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Under fairly general assumptions on the arrival and service time proces- ses, we prove fluid and heavy traffic limit theorems for the unfinished work, queue length, sojourn time and waiting time processes associated with a sin- gle station multiclass generalized processor sharing model. The fluid limit of the unfinished work process is characterized by the Skorokhod map asso- ciated with a Skorokhod problem formulation of the generalized processor sharing model, while the heavy traffic diffusion limit is characterized using the corresponding extended Skorokhod map. An interesting feature of the diffusion limits is that they may fail to be semimartingales. 1. Introduction. The efficient sharing of a single processing or transmission resource among traffic from several sources is a recurrent issue in high speed networks. One scheme that has been proposed is known as generalized processor sharing (GPS). GPS has several desirable characteristics that make it a reasonable candidate for a resource sharing policy. First, it provides a minimal guaranteed throughput rate to each source, regardless of the behavior of other sources. In addition, it allows for discrimination between the various sources in the sense that the minimal guaranteed throughputs can be different for each source. Furthermore, it ensures that the resource is fully utilized as long as there is a backlog for any of the sources. Given a single server that can process one unit of work per unit of time and that is being shared by J( 1 <J <∞) sources, the information needed to implement the GPS policy is contained in the weight vector α = (α1 ,...,α J ) .W hen all sources have a backlog of work, source j is allotted a fraction αj of the total capacity of the server. When some sources achieve no backlog by using less than their allotted capacity, the remaining service capacity of the server is split among the other sources in proportion to their αi 's. A more precise description of the model is given in Section 2, where a slight generalization is also considered. The GPS policy was introduced in (17) and (18), where worst case bounds on the delay and unfinished work for deterministically shaped traffic were also calculated. Estimates of the unfinished work for stochastically bounded traffic were derived in (29), and large deviation results were presented in (2), (8) and (16).

Fluid and heavy traffic diffusion limits for a generalized processor sharing model