Insensitivity to Service-time Distributions for Fluid Queueing Models

We study fluid limits based on ordinary differential equations (ODEs) for Markovian queueing models where nonexponential service times are fit by appropriate Coxian distributions to match their first and second moments. We focus on a heavy-load regime, whereby the fluid limit of the queue-length process of the nonexponential queue estimates a bottleneck situation. Under this condition, we show that the ODE solution admits a steady state which is insensitive to the service-time distribution: The ODE steady state only depends on the mean service times. By contrast, the steady-state average performance measures computed by Markovian analysis are in general dependent on the higher-order moments of the service-time distribution. A numerical investigation shows that, given any two Markovian queueing models with Coxian-distributed service times with the same mean and different variance, the model with lower variance converges more rapidly to the (same) fluid limit than the one with higher variance.