In queueing systems, effective scheduling algorithms are essential for optimizing performance. Optimal scheduling for the M/G/k queue has been explored in the heavy traffic limit, but much remains unknown in the intermediate load regime. In this paper, we give the first framework for proving nontrivial lower bounds on the mean response time of the M/G/k system under arbitrary scheduling policies. Our bounds tighten previous naive lower bounds by more than 60\%, yielding significant improvements particularly for moderate loads. Key to our approach is a new variable-speed queue, which more accurately captures the work completion behavior of multiserver systems. To analyze the expected work of this queue, we develop a novel manner of employing the drift method or the BAR approach, by developing test functions via the solutions to a differential equation. We validate our results numerically for systems with up to 5 servers and a range of job size distributions.

翻译：在排队系统中，有效的调度算法对于优化性能至关重要。M/G/k 队列的最优调度已在重负载极限下得到探索，但在中等负载区间仍存在大量未知问题。本文首次提出了一种框架，用于证明在任意调度策略下 M/G/k 系统平均响应时间的非平凡下界。我们的下界将先前朴素下界提高了 60% 以上，尤其在中等负载条件下实现了显著改进。我们方法的关键在于引入一种新的变速队列，该队列能更精确地刻画多服务器系统的工作完成行为。为分析该队列的期望工作量，我们通过微分方程的解构造测试函数，提出了一种运用漂移方法或 BAR 方法的新颖方式。我们通过数值实验验证了结果的有效性，实验涵盖最多 5 台服务器及多种作业规模分布场景。