Scaling the arrival rates and number of servers, in a dynamic rate multi-server queue with abandonment, arises naturally when updating a staffing schedule in response to a forecast of increased customer demand. This type of scaling gives us both fluid and diffusion limits.

The latter limit suggests a natural Gaussian approximation to the queueing behavior. The resulting mean and variance are easily computed from a two dimensional dynamical system that is derived from the fluid and diffusion limits.

We introduce a new three-dimensional dynamical system that improves the queueing approximation by constructing a quadratic function of a Gaussian random variable. The resulting equations estimate the mean, variance, and third cumulative moment, or skewness, for the queueing process.