1 The puzzle — same size, different health
Imagine two specialties, each with 2,000 patients waiting and 100 referrals arriving a week. In the first, bookings roughly follow arrival order: nobody is ancient, and 70% of the list is under 18 weeks. In the second, a few hundred long-waiters have quietly accumulated at the back: only 45% is under 18 weeks. Size alone cannot tell these lists apart. A model that worked from size and referrals only would treat them identically — and be wrong about both. k is how this model tells them apart using only published data.
2 The textbook shape — and k as the "how far from textbook" dial
Queueing theory has a natural default: if the next patient is picked effectively at random (a realistic stand-in for clinically-led mixed booking), the ages of people on the list follow the exponential curve — lots of recent joiners, a long thin tail. On that textbook shape, the size of the list and the referral rate fully determine the %<18wk. So the model asks each specialty one question: "given your size and referrals, is your published %<18wk what the textbook predicts?" The answer is k:
- k = 1 — the list behaves exactly as its size suggests (textbook shape).
- k > 1 — the list is younger than its size suggests: disciplined, FIFO-like booking keeps everyone moving (or the list was recently validated). It performs like a textbook list only 1/k as large.
- k < 1 — the list has a fatter long-wait tail than its size suggests: long-waiters are accumulating at the back.
Formally, for the 18-week window: k = −ln(1 − p) × L ÷ (18 × λ),
where p is the published %<18wk, L the list size and λ the effective referral rate. A worked
example with real numbers — the trust's largest list, straight from the seed:
3 Try it — booking discipline creates k
Drag the discipline slider from random selection towards strict first-in-first-out and watch the age mix change shape. FIFO-like booking clears the tail, so more of the list sits under the 18-week line at the same list size — that surplus over textbook is exactly k > 1. (The census-vs-flow explainer uses the same slider and calls the underlying distribution parameter k; on this page we keep k for the model's shape factor — the two are related but not the same number.)
Age mix of the waiting list (share of the list at each waiting age)
The implied-k card runs the same shapeFactor() function
the model calibrates with, fed by this demo queue's size, %<18wk and referral rate. Note the
last card: the sustainable list at 92% scales linearly with k — a list with
disciplined booking can be k times larger and still meet the standard, because meeting a census
standard is about the age mix, not the headcount.
4 The real NUH values (computed live from the seed)
Each specialty's k, calibrated from its published April 2026 list size, %<18wk and referral rate. Green bars (k ≥ 1) read younger than textbook; red bars (k < 1) carry a fatter tail.
5 What the model does with k — and what to watch
- Calibration: k is measured once per specialty at the model start and anchors each trajectory to the published %<18wk — so the model never claims a specialty is healthier or sicker than its published position.
- Held constant by default: booking discipline is assumed to stay as it is to 2030. That is a stated assumption, not a fact — the booking-discipline drift lever on the RTT page morphs k linearly by 2029 to test it, and the sensitivity tornado shows what ±20% of drift does to the headline (because the sustainable list scales linearly with k, drift towards FIFO shrinks the clearance ask and drift towards tail-neglect grows it).
- k conflates discipline with data quality. A very high k can mean genuinely disciplined booking — or a recently validated list. NUH's highest-k specialty sits just after the Feb–Mar 2026 EPR migration purge removed thousands of stale pathways, so treat extreme values with respect for both readings.
- One number, one distribution. k summarises a whole age distribution in a single ratio at the 18-week point. Two lists can share a k and differ elsewhere in the tail; per-patient PTL data would replace this approximation entirely.