Comparison of fluid approximations for service systems with state-dependent service rates and return probabilities

We compare two models of a multi-server queueing system with state-dependent service rates and return probabilities. In both models, upon completing service, customers are delayed prior to possibly returning to service. In one model, the determination of whether a customer will return occurs immediately upon service completion, at the beginning of the delay. In the other, that determination is made at the end of the delay, capturing the idea that it takes time for the customer’s condition and needs to evolve or assess, before it becomes known whether a return to service is needed. Our comparison focuses on fluid approximations of the two models. The fluid approximation for the first model, which has been studied previously, consists of a system of two ordinary differential equations. The fluid approximation for the second model, which is new, consists of a delay differential equation. We find that the two fluid approximations have the same set of equilibrium points, but their transient behavior can differ markedly. Both fluid approximations can exhibit bistability for certain parameter values. We use discrete event simulation to illustrate the extent to which the findings from the fluid approximations carry over to the underlying stochastic models.