Can customer arrival rates be modelled by sine waves?

Abstract

Customer arrival patterns observed in the real world typically exhibit strong seasonal effects. It is therefore natural to ask: Can a nonhomogeneous Poisson process (NHPP) with a rate function that is the simple sum of sinusoids provide an adequate description of reality? If so, how can the sinusoidal NHPP be used to improve the performance of service systems? We empirically validate that the model is consistent with arrivals data from two settings of great interest in service operations: patient arrivals to an emergency department and customer calls to a bank call centre. This finding provides rigorous justification for the use of the sinusoidal NHPP assumption in many existing queuing models. We also clarify why a sinusoidal NHPP model is more suitable than the standard NHPP when the underlying arrival pattern is aperiodic (e.g., does not follow a weekly cycle). This is illustrated using data from a car dealership, and also via a naturalistic staffing simulation based on the call centre. On the other hand, if the arrival pattern is periodic, we explain why both models should perform comparably. Even then, the sinusoidal NHPP is still necessary for managers to use to verify that the arrival pattern is indeed periodic, a step that is seldom performed in applications. Code for fitting the sinusoidal NHPP to data is provided on GitHub.

Keywords: queues; arrival rate estimation; spectral estimation; Fourier analysis; nonhomogeneous Poisson process; emergency departments; call centres

JEL Classification: C15, C22, C32, C44, C52, C53