Week 9: Don't Queue At All
Appointments, reservations and virtual queues do not remove the wait. They move it somewhere you cannot see — which is sometimes exactly right.
3 May 2027
A restaurant with no reservations has a queue on the pavement.
The same restaurant with a booking system has no queue at all.
Where did it go?
It has not become faster. It seats the same number of people in the same evening with the same kitchen.
What changed is that the waiting now happens at home, four days earlier, distributed across everybody who wanted a table — including the people who never got one, and who therefore appear in no statistic the restaurant keeps.
The wait did not disappear. It moved somewhere nobody measures.
- W at the door
- —
- W from wanting a table
- —
- turned away a night
- —
- seats filled
- —
- W at the door, walk-in
- —
Every system in this lecture is performing that trick.
Three ways to move a wait
The book above is doing something specific, and it is week 2’s mechanism: it is not creating capacity, it is deleting arrival variability. A walk-in room receives its guests when they feel like arriving. A booked room receives them on a schedule the kitchen wrote. That is the whole of the gain in seats filled, and it is why the same trick works on a drive-through.
Appointments convert waiting-in-line into waiting-in-calendar. The gain is real: you spend the interval living your life rather than standing in a corridor. Two costs. They are brittle under variance — one overrunning consultation cascades through the whole afternoon, which is why your 2:30 appointment is at 3:10 — and they demand commitment in advance, which serves worst exactly those people whose lives are least predictable.
Virtual queues keep the ordering and delete the standing. Take a ticket, go away, come back when called. Almost strictly better than a physical line — except that it silently requires a phone, a number, somewhere to wait, and the literacy to work the system. The queue has been made invisible, and so have the people who cannot join it.
Reservations with deposits are week 8’s mechanism wearing a different coat: allocation by willingness to commit money in advance. Every argument from last week applies unchanged.
Notice that all three keep the same ordering rule — arrival order — and change only where you stand while it is applied. That is why this week sits at the end of Act II rather than being a sixth rule. It is not a new answer to who goes first. It is a new answer to where the queue is, and it turns out that question does at least as much distributive work as the ordering did.
The mathematics of an appointment book
An appointment system looks like it has abolished queueing theory. It has done the opposite: it has moved the whole problem into a diary, where the same equations apply and nobody expects them.
Here is the tension, and it is unavoidable. A clinic has two costs it can trade: patients waiting in the room, and the doctor sitting idle between patients. Book appointments far apart and the doctor idles. Book them close and the patients queue. There is no schedule that eliminates both, because consultation lengths vary — and the size of the unavoidable trade is set by exactly the coefficient of variation from week 2.
This has been studied since the 1950s, and the classic result — the Bailey–Welch rule — is delightfully concrete: book two patients at the start of the session, then one per mean consultation length thereafter. The double booking at the top exists solely so that the doctor is never idle at the beginning, which is the one moment when idleness is guaranteed rather than probabilistic.
Three consequences follow, and all three are in your own experience.
Delay accumulates monotonically through a session. Any consultation that overruns pushes every subsequent appointment back, and nothing pushes them forward again except a cancellation. So the 9am appointment runs on time and the 4:30 does not, always, in every clinic in the world. Your appointment time is not a promise about when you will be seen; it is a promise about your position in a sequence.
No-shows are the reason for overbooking. Clinic no-show rates commonly run in the low tens of per cent, and an empty slot is capacity destroyed for ever. So clinics overbook, exactly as airlines do — and when the no-shows fail to materialise, the overbooking arrives as a queue in the waiting room. The people who pay for it are the ones who turned up.
The slack you removed was doing a job. A perfectly packed diary is a system at ρ = 1, and week 2’s table says what that means. Real schedulers protect themselves with catch-up slots, which look like waste on a utilisation report and are the only reason the afternoon ever recovers.
A clinic is running 40 minutes late every afternoon. What is the cheapest fix?
Two empty slots in the middle of the session.
Not faster consultations — the mean is not the problem, the variance is. Not more doctors, which is the expensive answer to a scheduling question. A deliberate gap partway through lets the accumulated delay drain, and it resets the second half of the afternoon to zero.
It costs one appointment’s worth of capacity and it is nearly always refused, because a gap in the diary is visible to management and a forty-minute delay is visible only to patients.
Which is week 4, in a diary: the cost that is inside the organisation gets optimised, and the cost that is outside it gets produced.
The virtual queue, and who cannot join one
The purest form of moving a wait is to keep the line and delete the standing.
Restaurant pagers. Take-a-ticket systems. Theme-park virtual queues, where you join a ride’s line from a bench half a mile away. Online waiting rooms — the software that sits in front of a ticket sale or a vaccine booking page and issues you a position rather than an error — which are now a standard piece of web infrastructure precisely because the alternative is a system that falls over.
These are close to a free win. The ordering is preserved, the fairness claim is identical to week 5’s, and the hours are returned to the people who own them. This is the single best intervention in the course by ratio of benefit to cost, and if your design brief has a physical line in it that could be a virtual one, say so.
Now the part that is always left out.
A virtual queue quietly changes the entry requirement. To join it you need a phone, charge, data, a number that receives messages, somewhere acceptable to wait, and enough confidence with the interface to trust that you have actually joined. Each of those is nearly universal and none of them is universal, and the people who fail one of them are not a random sample — they are, with depressing reliability, the same people who were disproportionately in week 4’s waiting room.
A physical queue excludes nobody who can stand in it. A virtual queue has an eligibility criterion that nobody wrote down.
The people who cannot join it are not visible in it, which is a different and worse thing than being at the back of it.The design response is not to abandon virtual queues. It is to keep a staffed physical path to the same service, and to measure what fraction of people use it — because that fraction is the size of the group your clever system was about to strand.
The trap this sets for measurement
A clinic moves from walk-in to booked appointments. Its published average wait falls from 90 minutes to 8. Did anything improve?
For some people, genuinely yes. For the published number, no — it is measuring a different thing and did not say so.
The clock now starts when you arrive for an appointment you were given four weeks ago. The four weeks are outside the definition, so they are not in the statistic.
Nobody lied. The figure measures exactly what the service defined it to measure. That is what makes this so hard to argue with, and so common.
The scene above draws exactly this, and the pair of bars is deliberately the same device as week 4’s bill: solid for the figure that reaches the account, a dashed empty outline for the one that does not. In week 4 the invisible bar was twice the visible one. Here it is a hundred and sixty times, and the bar the restaurant publishes is a sliver you have to be told is there.
A health service reporting an eight-minute average while people wait five weeks for the appointment is not committing fraud. It has drawn a boundary, and the boundary is the whole story.
Week 3 gave you the tool for catching this: apply L = λW at the boundary you actually care about, rather than the one the organisation chose to draw. That is your lab this week, on a real published statistic of your choosing.
When the waiting list becomes the policy
Once the wait has moved into a list, the list becomes a thing that can be managed — and that is where the most instructive failures in this entire course live.
Health systems set access targets: a maximum time from referral to treatment, published, with consequences attached. The intention is unimpeachable. The effect on the distribution of waits is well documented and is not what anyone intended.
A system sets an 18-week maximum and publishes the share of patients treated inside it. What shape does the distribution of waits take?
A pile-up just inside the threshold, and a sharp cliff just outside it.
The target creates a rule, the rule is optimised against, and the optimisation is entirely rational at every level. A manager with a patient at week 17 and a patient at week 20 has a strong reason to treat the one at 17, because the one at 20 has already breached and cannot be un-breached. The patient who has waited longest becomes the patient it is least valuable to treat.
Which is a priority queue, sorted by time remaining before a reporting boundary, that nobody designed and nobody would defend if asked.
Notice that there is no villain and no dishonesty. The target did exactly what a target does: it turned a continuous quantity into a threshold, and everything downstream reorganised around the threshold rather than the quantity.
The general lesson is worth writing in the margin of your design brief. A published metric becomes an ordering rule, whether or not you wanted one. If you can only report one number about a queue, you have effectively told the people running it which patients, tickets or customers to prioritise — and if the number is a threshold, you have told them to prioritise the ones nearest the threshold.
The defence, and it is only a partial one: report the whole distribution, or at minimum report the tail as well as the rate. A system publishing “the share seen within 18 weeks” and “the longest anyone waited” is much harder to game in the direction above, because the second number gets worse precisely when the first is being protected.
The honest case for doing it anyway
Do not read this as debunking. Moving a wait is often the best available move, and the reason is precise:
It can make the system genuinely better. Switch the scene above to bookings and watch the last two numbers: turned away falls from four a night to none, and the share of seats filled rises from 83% to 99%. Nobody is cooking faster. The book took a Saturday’s overflow and put it on a Tuesday.
And waiting is not uniformly costly. An hour at home is worth far more to you than an hour in a corridor. A system that converts the second into the first has created real value even though the clock says nothing changed — and week 4’s arithmetic, which priced all hours the same, cannot see that either.
The failure is not using appointments. It is using appointments and then believing your own waiting-time statistic.
What Act II gave you
Five ordering rules, and a sixth move that is not an ordering rule at all.
| Rule | Buys | Charges |
|---|---|---|
| Arrival order | Auditability, no information needed | Ignores length and need; tests free time |
| Job length | The best possible mean | Starves the longest job |
| Need | Lives, and near-universal consent | Everything rests on a 90-second classifier |
| Price | Revenue instead of burned time | Sorts by ability to pay; rewards congestion |
| Appointment | Variability, and the hours back | Brittle; demands commitment in advance |
Each one buys something. Each one charges somebody. You can now say which, for any of them, and that is the whole of Act II.
Act III is about what happens when the system you applied them to turns out to be connected to other systems — and full of people who have worked out what you are doing.
Teaching team
- Priya Raghunathan — tutor