A general model and convergence results for determining vehicle utilization in emergency systems

Emergency Medical Service (EMS) systems can be modeled as spatially distributed queueing systems. Each location in the area has a preference ordering for the servers (usually based on proximity of calls to servers). When a call arrives, the dispatcher scans the preference list and assigns the most preferred idle vehicle to the call. If all vehicles are busy, the call is sent to a private ambulance system that operates in parallel to the EMS system. A major problem in designing and operating EMS systems is to estimate vehicle utilizations and busy probabilities for a given set of base locations. In earlier work, various systems of nonlinear equations have been proposed to estimate the vehicle utilization in EMS systems. In this paper we present a general model structure that encompasses much of the past work. We develop convergence conditions for the general model and show that a simple bisection method can be used to find solutions. The bisection method also leads to a test for the uniqueness of the solution. We demonstrate the method on a problem with 5 vehicle bases and 300 demand locations.