An optimal non–uniform piecewise constant approximation for the patient arrival rate for a more efficient representation of the emergency departments arrival process

Overcrowding represents an increasing and well studied phenomenon which afflicts Emergency Departments all over the world. Shortage of staff, flu season and lack of hospital beds are among the possible causes. As consequence, waiting times are enlarged and life of critical patients can be endangered. This urges ED managers to improve performance of healthcare services. Modeling approaches used to tackle this problem are often based on Discrete Event Simulation, hence needing to accurately represent the patient arrival process to the ED. Since the arrival rate is time-dependent, suitable non stationary process models must be considered, such as the nonhomogeneous Poisson process. In this paper we focus on this arrival process, in order to determine the best piecewise-constant approximation of the arrival rate function. A proper number of non equally spaced intervals are used aiming at accurately representing the time-varying arrival rate. This is obtained by solving an integer non linear black box optimization problem with black box constraints. Data from a large Italian hospital ED are used to show the effectiveness of the proposed approach.