Optimal Capacity Overbooking for the Regular Treatment of Chronic Conditions

Patients suffering from a chronic condition often require periodic treatment. For example, patients with End-Stage Renal Disease (ESRD) require dialysis three times a week. These patients are also frequently hospitalized for complications from their treatment, resulting in idle capacity at the clinic. These temporary patient absences make overbooking at the clinic attractive. This paper develops a semiclosed migration network to capture patient flow into the clinic and between the clinic and hospital. We consider a simple class of stationary control policies for patient admissions and provide algorithms for selecting one that maximizes long-run average earnings. Local diffusion approximations were constructed to provide square-root loading formulas for the optimal capacity level and patient overbooking level: as the total patient arrival rate increases, the deviation between the optimal and fluid-limit capacity and overbooking levels scale up with the square root of the total arrival rate. We find that high hospitalization rates and long inpatient stays allow for more overbooking. Numerical examples based on the typical dialysis clinic in the United States suggest an increase in earnings of 11%--14% over policies derived from traditional M / M / N models that do not account for hospitalizations and do not allow overbooking, while keeping the probability of capacity shortage arbitrarily small.