Week 12: There Is No Neutral Queue
Mean, tail, fairness or revenue — a queue can only optimise one. The closing lecture, on the system where society had to choose out loud.
24 May 2027
Almost every queue you have met this semester was optimising something without saying so.
This one had to say so, in public, and argue about it for two decades.
A queue with no throughput to find
There are more people needing a kidney than there are kidneys.
No amount of process improvement fixes that. It is not a queueing problem in the sense of the first nine weeks, because there is no efficiency left to extract — every organ gets used. The only question is order, and here order decides who lives.
Every rule you have learned has been seriously proposed for it. So the unit below runs all five, on the same list, and you can watch each one pick.
Run your five rules on it
First come, first served — allocate by time on the list. Transparent, ungameable, requires no judgement about whose life is worth more. It also wastes organs: a kidney given to a 70-year-old yields far fewer life-years than the same kidney given to a 25-year-old, and under pure FCFS that difference is invisible by design.
Shortest job first becomes allocation by expected benefit — most life-years per organ. Maximises the good done. It also systematically deprioritises the old, the diabetic and the already-sick, which is to say it allocates least to those who are worst off.
Triage — sickest first. Matches the medical instinct and week 7’s consent. It is also the least efficient rule available, because the sickest patients have the worst outcomes: it maximises organs used and minimises life-years produced.
Price. Illegal almost everywhere. Iran is the notable exception and the literature is worth reading rather than dismissing. Notice that the objection here is not efficiency — a market would clear — but that we have decided this is not a thing that should be allocated by ability to pay at all.
So which one did the United States pick?
None of them, and that is the answer.
After two decades of public argument it runs a weighted composite that pleases nobody entirely, gets revised repeatedly, and publishes its weights.
That is not a failure to find the right answer. It is what choosing looks like when you are forced to do it in the open, with everybody watching, and cannot retreat into the pretence that the current rule is just how things are done.
The fifth setting below is that composite: four points of waiting time against four of expected benefit against two of urgency. It is second best at everything, and that is not a compromise — it is the design.
- life-years an organ
- —
- W on the list
- —
- slowest tenth waited over
- —
- organs to the over-60s
- —
- died on the list
- —
The four objectives, and why you cannot have them all
The readout above is the matrix below, live. Life-years an organ is efficiency; W is the mean; the slowest tenth is the tail; organs reaching the over-sixties is one concrete reading of fairness.
Read across the five settings and the shape is unmistakable. Sorting by benefit produces nearly a third more life-years than arrival order — 12.6 against 9.7 — and cuts the share of organs reaching the over-sixties from 43% to 24%.
Then look at who is in the room. Under arrival order the over-sixties are 42% of the list. Under most-life-years they are 83% of it — because they are the people it never picks, so they stay, and they keep getting older while they do. The rule does not merely under-serve them. It fills the unit with them and then declines them faster.
Sickest-first has the most medical consent of any rule here and is the least efficient on offer, at 9.0, with a slowest tenth of 12.0 years against arrival order’s 4.8. That is week 7’s starved bottom category exactly, except that here the bottom category is whoever is coping best — which is why, if you leave the scene running on that setting, the pink mark lands on somebody in their thirties who has been on the list for decades.
| Rule, and who pays for it | Mean wait | Worst case | Fairness | Revenue |
|---|---|---|---|---|
| First come, first servedIgnores both urgency and job length | poor at mean wait | mixed on worst case | good at fairness | poor at revenue |
| Shortest job firstStarves the longest job indefinitely | good at mean wait | poor at worst case | poor at fairness | poor at revenue |
| Triage by needOnly as good as a 90-second judgement | mixed on mean wait | good at worst case | mixed on fairness | poor at revenue |
| Pay to skipSorts by ability to pay, and rewards congestion | mixed on mean wait | poor at worst case | poor at fairness | good at revenue |
| AppointmentsMoves the wait somewhere nobody measures | good at mean wait | mixed on worst case | mixed on fairness | mixed on revenue |
Week 6 showed mean and worst-case pulling against each other. Week 8 showed revenue pulling against both. Week 7 showed that fairness is not even one thing.
There is no setting that maximises all four. Which means every queue has already made this trade — including, especially, the ones that have never discussed it.
One last trap, and one last conservation law
Two things in that readout are worth stopping on, because both are traps you were taught to spot.
The wait is measured over everybody who left the list, not everybody who got an organ. Measure only the transplanted and sorting by benefit appears to shorten the queue: 0.9 years, against the 2.2 the readout reports. It does, for the people it picks. The people it never picks sit there until they die and leave the statistic without ever having entered it. That is week 9’s boundary, drawn one last time around the people a rule excludes.
Under arrival order the same two figures are 2.9 and 2.7 — no gap worth mentioning, because that rule is not selecting on anything. So the gap between those two numbers is not noise. It is the sorting, measured.
And the last number barely moves. Thirty-five to thirty-eight per cent leave without an organ under every single rule.
Five rules, five different orderings, and the same share die. Why?
Because ordering is conservative, and this is week 8’s theorem with the stakes changed.
More people join than there are organs. Nobody is served faster, no organ goes unused, and the surplus has to leave the list some other way — so the share who never get one is fixed by arithmetic before any rule is chosen. Reordering can only decide which people are in it.
The only control in the room that changes how many people die is the supply.
Drag it from three to six and the share who die goes from about half to about one in nine, under whichever rule is selected. Nothing else in the scene comes close.Which is the honest end of the course. Every ordering rule is a way of deciding who bears a shortage, and none of them is a way of ending one. If your design brief can increase capacity, that dominates every clever rule in here.
What the real system actually does
The scene above runs five pure rules because pure rules are what a course can teach. No real allocation system runs one, and the US kidney system is worth describing in outline because it is the most argued-over queue on earth and every fix in it is a response to a failure this course has already covered.
It is a points system. Candidates accumulate points, and the components are recognisable:
Waiting time — week 5, and the fix applied to it is the most instructive detail in the whole system. The clock used to start when you joined the list, which quietly rewarded whoever had a doctor who referred them early. It now runs from the start of dialysis, which is a fact about the patient rather than about their access to a referral. That single change is week 5’s lecture in practice: arrival order is an allocation that already happened somewhere else.
Sensitisation — a measure of how hard a candidate is to match. Points rise steeply for the hardest-to-match, because otherwise they never get an offer at all. This is an ageing rule, in exactly week 6’s sense, applied to a starvation problem the pure rules produce.
Paediatric status, and priority for prior living donors: explicit ethical commitments, written into the arithmetic, and defensible as such.
Longevity matching — the most contested part, introduced in 2014. Roughly the best fifth of kidneys are offered first to roughly the fifth of candidates with the best expected post-transplant survival. That is week 6’s rule, limited on purpose to a fraction of the supply rather than applied to all of it.
And in 2021 the geography changed: fixed regional boundaries were replaced with distance-based circles around the donor hospital, because a boundary is a discontinuity and people on the wrong side of one were waiting years longer for no defensible reason.
Every component in that list is a correction to a failure you have already met in this course.
Which is what a designed queue looks like after thirty years of public argument: not one rule, but five, each limited so that no single objective can run away with the supply.Notice the design move that makes it work, because it is the one you should steal. Longevity matching is not applied to every organ. It is applied to a capped fraction of them. Bounding how much of the supply any one objective may claim is how a system gets most of the efficiency without any of the starvation — and it is available to you in almost any design, at the cost of being explicit about the fraction.
Other places society had to say it out loud
Organ allocation is unusual only in how long the argument has been running. Once you can see the shape, you can see the same public reckoning in several places within living memory.
Vaccine prioritisation, 2020–21. Every country published an ordering, in advance, in a document anybody could read: by age, by clinical risk, by occupation. The arguments were ferocious and they were arguments about objectives — lives saved against years saved against keeping hospitals staffed — which are precisely the columns of the matrix above. It is the largest triage exercise ever conducted in public.
ICU protocols in a pandemic. Several health systems wrote down, in advance, how ventilators would be allocated if there were not enough. Publishing it was agonising and correct: a rule written in advance can be argued with, and a rule improvised at the bedside cannot.
School places. Catchment, sibling priority, lottery, or a mixture. A lottery is chosen specifically because it cannot be gamed by moving house — it trades away a fairness claim to buy immunity from a form of gaming, which is week 11’s trade with children attached.
Social housing. A points system, published, with the weights visible. The same structure as the transplant list and the same objections.
What those four have in common is not that they got the answer right. It is that they were made to publish the rule — and publishing turns out to be the intervention, because a published rule can be objected to by the people it sorts.
Set that against everything else in this course. The café’s layout, the supermarket’s express lane, the call centre’s hold message, the airline’s boarding zones: every one of those is an allocation rule affecting thousands of people a day, and not one of them is written down anywhere the people in the queue can read it.
The thesis, finally
There is no neutral queue. A system that has not chosen has not escaped the choice.
It has inherited one, usually from whoever laid out the room, and it is enforcing that choice on everybody who walks in without the courtesy of having said what it is.This is why the course spent four weeks on measurement before touching a single ordering rule. You cannot argue about what a queue should optimise until you can say what it currently does, in numbers, and name who is paying.
That is the whole method: observe, model, experiment, cost, decide — and then say out loud what you decided. The board on the wall of that unit is doing the only genuinely admirable thing available to it: it has written the rule down where the people it sorts can read it.
What you can do now that you could not in February
You can walk into a building, watch a line for ninety seconds, and state the rule it is running.
You can measure it from outside with a stopwatch. You can put a number on what it costs and say who is holding the bill. You can propose a different rule and predict, quantitatively, who wins and who loses. And you can tell the difference between a queue that is badly designed and a queue that is designed against you.
Your brief is due Friday. The final page asks you to state what your queue optimises for.
Everything since week 1 has been preparation for being able to finish that sentence honestly.
Teaching team
- Adaeze Nkemelu — convenor
- Priya Raghunathan — tutor