Stability of the simultaneous processor sharing model

We study the phenomenon of entrainment in processor sharing networks, whereby, while individual network resources have sufficient capacity to meet demand, the requirement for simultaneous availability of resources means that a network may nevertheless be unstable. We show that instability occurs through a poor control strategy, and that, for a variety of network topologies, only small modifications to control strategies are required in order to ensure stability. For control strategies which possess a natural monotonicity property, we give some new results for the classification of the corresponding Markov processes, which lead to conditions both for stability and for instability. Finally, we study the effect of variation of call size distribution on stationary distributions and stability.