Queues with time-varying arrivals and inspections with applications to hospital discharge policies

In order for a patient to be discharged from a hospital unit, a physician must first perform a physical examination and review the pertinent medical information to determine that the patient is stable enough to be transferred to a lower level of care or be discharged home. Requiring an inspection of a patient's "readiness for discharge" introduces an interesting dynamic where patients may occupy a bed longer than medically necessary. Motivated by this phenomenon, we introduce a queueing system with time-varying arrival rates in which servers who have completed service cannot be released until an inspection occurs. Weexamine how such adynamic impacts common system measures such as stability, expected number of customers in the system, probability of waiting, and expected waiting time. Leveraging insights from an infinite-server model, we're able to optimize the timing of inspections and find via theoretical and numerical analysis that (1) optimizing a single inspection time could lead to significant improvements in system performance when the amplitude of the arrival rate function is large, (2) multiple inspections should be uniformly distributed throughout the day, and (3) the marginal improvements of adding additional inspection times is decreasing.