  In previous papers we developed a deterministic fluid approximation for an
overloaded Markovian queueing system having two customer classes and two
service pools, known in the call-center literature as the X model. The system
uses the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed
in a recent paper as a way for one service system to help another in face of an
unexpected overload. Under FQR-T, customers are served by their own service
pool until a threshold is exceeded. Then, one-way sharing is activated with
customers from one class allowed to be served in both pools. The control aims
to keep the two queues at a pre-specified fixed ratio. We supported the fluid
approximation by establishing a many-server heavy-traffic functional weak law
of large numbers (FWLLN) involving an averaging principle. In this paper we
develop a refined diffusion approximation for the same model based on a
many-server heavy-traffic functional central limit theorem (FCLT).
