Based on Raffensperger, John F., Marygail K. Brauner, and
R. J. Briggs, *Planning Hospital Needs for Ventilators and Respiratory Therapistsin the COVID-19 Crisis*. Santa Monica, CA: RAND Corporation, 2020.

In this post, I explain how to
implement our method using a spreadsheet.

Medical administrators face the problem
of allocating resources such as ventilators among different facilities when demands
vary extremely. Administrators may be able to forecast average demands at each facility
by day or week, but must still manage the variability of demand over minutes and
hours. Different average demands among smaller and larger facilities complicates
the allocations further. Queuing processes are non-linear, so proportional allocation
methods (such as allocating based on average ventilators per patient) can result
in inefficient and unfair outcomes.

To efficiently and fairly ration resources
between facilities, we propose application of a classical queuing model, enabling
administrators to define fairness as equal waiting times across facilities. Administrators
can easily run the model in a spreadsheet or other computer system. Using the queuing
model, the administrator allocates resources to equalize the wait time among facilities.
Due to the non-linearity of waiting time as a function of the number of resources,
equalizing wait times across facilities also minimizes total waiting time.

This method of equalizing wait times
across facilities is not meant for use in prioritizing patients within a facility,
nor for rationing resources between patient types, e.g., by age or socio-economic
status. Our method assumes that the population of patients at one facility are clinically
similar to the population of patients at other facilities.

With clinically similar patient
groups, we can use a classical queuing model to allocate resources. The Erlang C
formula was developed by Agner Krarup Erlang in the early 1920s to determine the
number of telephone circuits required in a telephone exchange. The formula is appropriate
for multi-server queues (where servers are the resources such as ventilators within
a hospital), and where arrivals and service times follow Poisson distributions.

λ is the arrival rate of patients, e.g.,
arriving patients per hour at a hospital.

μ is the treatment time (length of stay)
on a single server, e.g., average hours that a patient is on a ventilator.

*N* is the number of servers,
e.g., ventilators at a hospital.

*Poisson*(*N*,
ρ) is the probability of exactly *N* events occurring in a fixed interval of
time when the average occurrence rate is ρ, such as λ*μ in this case.

The probability of wait is as follows:

Prob(wait > 0) = *Poisson*(*N*,
λ*μ)/[*Poisson*(*N, *λ*μ)* + *(1* – *λ*μ/*N*))∑_{k}_{=0}^{N}^{–1} *Poisson*(*k*,
λ*μ)]

As the probability of wait approaches
100%, the average queue gets longer. If the probability of wait reaches 100% or
greater, patient load exceeds hospital capacity, and the queue length will explode.
When the formula exceeds 100%, the reader may not find the value to have
intuitive meaning, so we can restrict the calculation to a maximum of 100%:

Prob(wait > 0) = MIN(1,
POISSON.DIST (*N*, λ*μ, FALSE) /

**(**POISSON.DIST
(*N*, λ*μ, FALSE) + (1 – λ***μ*/N*))*POISSON.DIST (*N* – 1,
λ*μ,TRUE)**))**

To build the full spreadsheet, proceed
as follows:

1. Write a spreadsheet
row for each facility. Each row has the facility’s current number of resources (e.g.,
ventilators), the average arrival rate of patients per day, and the average service
time per patient (e.g., days on ventilator).

2. In each row,
add a cell with the expected waiting time for each patient, using the Erlang C formula
we give below.

3. Given the
number of new resources to allocate across facilities, give the next available resource
to the facility with the longest patient waiting time. Of course, with a new resource
such as a ventilator, the facility may also need more resources of other types,
e.g., respiratory therapists. The spreadsheet can calculate these other
resources as well, if appropriate data (e.g., ventilators per therapist) is
available.

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