  We study an ordinary differential equation (ODE) arising as the many-server
heavy-traffic fluid limit of a sequence of overloaded Markovian queueing models
with two customer classes and two service pools. The system, known as the X
model in the call-center literature, operates under 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
unanticipated overload. Each pool serves only its own class until a threshold
is exceeded; then one-way sharing is activated with all customer-server
assignments then driving the two queues toward a fixed ratio. For large
systems, that fixed ratio is achieved approximately. The ODE describes system
performance during an overload. The control is driven by a queue-difference
stochastic process, which operates in a faster time scale than the queueing
processes themselves, thus achieving a time-dependent steady state
instantaneously in the limit. As a result, for the ODE, the driving process is
replaced by its long-run average behavior at each instant of time; i.e., the
ODE involves a heavy-traffic averaging principle (AP).
