Modeling Time-Dependent Arrivals to Service Systems: A Case in Using a Piecewise-Polynomial Rate Function in a Nonhomogeneous Poisson Process

We consider the use of a nonhomogeneous Poisson process in modeling time-dependent arrivals to service systems. In analyzing a set of actual arrival times corresponding to epochs of calls for on-line analysis of electrocardiograms, we found that approximating the rate function by an exponential polynomial (or exponential trigonometric polynomial) suggested in the literature inadequate. This was caused by the nature of our data which contained frequent cyclic abrupt changes in arrival rates. In this paper, we propose the use a piecewise polynomial to represent the rate function. We present two maximum likelihood estimators for estimating the parameters of the piecewise polynomial---one based on arrival times, and another based on aggregated counts, and a numerical method for carrying out the computation. We use a procedure based on thinning for generating arrival times from such a process. For hypothesis testing, these are combined to produce critical values for the Kolmogorov Smirnov statistic by a Monte Carlo simulation. While our approach is presented and discussed in the context of a specific case, the results are applicable and observations relevant to many other systems sharing similar patterns of variation.