Modeling Yellow and Red Alert Durations for Ambulance Systems

Emergency systems are designed to almost always have enough capacity to respond to emergencies. However, capacity shortage periods do occur and these systems need to recover quickly. In this study, we apply queueing models and study whether it is better for an emergency system to add or to expedite servers, in order to quickly recover from a capacity shortage period. We focus on emergency medical service (EMS) systems and use Erlang loss models to study Red Alerts (when all ambulances are busy) and Yellow Alerts (when the number of available ambulances falls below a threshold). We analyze two loss models: one with Markovian state‐dependent service rates and one with generally and independently distributed service times. We validate the two models against EMS data sets from two cities. Despite the fact that the distribution of ambulance service times is a mixture of lognormal distributions, which is far from being exponential, we find that the loss model with Markovian state‐dependent service rates provides a better representation of empirical Yellow and Red alert statistics. We build on the model with state‐dependent rates and use the theory of absorbing Markov chains to quantify the impact of adding or expediting ambulances, with respect to two performance measures: (i) the duration of alert periods, and (ii) the number of lost calls. This quantification helps EMS staff (dispatchers and supervisors) to make better decisions to avoid, and to recover from, alert periods. For example, staff should not wait until a Red Alert before adding ambulances, which is a common practice, because the expected number of lost calls rapidly increases as the number of available ambulances at the action epoch decreases.