Multiple Unit Dispatches in Emergency Services: Models to Estimate System Performance

One emergency service system model that has undergone extensive development and testing is the hypercube queuing model and its approximate analog, which were developed by Larson. The model estimates key system performance measures, such as individual unit workloads and travel times. One limitation of this model, as well as other probabilistic deployment models, is that they assume only a single unit is dispatched to each call. In the police environment, if two officers are assigned to each unit, this assumption is generally valid. However, most fire services dispatch multiple vehicles to a serious fire, and in police departments with one-officer units, a second unit will be sent to all potentially dangerous calls. The need for a new model goes beyond an interest in more accurately estimating standard performance measures when multiple units are dispatched. New models are required to predict new performance measures that are relevant only when multiple units are dispatched. For example, what are the paired travel times of the first- and second-arriving units at an emergency? How long will the first unit be exposed at a potentially dangerous situation until a backup unit arrives? Which unit most frequently arrives first at calls requiring multiple units? In this paper we build on Larson's work and present two models, one exact, the other, approximate, which can capture the simultaneous response of two identical units dispatched to a single call. We present an example from the police context to illustrate both models and compare the speed and accuracy of the approximate model to the exact one. The workload and travel time estimates of both models differ on the average by only two percent. We conclude our presentation with a brief discussion of model extensions with a focus on the analysis of a merger of police, fire and emergency medical services into a public safety service.