Cubicles are a shared, finite resource, and held by patients waiting for an inpatient bed. Set the size of the department, arrivals, and how long admitted patients wait for a bed. Every length of stay below — including that of the patients who are not being admitted at all — will follow these
The department supplies cubicle-hours: one cubicle, open for one hour. Each patient consumes cubicle-hours equal to their length of stay, so the average number present is the arrival rate multiplied by that length of stay. The bar below sets the day's demand against the day's supply.
An extra attendance and an extra hour of waiting for a bed both consume cubicle-hours. This converts one into the other at your current settings.
Each curve varies one driver by the percentage on the horizontal axis, holding the others at your current settings. The horizontal rule marks the point at which patients present equals cubicles available.
A patient waiting for an inpatient bed is not waiting in a corridor of their own. They are holding a cubicle, and that cubicle is the same one a low-acuity patient needs in order to be seen at all. Because admitted patients are triaged first, low-acuity patients take what is left over — so the bed wait sets their length of stay too. Past a certain point it stops setting their length of stay and starts deciding whether they are treated.
Decanting a boarder into a corridor is the only move here that protects cubicle flow without treating fewer people, which is why departments reach for it. It works: the figures above are as flat as they are largely because of it. What it does not do is make the wait shorter. The patient still waits the same hours for the same bed, but does so on a trolley in a thoroughfare, without a call bell, oxygen point, or privacy. Ambulance holding does not even buy that — it frees no cubicle, it only moves the queue outside and takes a crew off the road.
Corridor care is counted here, not scored. The mortality estimate below applies the Jones coefficients to total time from arrival regardless of where the patient spends it, because that is what the study measured. There is no published number-needed-to-harm for corridor care specifically, so attaching a multiplier to it would be inventing a figure. The counts above should be read as a measure of how much care is being delivered in a space never designed for it.
Total time in department for admitted patients is modelled as log-normal around the median implied by your settings. The number-needed-to-harm coefficients from Jones et al. (2022) are applied to that distribution, following the method used by the Royal College of Emergency Medicine on national data.
These are associations from observational data adjusted for case mix, not causal effects. Jones et al. estimated no coefficient beyond 12 hours; the 8–12 hour value is carried forward here, which assumes no further increase in risk after that point. The output is an order of magnitude rather than a count.
A deterministic fluid queue in 15-minute steps over a repeating day, iterated until the day-to-day state stops changing. Arrivals follow a conventional UK ED profile (trough at 05:00, plateau from late morning). Cubicles are a shared, finite resource: an admitted patient holds one for work-up + bed wait, a low-acuity patient for their treatment time. Neither can start without a cubicle, so length of stay is an output of the model rather than an input.
When a cubicle frees, patients needing admission are placed ahead of low-acuity patients, and allocation within each stream is first-come-first-served. Priority is non-preemptive: nobody already in a cubicle is moved out. This is why a rising bed wait lengthens the low-acuity stay before it lengthens the admitted one.
A low-acuity patient who has waited longer than the patience setting leaves untreated. Reneging is deterministic — every patient waits exactly the same length of time before giving up — so the modelled count is a threshold effect rather than the smooth curve real data show. Because the reported wait is the mean among those actually seen, it can never exceed the patience setting: that figure is a ceiling on the waits this page can display, not only a driver of how many people leave.
Contact time and cubicle time are separated, because they are not the same quantity and only the second competes with boarders. Fifteen minutes of assessment may sit inside a visit that occupies the space for three hours while bloods, imaging and review are awaited. Set the second input to zero for a department where minors return to the waiting room between steps; the resulting waits fall sharply, which is itself the argument for having somewhere else for them to sit.
A single repeating day makes a department either cope every day or fail every day, with a sharp threshold between. Real departments cross that line on bad days well before their average crosses it. The model therefore repeats a fixed fortnight of busier and quieter days with the same mean attendance, drawn at the quantiles of a normal distribution and ordered so that busy days sometimes fall together. The sequence is fixed, so the model remains deterministic and reproducible, but the response to a lengthening bed wait becomes a gradient rather than a cliff. Setting the variation to zero recovers the single-day behaviour.
Patients needing admission never give up waiting, so if work-up plus the boarding the corridor cannot absorb exceeds what the cubicles can supply, the backlog grows every day and never settles. The page reports that as a system-wide response being required, together with the rate at which the backlog is growing and the maximum admission rate the department could sustain, min(24 × cubicles / work-up, 24 × (cubicles + corridor) / (work-up + bed wait)). Figures that depend on a stable queue — median time in department, mortality — are shown as rising rather than given a value, because any number would depend on how long you had been watching.
Occupancy is every patient physically present divided by the number of cubicles, the way departments normally quote it. It therefore exceeds 100% whenever anyone is in a corridor, an ambulance or the waiting room, and it is not capped. Cubicle occupancy on its own can never exceed 100% because the model will not put two patients in one cubicle.
The model iterates whole days until the state of the cubicles and the corridor stops changing, detecting repeating cycles up to eight days long. Service times that do not divide into 24 hours produce much longer cycles — a 34-hour total stay against a 24-hour day repeats only every 17 days — so where no cycle is found the reported figures are averaged over 40 days rather than taken from a single day, which would otherwise depend on which phase of the cycle you happened to land on.
No stochastic arrivals, no staffing constraint, no majors/minors split, no ambulance offload model, and no corridor care — a patient is either in a cubicle or waiting. The model is deterministic throughout. Real departments have more relief valves than this one does, so the modelled cliff edge is sharper than the observed one; the direction of every omission is to understate how gradually a department degrades, not to overstate the harm.
Weiss et al. (2004), computed at the peak census, with ventilated patients fixed at one. Waiting-room time is derived from the overflow queue by Little's Law; longest admitted wait is the 95th centile of the modelled distribution. Bands: 101–140 overcrowded, 141–180 severe, 181+ dangerous.
Jones S, Moulton C, Swift S, et al. Association between delays to patient admission from the emergency department and all-cause 30-day mortality. Emerg Med J 2022;39:168–173. One additional death per 191 patients at 4–6 hours, 82 at 6–8 hours, 72 at 8–12 hours, measured from arrival.
Asplin BR, Magid DJ, Rhodes KV, et al. A conceptual model of emergency department crowding. Ann Emerg Med 2003;42:173–180. Input, throughput and output enter this model as terms in a single product; each can be varied independently above.