The Heavy Traffic Limit of an Unbalanced Generalized Processor Sharing Model

This work considers a server that processes J classes using the generalized processor sharing discipline. The invariant manifold of the so- called fluid limit associated with this model is shown to have a particular form and we explicitly identify the set of strictly subcritical classes in terms of the base weight vector and the vector of long-run average workload arrival rates. In addition, under general assumptions it is shown that when the heavy traffic condition holds, the functional central limit of the scaled unfinished work process is a reflected diffusion process that lies in the invariant manifold. In particular, the result identifies precisely how the covariance structure of the unfinished work of the critical classes is influenced by the variance of the cumulative workload arrival processes of the strictly subcritical classes. The reflected diffusion limit is characterized by the so-called extended Skorokhod map and may fail to be a semimartingale. This generalizes previous results obtained for the simpler balanced case for which there is no state-space collapse. Along the way, this work also establishes a comparison principle for solutions to the extended Skorokhod map associated with this model, which may be of independent interest.