Largest weighted delay first scheduling: Large deviations and optimality

We consider a single server system with N input flows. We assume that each flow has stationary increments and satisfies a sample path large deviation principle, and that the system is stable. We introduce the largest weighted delay first (LWDF) queueing discipline associated with any given weight vector alpha = (alpha (1),...,alpha (N)). We show that under the LWDF discipline the sequence of scaled stationary distributions of the delay (w) over cap (i) of each flow satisfies a large deviation principle with the rate function given by a finite-dimensional optimization problem. We also prove that the LWDF discipline is optimal in the sense that it maximizes the quantity min(i=1,...,N) [alpha (i) lim(n --> proportional to) -1/n log P((w) over cap (i)>n)] , within a large class of work conserving disciplines.