Cross-Trained Fire-Medics Respond to Medical Calls and Fire Incidents: A Fast Algorithm for a Three-State Spatial Queuing Problem

Problem definition: We focus on modeling and evaluating an emergency service system in which cross-trained fire-medics are pooled and respond to both fire calls and medical emergencies. Academic/practical relevance: Fire demand in the United States has decreased dramatically in the last nearly four decades, whereas emergency medical calls have surged. With this changing landscape, cities are under pressure to reduce their budgets by closing fire stations. We evaluate the alternative of implementing a fire-medic system in terms of cost savings and response time performance. Methodology: A cross-trained fire-medic unit may be in one of three states: available, busy at an emergency medical incident, or busy at a fire call. An exact model for the fire-medic system has exponential complexity. We develop a fast approximation algorithm that has linear complexity and can be used to solve three-state problems of any size. Results: Our approximation algorithm yields accurate predictions of response times and unit workloads and provides rapid solution times. We apply our model to the fire-medic system in St. Paul, MN, and find close agreement between predicted and actual average response times. A traditional system would require 33% more personnel to achieve about the same average response times. In sensitivity analyses, we show that the fire-medic system outperforms a traditional system over a wide range of values for call rates and number of joint units. Managerial implications: The fire-medic system in St. Paul, MN, saves more than three million dollars annually. The greatest benefit of the fire-medic system is in reducing response times to medical emergencies. We also show that in a fire-medic system that includes separate engine units, it is advantageous to convert the separate engines to fire-medic units. Our fast approximation algorithm can be applied even in the largest cities that implement fire-medic systems to improve resource deployment and reduce costs. Funding: This work was supported by the Shanghai Science and Technology Development Foundation [Grant 21YF1420200]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/msom.2022.1140 .