisation, variability, and service time. The utilisation term is the one that matters here, and it takes the form of utilisation divided by one minus utilisation.
Run the numbers:
At 50% utilisation, that term equals 1
At 80%, it equals 4
At 90%, it equals 9
At 95%, it equals 19
At 98%, it equals 49
A team operating at 95% capacity waits roughly five times as long as the same team at 80%. Same people, same skills, same tools, same work. The only thing that changed is how full the system is.
This is not a study or a heuristic. It is a mathematical property of any system with variability in when work arrives and how long it takes — which is every system.
Past roughly 80 to 85% utilisation, queue times rise steeply rather than proportionally. A common operational rule of thumb is to hold around 25% more capacity than average demand. Most organisations do the reverse: they treat spare capacity as waste and push utilisation toward 100%, then wonder why everything takes so long.
Which is why adding people rarely helps
The instinct when delivery slows is to add capacity. It rarely works, for two reasons that compound.
Capacity added away from the constraint does nothing. A system's throughput is set by its single binding limit. Adding designers when the bottleneck is engineering review produces more designs waiting for review. The work in progress rises, the queue lengthens, and throughput does not move.
Capacity added to a loaded system arrives slowly. New people consume the time of existing people to become productive. In an already saturated system, that borrowing comes directly out of the constraint's capacity, so delivery gets worse before it gets better — which is the mechanism behind the observation that adding people to a late project makes it later.
The productive question is not how to add capacity. It is which single point is limiting throughput, and what would move it.
The five focusing steps
Eliyahu Goldratt set out the method in The Goal in 1984. Five steps, run as a loop.
1. Identify the constraint. The one resource, process, or decision point that limits throughput. There is usually exactly one that matters.
2. Exploit the constraint. Get more out of it without spending anything. Remove work from it that something else could do. Stop it idling. Protect it from interruption. Make sure everything reaching it is defect-free, so no constraint time is spent on rework.
3. Subordinate everything else. The rest of the system runs at the pace the constraint can absorb, deliberately. This is the step organisations refuse, because it means telling productive people to slow down. Running non-constraints at full speed produces inventory, not output.
4. Elevate the constraint. Now add capacity — hire, buy, automate, outsource. This step costs money, which is why it comes fourth rather than first.
5. Repeat. Once the constraint moves, it is somewhere else. Finding the new one is permanent work.
The order is the framework. Most organisations jump to step four, having skipped the two free steps that frequently make it unnecessary.
The evidence
Kingman's formula is mathematics. It was published in 1961 and is a proven approximation for waiting time in a general single-server queue, known to be particularly accurate near saturation. It is not a management theory and does not require validation studies.
The applied methodology is empirically supported, with a caveat. Mabin and Balderstone's meta-analysis in the International Journal of Operations and Production Management examined more than 80 documented applications across manufacturing, distribution, and services. Average lead times fell around 69%, cycle times around 66%, inventory around 49%, and financial performance improved by over 60%.
The caveat is one the authors raise themselves: despite extensive searching, they found no reports of failure. A literature containing only successes indicates publication bias. Read the magnitudes as what good implementations achieved rather than as expected returns.
Little's Law completes the picture. Work in progress equals throughput multiplied by lead time. Rearranged: if throughput is fixed by the constraint, adding work in progress only extends lead time. Starting more work does not finish more work, and this is arithmetic rather than opinion.
Finding the constraint
Four practical signals, none requiring instrumentation.
Where work piles up. The constraint has a queue in front of it and everything downstream of it is periodically idle. Look for the stage where things wait longest, not the stage where people look busiest.
Where expediting happens. Whatever gets escalated, chased, or personally intervened on is usually the constraint. Expediting is the organisation routing around a limit it has not named.
Who is in every meeting. In service and creative businesses the constraint is often a person — the one approver, the one technical lead, the one founder. If a name appears in every critical path, that is the constraint.
Where decisions wait. Decision latency is invisible in most tracking systems and is frequently the largest queue in the business. Work sitting idle awaiting a decision is indistinguishable, in effect, from a machine standing idle.
Constraints that are not people or machines
For most businesses reading this, the binding constraint is not a production line.
Approval and decision points. One person who must sign off, available two days a week.
A single specialised skill. The only person who can do a particular category of work, in every project simultaneously.
Context switching. When everyone works on four things, the effective constraint is attention, and the cure is limiting work in progress rather than adding people.
The market. When the business can produce more than it can sell, the constraint has moved outside operations entirely — and the entire focus should shift to demand rather than delivery. This is a real and frequently missed diagnosis.
Where the framework is misapplied
Elevating before exploiting. Buying capacity when the constraint is idle 30% of the time because work reaches it in the wrong order.
Optimising everything. Efficiency programmes applied across all functions equally. Improvement anywhere except the constraint produces no throughput gain and often produces more inventory.
Treating utilisation as the goal. High utilisation of non-constraints is not a virtue. It is the mechanism that creates the queues.
Refusing to subordinate. The step organisations skip most, because deliberately idling capable people feels wasteful. It is the step that converts an identified constraint into actual throughput.
Assuming the constraint is permanent. It moves as soon as it is elevated. A business running the same improvement programme for two years is probably optimising something that stopped being the limit long ago.
Diagnostic: does the business know its constraint?
Six tests.
The current constraint can be named in one sentence, and two leaders name the same one.
Someone owns moving it, with a date.
The constraint's idle time is known, and it is close to zero for reasons other than overload.
Work in progress is limited somewhere, deliberately.
Target utilisation for non-constraint resources is below 100% on purpose.
Capacity has been added at the constraint rather than around it, or not added at all.
Failing test five means the organisation is fighting queueing mathematics with willpower.
What this produces
Throughput that responds to intervention.
That is the practical value. Once the constraint is named, improvement becomes targeted rather than general, and the effect is visible. Before it is named, effort spreads evenly across a system and produces work in progress that feels like progress and is not.
The uncomfortable corollary is that most efficiency work in most organisations produces nothing measurable, because it was applied where there was spare capacity. Finding the constraint is what makes the next investment worth making — including the ones the budget has already committed to.
Frequently asked questions
Why does adding people often slow delivery down?
Two effects compound. Capacity added away from the constraint produces work in progress rather than throughput, and new people consume time from existing people — which in a saturated system comes out of the constraint's own capacity.
What utilisation should a team run at?
Below saturation, deliberately. Queue times rise steeply past roughly 80 to 85% utilisation because of the mathematics of queueing, and a common operational guideline is to hold around 25% more capacity than average demand.
How do I identify my constraint?
Look for where work waits longest rather than where people look busiest, where expediting happens, whose name appears on every critical path, and where decisions sit unmade. The constraint has a queue in front of it and idle capacity behind it.
Does Theory of Constraints work outside manufacturing?
Yes. The meta-analysis covered distribution and services alongside manufacturing, and the underlying mathematics applies to any system with variability. In service and creative businesses the constraint is usually a person, a decision point, or attention itself.
Is the evidence for Theory of Constraints reliable?
The applied methodology has a supportive meta-analysis of over 80 cases, with the authors noting they found no reported failures — an indication of publication bias. The underlying queueing mathematics is proven and does not depend on that literature.
What if we can produce more than we can sell?
Then the constraint is the market, not operations, and improving delivery will produce nothing. This is a common and frequently missed diagnosis, and it moves the entire focus to demand.
How often does the constraint move?
Every time it is successfully elevated. Repeating the cycle is the fifth step for that reason. An improvement programme running unchanged for years is likely addressing something that stopped being the limit.
Sources
Kingman, J.F.C., The Single Server Queue in Heavy Traffic (1961) — the VUT approximation for mean waiting time
Goldratt, E., The Goal (1984); the five focusing steps
Mabin, V.J. and Balderstone, S.J., The Performance of the Theory of Constraints Methodology, International Journal of Operations and Production Management (2003)
Little's Law, standard queueing theory
Structure your next phase
Zerologic works with leadership teams to identify what is actually limiting throughput — before capacity gets added to the parts of the system that already have spare.
Talk to us: partners@zerologic.io · zerologic.io
