Asymptotic Optimality of the Max-Min Fair Allocation

Multiple classes of jobs are processed in a stochastic network that consists of a set of servers. Each class of jobs requires a concurrent occupancy of a subset of servers to be processed, and each server is shared among the job classes in a head-of-the-line processor-sharing mechanism. In each state of the network, the server capacities are allocated among the job classes according to the so-called max-min fair policy. We derive the fluid and diffusion limits of the network under this resource control policy. Furthermore, we provide a characterization of the fixed-point state associated with the fluid limit, and identify a cost function that is minimized in the diffusion regime