Stochastic Location Models with Congestion

In this chapter we describe facility location models where consumers generate streams of stochastic demands for service, and service times are stochastic. This combination leads to congestion, where some of the arriving demands cannot be served immediately and must either wait in queue or be lost to the system. These models have applications that range from emergency service systems (fire, ambulance, police) to networks of public and private facilities. One key issue is whether customers travel to facilities to obtain service, or mobile servers travel to customer locations (e.g., in case of police cars). For the most part, we focus on models with static (fixed) servers, as the underlying queueing systems are more tractable and thus a richer set of analytical results is available. After describing the main components of the system (customers, facilities, and the objective function), we focus on the customer-facility interaction, developing a classification of models based on the how customer demand is allocated to facilities and whether the demand is elastic or not. We use our description of system components and customer-response classification to organize the rich variety of models considered in the literature into four thematic groups that share common assumptions and structural properties. For each group we review the solution approaches and outline the main difficulties. We conclude with a review of some important open problems.