We describe an optimization model for ambulance location that maximizes the expected system wide coverage, given a total number of ambulances. The model measures expected coverage as the fraction of calls reached within a given time standard and considers response time to be composed of a random delay (prior to travel to the scene) plus a random travel time. Pre-travel delays at dispatch and activation stages can be significant, and models that do not account for such delays can severely overestimate the possible coverage for a given number of ambulances and underestimate the number of ambulances needed to provide a specified coverage level. By explicitly modeling the randomness in the delays and the travel time, we arrive at a more realistic model for ambulance location. In order to capture the dependence of ambulance busy fractions on the allocation of ambulances between stations, we iterate between solving the optimization model and using the approximate hypercube model to calculate busy fractions. We illustrate the use of the model using actual data from Edmonton.

W e compare the performance of seven methods in computing or approximating service levels for nonstationary M t /M/s t queueing systems: an exact method (a Runge-Kutta ordinary-differential-equation solver), the randomization method, a closure (or surrogate-distribution) approximation, a direct infinite-server approximation, a modified-offered-load infinite-server approximation, an effective-arrival-rate approximation, and a lagged stationary approximation. We assume an exhaustive service discipline, where service in progress when a server is scheduled to leave is completed before the server leaves. We used all of the methods to solve the same set of 640 test problems. The randomization method was almost as accurate as the exact method and used about half the computational time. The closure approximation was less accurate, and usually slower, than the randomization method. The two infinite-server-based approximations, the effective-arrival-rate approximation, and the lagged stationary approximation were less accurate but had computation times that were far shorter and less problem-dependent than the other three methods.

Using administrative data for high-priority calls in Calgary, Alberta, we estimate how ambulance travel times depend on distance. We find that a logarithmic transformation produces symmetric travel-time distributions with heavier tails than those of a normal distribution. Guided by nonparametric estimates of the median and coefficient of variation, we demonstrate that a previously proposed model for mean fire engine travel times is a valid and useful description of median ambulance travel times. We propose a new specification for the coefficient of variation, which decreases with distance. We illustrate how the resulting travel-time distribution model can be used to create probability-of-coverage maps for diagnosis and improvement of system performance.ambulance service, probability-of-coverage map, nonparametric, travel

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To calculate many of the important performance measures for an emergency response system one requires knowledge of the probability that a particular server will respond to an incoming call at a particular location. Estimating these "dispatch probabilities" is complicated by four important characteristics of emergency service systems. We discuss these characteristics and extend previous approximation methods for calculating dispatch probabilities, to account for the possibilities of workload variation by station, multiple vehicles per station, call and station dependent service times, and server cooperation and dependence.