{'fraction_non_alphanumeric': 0.021649484536082474, 'fraction_numerical': 0.0, 'mean_word_length': 4.993827160493828, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
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.