  A many-server heavy-traffic FCLT is proved for the $G_t/M/s_t+\mathit {GI}$
queueing model, having time-varying arrival rate and staffing, a general
arrival process satisfying a FCLT, exponential service times and customer
abandonment according to a general probability distribution. The FCLT provides
theoretical support for the approximating deterministic fluid model the authors
analyzed in a previous paper and a refined Gaussian process approximation,
using variance formulas given here. The model is assumed to alternate between
underloaded and overloaded intervals, with critical loading only at the
isolated switching points. The proof is based on a recursive analysis of the
system over these successive intervals, drawing heavily on previous results for
infinite-server models. The FCLT requires careful treatment of the initial
conditions for each interval.
