This article presents a fuzzy location–allocation model for congested systems. In service networks, such as health and emergency services, public safety, fire fighting and so on, the location of servers and allocation of demand nodes to them have a strong impact on the congestion at each server and as such, on the quality of service. The previous efforts in this area have concentrated on enhancing the reliability and quality of service with a probabilistic orientation. In this paper we utilize fuzzy theory to develop a queuing maximal covering location–allocation model which we call the fuzzy queuing maximal covering location–allocation model. We consider fuzzified queuing parameters as well as fuzzified constraints to develop a new mathematical model which we convert to a single objective integer programming model. Our model considers one type of service call, one type of server and includes one constraint on the quality of service in the form of a service time or a queue length constraint. A genetic algorithm is developed to solve and test the model using up to 50-node networks. We also propose extensions to our model.