The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers gives us fluid and diffusion limits. The diffusion limit suggests a Gaussian approximation to the stochastic behavior. The fluid mean and diffusion variance can form a two- dimensional dynamical system that approximates the actual transient mean and variance for the queueing process. Recent work showed that a better approximation for mean and variance can be computed from a related two-dimensional dynamical system. In this spirit, we introduce a new three-dimensional dynamical system that estimates the mean, variance, and third cumulant moment. This surpasses the previous two approaches by fitting the number in the queue to a quadratic function of a Gaussian random variable.

* This is based on a paper published in QUESTA and is joint work with Jamol Pender of Cornell University.