  Amid unprecedented times caused by COVID-19, healthcare systems all over the
world are strained to the limits of, or even beyond, capacity. A similar event
is experienced by some healthcare systems regularly, due to for instance
seasonal spikes in the number of patients. We model this as a queueing system
in heavy traffic (where the arrival rate is approaching the service rate from
below) or in overload (where the arrival rate exceeds the service rate). In
both cases we assume that customers (patients) may have different priorities
and we consider two popular service disciplines: static priorities and
accumulating priorities. It has been shown that the latter allows for patients
of all classes to be seen in a timely manner as long as the system is stable.
We demonstrate however that if accumulating priorities are used in the heavy
traffic or overload regime, then all patients, including those with the highest
priority, will experience very long waiting times. If on the other hand static
priorities are applied, then one can ensure that the highest-priority patients
will be seen in a timely manner even in overloaded systems.
