Utilization
The last ten percent of utilization costs more than the first eighty
A desk at eighty percent utilization takes five times its own handling time to answer. At ninety it takes ten. The step between them is one extra request an hour.
Somebody asks why the work is late and gets shown a utilization report. Everyone is at eighty five percent. Nobody is idle. There is no obvious slack anywhere, no station standing empty, and on the evidence of that report the only remaining explanation is that the team is not trying hard enough.
The report is correct and the conclusion drawn from it is wrong. High utilization is not the sign of a process working well. It is the sign of a process that has nowhere to put a surprise, and the arithmetic that connects the two is old, well established, and almost never shown to the person reading the report.
The desk
One approvals officer. One queue. Requests arrive at six an hour on average, at unpredictable times, and each one takes eight minutes of work on average, some quicker and some slower.
Eight minutes a request means the officer can clear seven and a half an hour if nothing ever waits. Six arrive. So the desk is busy eighty percent of the time and idle twenty percent of it, which is a figure most managers would be pleased with.
Time in the system = handling time ÷ (1 − utilization)
| λ | Requests arriving, on average | 6 an hour |
|---|---|---|
| μ | Requests the desk can clear, at 8 minutes each | 7.5 an hour |
| ρ | Utilization, arrivals divided by capacity | 0.80 |
| W | Average time from arriving to being answered | 40 min |
Illustrative figures, and both inputs are countable without instrumentation: an arrival log and a stopwatch on a dozen requests. The relationship holds for a single queue with one server, random arrivals and variable handling times. What changes when those assumptions do not hold is set out below, and it changes the size of the answer rather than its direction.
Forty minutes. Eight of those are work and thirty two are waiting, and the officer was idle a fifth of the day while it happened. Both facts are true at once, and that is the part people find hard to accept: there is spare capacity in the day and the queue exists anyway, because the spare capacity and the arrivals do not line up in time.
What one more request an hour does
Now the volume rises. Not by much. One more request an hour, which is a seventeen percent increase and the sort of thing that arrives with a new product line or one team’s reorganisation.
Time in the system = handling time ÷ (1 − utilization)
| λ | Requests arriving, up by one an hour | 7 an hour |
|---|---|---|
| μ | Capacity, unchanged | 7.5 an hour |
| ρ | Utilization, now | 0.93 |
| W | Average time from arriving to being answered | 120 min |
Nobody slowed down. The handling time is still eight minutes and the officer is still the same person. Seventeen percent more demand, three times the wait.
Seventeen percent more work, three times the delay. That is the whole piece, and it is why the report showing eighty five percent utilization across the board is a warning rather than a reassurance.
The curve, printed once
The relationship is a division by the idle fraction, so the closer utilization gets to one, the smaller the number being divided by, and the answer runs away. Here it is at the same eight minute handling time.
| Utilization | Multiplier | Time in the system |
|---|---|---|
| 50% | 2.0 | 16 min |
| 70% | 3.3 | 27 min |
| 80% | 5.0 | 40 min |
| 90% | 10.0 | 80 min |
| 95% | 20.0 | 160 min |
Read the last two rows against the first three. Getting from an empty desk to eighty percent utilization added thirty two minutes of delay. Getting from eighty to ninety added forty. The last ten points of utilization cost more than the first eighty, and there is no version of this curve where they do not, because the shape is set by the division and not by the industry.
This also settles a common argument about targets. A target of ninety five percent utilization is not an ambitious version of eighty. It is a different operating regime, in which a queue of twenty times the handling time is the designed outcome rather than a failure, and anyone setting that target should be asked whether they intended it.
The lever that does not look like one
Read the formula again with an eye on what is actually adjustable. The arrival rate is the customer’s. The handling time is the work. Utilization is the ratio of the two, so anything that moves either moves the answer, but the movement is not proportional and that is what gets missed.
Take one minute out of the handling time. Eight becomes seven, which is a twelve and a half percent improvement and the sort of thing a decent morning of standard work produces.
Time in the system = handling time ÷ (1 − utilization)
| λ | Requests arriving, unchanged | 6 an hour |
|---|---|---|
| μ | Capacity, at 7 minutes each | 8.6 an hour |
| ρ | Utilization, was 0.80 | 0.70 |
| W | Average time from arriving to being answered, was 40 min | 23 min |
One minute of work removed. Seventeen minutes of delay removed. The minute is worth seventeen because of where on the curve it was taken from, not because of what it was.
One minute of work bought seventeen minutes of turnaround. The same minute taken out of a desk running at fifty percent utilization would have bought about three. The improvement is identical and its value is not, which is why an improvement program that ranks its projects by minutes saved will systematically fund the wrong ones.
What this model does not know
The arithmetic above is the simplest queueing model there is, and it earns its place by being reproducible on the back of an envelope rather than by being complete. Four things it assumes, and what happens when they are not true.
It assumes handling times vary a lot. Specifically, it assumes them exponentially distributed, which is a strong assumption and rarely exactly right. If every request took exactly eight minutes with no variation at all, the waiting at eighty percent utilization would be sixteen minutes rather than thirty two. Exactly half. Real work sits between the two, and where it sits is worth measuring, because the spread of the handling time is a lever on the queue in its own right.
It assumes one server. Two officers sharing one queue at the same overall utilization wait considerably less than one officer at that utilization, because a free officer can take the next arrival whenever either one becomes free. Pooling is genuinely worth something. What pooling does not do is repeal the curve: two officers at ninety five percent are still in trouble, they are just in slightly less of it.
It assumes arrivals are unscheduled. An appointment book is a machine for removing the randomness from the arrival rate, and it is the main reason a clinic can run at a utilization a walk in service could not survive. If you can move arrivals, do that before you buy capacity.
It assumes nobody gives up. Once people start abandoning the queue and coming back tomorrow, arrivals stop being independent of the wait, the model no longer describes the system, and the reported demand quietly understates the real demand.
None of those change the direction. They change the size of the number, and they are the reason to measure your own process rather than to quote this one.
What to do with it
Three things, and none of them needs a project.
Compute your own utilization properly. Arrivals over a window, divided by what the process could do in that window if nothing waited. Not the percentage of the day somebody was doing something, which is a different quantity and is nearly always higher.
Decide where on the curve you intend to run, and say so out loud. Eighty percent is a choice to hold a queue of about four handling times. Ninety is a choice to hold ten. Both are legitimate and only one of them is usually intended.
Stop reading a high utilization figure as good news. It is the price of a short queue that some capacity goes unused, and a process with no unused capacity anywhere has bought its utilization figure with the customer’s time.
What the arithmetic here will not tell you is how much unfinished work is sitting in the process while all this waiting happens. That is a different relationship, it needs two numbers most teams already have, and it is Little’s Law.