Rate-Based Daily Arrival Process Models with Application to Call Centers

We propose, develop, and compare new stochastic models for the daily arrival rate in a call center. Following standard practice, the day is divided into time periods of equal length (e.g., 15 or 30 minutes), the arrival rate is assumed random but constant in time in each period, and the arrivals are from a Poisson process, conditional on the rate. The random rate for each period is taken as a deterministic base rate (or expected rate) multiplied by a random busyness factor having mean 1. Models in which the busyness factors are independent across periods, or in which a common busyness factor applies to all periods, have been studied previously. But they are not sufficiently realistic. We examine alternative models for which the busyness factors have some form of dependence across periods. Maximum likelihood parameter estimation for these models is not easy, mainly because the arrival rates themselves are never observed. We develop specialized techniques to perform this estimation. We compare the goodness-of-fit of these models on arrival data from three call centers, both in-sample and out-of-sample. Our models can represent arrivals in many other types of systems as well. Estimating a model for the vector of counts (the number of arrivals in each period) is generally easier than for the vector of rates, because the counts can be observed, but a model for the rates is often more convenient and natural, e.g., for simulation. We examine and provide insight on the relationship between these two types of modeling. In particular, we give explicit formulas for the relationship between the correlation between rates and that between counts in two given periods, and for the variance and dispersion index in a given period. These formulas imply that for a given correlation between the rates, the correlation between the counts is much smaller in low traffic than in high traffic.