On the large-deviations optimality of scheduling policies minimizing the drift of a Lyapunov function

We show that for a large class of scheduling algorithms, when the algorithm minimizes the drift of a Lyapunov function, the algorithm is optimal in maximizing the asymptotic decay-rate of the probability that the Lyapunov function value exceeds a large threshold. The result in this paper extends our prior results to the important and practically-useful case when the Lyapunov function is not linear in scale, in which case the evolution of the fluid-sample-paths becomes more difficult to track. We use the notion of generalized fluid-sample-paths to address this difficulty, and provide easy-to-verify conditions for checking the large-deviations optimality of scheduling algorithms. As an immediate application of the result, we show that the log-rule is optimal in maximizing the asymptotic decay-rate of the probability that the sum queue exceeds a threshold B.