A fluid approximation for service systems responding to unexpected overloads

In a recent paper we considered two networked service systems, each having its own customers and designated service pool with many agents, where all agents are able to serve the other customers, although they may do so inefficiently. Usually the agents should serve only their own customers, but we want an automatic control that activates serving some of the other customers when an unexpected overload occurs. Assuming that the identity of the class that will experience the overload or the timing and extent of the overload are unknown, we proposed a queue-ratio control with thresholds: When a weighted difference of the queue lengths crosses a prespecified threshold, with the weight and the threshold depending on the class to be helped, serving the other customers is activated so that a certain queue ratio is maintained. We then developed a simple deterministic steady-state fluid approximation, based on flow balance, under which this control was shown to be optimal, and we showed how to calculate the control parameters. In this sequel we focus on the fluid approximation itself and describe its transient behavior, which depends on a heavy-traffic averaging principle. The new fluid model developed here is an ordinary differential equation driven by the instantaneous steady-state probabilities of a fast-time-scale stochastic process. The averaging principle also provides the basis for an effective Gaussian approximation for the steady-state queue lengths. Effectiveness of the approximations is confirmed by simulation experiments.