Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Erlang loss sizing function for hospital beds with refused admissions

c(lambda, L_bar, N) = (a, P_block, R, o_ach)

Maps a ward's daily arrival rate, mean length of stay and number of staffed beds to the offered load, the share of arriving patients who find every bed full and are refused, the refusals a year and the occupancy the beds then achieve, and uses these to find the beds needed for a refusal target or at the lowest total cost of beds and refusals. The first estimate of beds, expected bed-days divided by the bed-days one bed supplies at a target occupancy, is the live throughput identity HE-FM-ALOS-002 solved for the beds with admissions in place of discharges, and the average number of occupied beds is Little's law, HE-FM-DES-003; neither is drafted again. The loss probability is not the Poisson overflow of HE-CF-ARRP-001, which gives the chance that more patients arrive in one hour than the service can absorb, a count of arrivals rather than of occupied beds, and not the single-server wait of HE-FM-ARRP-005, in which patients queue. HE-FM-CAPP-001 writes the loss model term by term for a six-bed unit, and HE-CF-CAPP-001 runs it for any number of beds by the Erlang B recursion and finds the beds for a refusal target and the number beyond which one more bed costs more than the refusals it avoids. Notation follows the Capacity Planning article.

  • Erlang loss refusal probability, annual refusals and achieved occupancy of a six-bed unit

    a = lambda * L_bar; P_block = (a^6 / 720) / (1 + a + a^2 / 2 + a^3 / 6 + a^4 / 24 + a^5 / 120 + a^6 / 720); R = 365 * lambda * P_block; o_ach = a * (1 - P_block) / 6

    Gives, for a unit of six staffed beds with Poisson arrivals, the share of arriving patients who find all six beds occupied and are refused, the Erlang loss formula of de Bruin and colleagues written out term by term, with the offered load a equal to arrivals per day times the mean stay. Refusals a year are the annual arrivals times that share, and the achieved occupancy is the load actually carried, a times one minus the refusal probability, over the six beds, de Bruin and colleagues' occupancy rate for the loss model, which takes the form of HE-FM-ALOS-002 solved for o with the patients admitted, not the patients arriving, as the discharges. Achieved occupancy is always below a / 6 because refused patients use no bed-days. The model needs only the mean stay, not the shape of the length of stay distribution. Other numbers of beds follow the same pattern with terms up to a^N / N!, which HE-CF-CAPP-001 computes by recursion.