Online Supplement to “A Fluid Approximation for Service Systems Responding to Unexpected Overloads”

of the Main Paper 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 we want the agents to 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 we will not know which class will experience the overload or the the timing and extent of the overload, we proposed a queue-ratio control with thresholds: When a weighted difference between the queue lengths crosses a pre-specified 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, by only observing the queue lengths. We then developed a simple deterministic fluid approximation in 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 develop its transient equations, which depend on a heavy-traffic averaging principle. Although deterministic, the new fluid model includes stochastic balance equations. The full fluid model here enables us to analyze scenarios we could not analyze previously. We also use this new fluid approximation to refine our previous results. Simulation experiments show that the refinements greatly increases the accuracy of our performance predictions.