AbstractIn urban fire departments, fire companies are dispatched to respond to alarms which occur spatially and temporally in a generally unpredictable way. Also, the time during which one of the responding units is busy on a call, and hence unavailable to other alarms, is itself a random variable. This variability in demand and service time makes it difficult to maintain a balance between the need for effective response to alarms which occur now and those which may arrive in the future. In this paper, we discuss two specific mathematical models, based on work done by the Rand Institute in New York City, for determining not only how many but also which of the available fire-fighting units to deploy to any given alarm. Each is a Markovian decision model in which the conflicting objectives of adequate response to present or future incidents are explicity accounted for. Similar considerations are applicable to other municipal emergency services.