Little Sums

My local supermarket is pretty good at managing crowding. They seem to staff the various areas according to predicted demand, while holding some extra staff in reserve tasks to cover surges in essential activity. They also know exit block at the tills is usually the major bottleneck, and understand the need for escalation actions (like opening more tills) when impeded flow is causing crowding in the aisles. Crowding quickly impairs the core business of the supermarket, and shoppers become frustrated and leave, so the duty manager must respond effectively. The formula that the retail industry uses to understand crowding in their stores is Little’s Law: Average number customers in store = average rate arrival X average length of stay.

In the much more complex environment of A&E, control of length of stay is even more essential, as the consequences are not lost business revenue or someone running out of milk, but unnecessary deaths. In an Emergency Department setting, Little’s Law can be adapted to: average number patients = rate of arrival X length of stay. Some patients may be suitable for a waiting room, redirection elsewhere or “see and treat” (the equivalent of the supermarket 12 items or less queue), and can be excluded from the space calculations. However, the “majors” need not only a clinical space for assessment and treatment to occur with any dignity, they also require timely and potentially life-saving care. The rate of arrival of such patients is surprisingly predictable for most departments, with a steady peak between 11am and 11pm. Therefore:

Number staffed A&E trolley cubicles needed = average peak majors arrival rate x average length of stay.

(Example: If there are 31 suitable staffed spaces and average peak arrival rate of 12 patients per hour. The average length of stay must be 2.58 hours or queues for spaces will develop and persist until the rate of arrival subsides. If there are 61 staffed spaces and the same average peak arrival rate then the average length of stay can be 5.08 hours.)

What dictates the balance between number of cubicles and length of stay? Most discharged A&E “major” patients need a minimum time of 2 hours (for triage, assessment, investigation, review, discharge). 4 hours is thought to be a reasonable time frame for the majority of the admitted patients to be worked up by A&E, referral to in-patient units made and patient transferred out. Clearly there are important clinical exceptions, but the concept of a maximum 4-hour target length of stay sets an overall reasonable mean length of stay for majors of around 3 hours. If A&E departments are properly configured in terms of trolley space by Little’s Law and staffed for their expected workload to meet these timeframes, then the transfer out of A&E to in-patient beds remains the only significant variable affecting length of stay (equivalent to queues at the supermarket tills). This figure is closely related to hospital occupancy as it relies on the ready availability of the in-patient beds.

In a full hospital, patients awaiting admission cannot be moved out of the ED. Once average length of ED stay then exceeds the calculated required average for the number of staffed majors cubicles, there are no longer safe and appropriate spaces for new patients and A&E becomes overcrowded.  In an overcrowded,A&E there are multiple well known adverse consequences. Violence to staff, increased complaints and critical incidents, missed diagnoses, missed treatments, complications of long trolley stays (bedsores, infections, DVT, falls), all resulting in 30% increased mortality in admitted patients and 70% in discharged patients.

Despite all the evidence of overall positive effect of time targets from around the world, there are now voices in the UK health sector calling for the abandonment of the A&E 4-hour time target. The emphasis it places on rapid treatment of acutely ill and injured patients requiring admission is no longer being portrayed as an essential safety requirement, but as an encouragement to over utilise the allegedly overpriced and inefficient free acute care supermarket.  The future, we are told, is in developing health delivery alternatives analogous to small local stores and home delivery services. However it is very unlikely that emergency departments will be short of business as a result of any changes in that direction, and without ED crowd control time targets, every day in them will feel like Black Friday.