Novel Lower Bounds on M/G/k Scheduling

TL;DR

This paper introduces a new framework for deriving nontrivial lower bounds on mean response time in M/G/k queueing systems, especially effective at moderate loads, using a novel variable-speed queue and advanced analytical techniques.

Contribution

It provides the first systematic method for establishing lower bounds on M/G/k response times under arbitrary policies, improving previous bounds by over 60%.

Findings

Bounds are tighter than naive estimates by more than 60%.

Framework validated numerically for systems with up to 5 servers.

New variable-speed queue accurately models multiserver work completion.

Abstract

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.