Sample frameDay 220·Operations · Queues

Good morning. Push a step from eighty per cent loaded to ninety-five and the queue in front of it grows almost fivefold. That's arithmetic, not effort.

Operations.

Utilisation and Variability

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The question

Take your most loaded person or team — the one absorbing something like ninety per cent of everything they can handle. There's a queue sitting in front of them right now. If you handed back a fifth of their capacity, how much of that queue would clear? Far more than a fifth. The relationship isn't linear, and no amount of effort from them is going to make it so.

The idea

Day 42 made the case that busy is not the same as productive — that work piling up in front of a bottleneck flatters everyone's utilisation and delivers nothing out the far end. What it didn't give you was the arithmetic. In 1961 the British mathematician John Kingman published a short paper in the Proceedings of the Cambridge Philosophical Society on what happens to a single queue as it gets busy. Wallace Hopp and Mark Spearman, both then professors of industrial engineering at Northwestern, restated it for operations managers in Factory Physics in 1996 as the VUT equation: waiting time is variability multiplied by utilisation multiplied by the time the work itself takes. The utilisation term is u divided by one minus u, and it behaves like nothing in a manager's intuition. At fifty per cent busy it sits at 1. At eighty per cent it's 4. At ninety it's 9, and at ninety-five it's 19. Moving from ninety to ninety-five doubles the wait and then some, without one thing about the work changing. And because the three terms multiply, a system with genuinely zero variability would queue not at all even when completely full — queues aren't built by load, they're built by load meeting irregularity. The trap is reading a resource at eighty per cent as a fifth wasted. That last fifth isn't slack. It's what stops the queue.

The framework

01
Measure utilisation where the queue is, not across the team.
Utilisation only means anything at a single resource. A team averaging seventy per cent tells you nothing if one person inside it is running at ninety-eight and the rest are at fifty, because the queue you're staring at belongs entirely to that one person. So pick the step where work visibly waits and work the fraction out honestly: hours of work arriving in a period, divided by hours genuinely available in that period. Available means after meetings, admin, leave and the other job they've also been given. The real figure is almost always higher than the one being quoted.
02
Variability is the cheaper half of the equation.
There are two ways to shorten a queue and only one of them costs money. You can lower utilisation, which means buying capacity or turning work away. Or you can lower variability, which is usually close to free and almost always ignored: work arriving in Monday heaps rather than a steady trickle, briefs that reach the step incomplete and bounce back, jobs that take an hour one week and a day the next because nobody standardised the input. Because the equation multiplies, a cut in variability pays hardest exactly when you're busiest.
03
The eighty-five per cent line.
Nothing special happens at eighty-five per cent — the curve is perfectly smooth — but it's a useful line to manage against, because it marks roughly where the arithmetic stops being forgiving. Below it, another point of load costs a little more waiting. Above it, each point costs visibly more than the one before, and the last few cost enormously. This is why hospitals treat full bed occupancy as a warning rather than an achievement, why call centres are deliberately staffed with slack, and why a firm that sells every available hour of its best people delivers late for the rest of the year.
04
Who actually pays for the queue?
Price the queue on Day 90's worksheet — the wait converted to a weekly number, the four invoices, the sequencing — because none of that method changes when the thing waiting is work rather than a decision. What Day 90 doesn't ask is who carries the cost, and that's the question that explains why nothing gets fixed. High utilisation is visible, praised, and sits on somebody's report as an achievement. The queue it creates is paid somewhere else entirely: by a customer waiting, by a salesperson re-explaining, by you on a Sunday. The person who benefits from the ninety-five per cent never sees the bill.

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