Characterization and asymptotic analysis of the stationary probabilities in Discriminatory Processor Sharing Systems

In this paper, we establish two different results. The first result is a characterization theorem saying that if the stationary state probabilities for originally described Markovian discriminatory processor sharing (DPS) system have a closed product geometric form (the exact definition is given in the paper), then the system must only be Egalitarian, i.e. all flows in this system must have equal priorities. The second result is the tail asymptotics for the stationary probabilities. We provide a detailed asymptotic analysis of the system, and obtain the exact asymptotic form of the stationary probabilities in DPS systems when the number of flows in the system is large.