  We consider an automatic overload control for two large service systems
modeled as multi-server queues, such as call centers. We assume that the two
systems are designed to operate independently, but want to help each other
respond to unexpected overloads. The proposed overload control automatically
activates sharing (sending some customers from one system to the other) once a
ratio of the queue lengths in the two systems crosses an activation threshold
(with ratio and activation threshold parameters for each direction). To prevent
harmful sharing, sharing is allowed in only one direction at any time. In this
paper, we are primarily concerned with ensuring that the system recovers
rapidly after the overload is over, either (i) because the two systems return
to normal loading or (ii) because the direction of the overload suddenly shifts
in the opposite direction. To achieve rapid recovery, we introduce lower
thresholds for the queue ratios, below which one-way sharing is released. As a
basis for studying the complex dynamics, we develop a new six-dimensional fluid
approximation for a system with time-varying arrival rates, extending a
previous fluid approximation involving a stochastic averaging principle. We
conduct simulations to confirm that the new algorithm is effective for
predicting the system performance and choosing effective control parameters.
The simulation and the algorithm both show that the system can experience an
inefficient nearly-periodic behavior, corresponding to an oscillating
equilibrium (congestion collapse), if the sharing is strongly inefficient and
the control parameters are set inappropriately.
