Probabilistic Coverage

Much of the book is focused on facilities of various types, represented as points, nodes, lines, arcs, paths, tours, areas, etc., providing a wide range of services. The underlying assumption has generally been that facilities, or personnel at/from the facility, are available to serve when needed. This is not too surprising because a sizable number of location covering models are firmly rooted in the initial work Toregas and ReVelle (1972) on the location set covering problem (LSCP), where they were specifically interested in public sector issues involving equity of access to service. In the LSCP facilities were viewed as available for service when needed. In particular, the application of the LSCP to site emergency services, like fire, ambulance and police response, helped to design and relocate such services so that they provide coverage to all. Little has changed, in fact, over the intervening years as emergency service contexts remain of great interest and coverage models have time and again been instrumental in helping to both understand existing service systems as well as develop management plans for emergency response while promoting fairness and equity in service access. Of course, there are many other areas of application for coverage models as well, but the emergency response context has continued to be both challenging and interesting as we better understand such systems and have better supporting data. The focus of this chapter involves the fact that facilities (or personnel) may not always be available when needed. That is, there is a non-zero probability that facility service coverage may not be provided even when every demand is within a desired maximal service standard of a facility. There are clearly many ways in which a facility would be unavailable for service. One situation is that personnel are already busy serving another demand. This is depicted in Fig. 4.1, where the fire engine has traveled from the fire station in response to a fire. However, while busy fighting this fire, another incident (vehicle crash and fire) has occurred across town). It is therefore not possible for a fire crew to respond immediately. Another situation is that a facility may be unavailable due to a failure of some sort, such as equipment being broken, a power outage, a flood, or even an accident.