The restricted Erlang-R queue: finite-size effects in service systems with returning customers

We study a restricted version of the Erlang-R queueing model that has both server and space constraints, in which the space restriction is administered by two types of admission policies: blocking or holding in a pre-entrant queue. This model, in which customers occupying spaces may be either “needy” (requiring attention from servers) or “content,” captures a wide range of service environments, such as medical units, contact centers, and courts. We study the interplay between the two resource constraints and the reentrant queueing structure. We analyze the connection between staffing, space, and policy decisions and the system performance as a tool for optimizing resource utilization and in determining stability and performance. We develop product form and many-server approximations of performance measures when the blocking policy applies, and then utilize these results to determine approximations for the holding policy as well. We show that system dynamics are captured by a single measure of the fraction time a customer is “needy” in the network, and that returning customers should be explicitly accounted for both in steady-state and transient environments. We show that capacity allocation of both resource constraints (servers and space) should be considered jointly, and derive methodology to determine it. We show that applying a holding policy requires more resources than a blocking policy and demonstrate the application of our policies in two case studies of resource allocation in hospitals.