  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 with the aim 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 the 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 hours into a suitable
number of non 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 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.
