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 with base weight vector alpha = (alpha1,...,alphaJ) and redistribution weight vector beta = )beta1,...,betaJ). This paper is organised as follows. In the remainder of this section we describe our notational convention. Section 2 contains a detailed description of the GPS discipline, a characterization of the unfinished work process, and definitions of the SKorokhod and extended Skorokhod problems, culminating in a representation of the unfinished work process in terms of a Skorokhod problem. The fluid limit of the unbalanced GPS model is investigated in Section 3, where the invariant manifold of the fluid limit is characterized. Section 4 contains some properties of the GPS Skorokhod map used in characterizing the heavy traffic limit and proving the heavy traffic limit theorem. In particular, a comparison principle for the extended Skorokhod map associated with GPS is established. Finally, Section 5 contains the statement and proof of the heavy traffic limit theorem for the unbalanced GPS model.