Determining the optimal piecewise constant approximation for the nonhomogeneous Poisson process rate of Emergency Department patient arrivals

Modeling the arrival process to an Emergency Department (ED) is the first step of all studies dealing with the patient flow within the ED. Many of them focus on the increasing phenomenon of ED overcrowding, which is afflicting hospitals all over the world. Since Discrete Event Simulation models are often adopted to assess solutions for reducing the impact of this problem, proper nonstationary processes are taken into account to reproduce time–dependent arrivals. Accordingly, an accurate estimation of the unknown arrival rate is required to guarantee the reliability of results. In this work, an integer nonlinear black–box optimization problem is solved to determine the best piecewise constant approximation of the time-varying arrival rate function, by finding the optimal partition of the 24 h into a suitable number of not equally spaced intervals. The black-box constraints of the optimization problem make the feasible solutions satisfy proper statistical hypotheses; these ensure the validity of the nonhomogeneous Poisson assumption about the arrival process, commonly adopted in the literature, and prevent mixing overdispersed data for model estimation. The cost function of the optimization problem includes a fit error term for the solution accuracy and a penalty term to select an adequate degree of regularity of the optimal solution. To show the effectiveness of this methodology, real data from one of the largest Italian hospital EDs are used.