Control problems and management policies in health systems: application to intensive care units

The stochastic nature of both patient arrivals and lengths of stay leads inevitably to periodic bed shortages in healthcare units. Physicians are challenged to fit demand to service capacity. If all beds are occupied eligible patients are usually referred to another ward or hospital and scheduled surgeries may be cancelled. Lack of beds may also have consequences for patients, who may be discharged in advance when the number of occupied beds is so high as to compromise the medical care of new incoming patients. In this paper we deal with the problem of obtaining efficient bed-management policies. We introduce a queuing control problem in which neither the arrival rates nor the number of servers can be modified. Bed occupancy control is addressed by modifying the service time rates, to make them dependent on the state of the system. The objective functions are two quality-of-service components: to minimize patient rejections and to minimize the length of stay shortening. The first objective has a clear mathematical formulation: minimize the probability of rejecting a patient. The second objective admits several formulations. Four different expressions, all leading to nonlinear optimization problems, are proposed. The solutions of these optimization problems define different control policies. We obtain the analytical solutions by adopting Markov-type assumptions and comparing them in terms of the two quality-of-service components. We extend these results to the general case using optimization with simulation, and propose a way to simulate general length of stay distributions enabling the inclusion of state-dependent service rates.