Novel Lower Bounds on M/G/k Scheduling

TL;DR

The paper tackles the long-standing problem of deriving nontrivial lower bounds on mean response time for M/G/k queues under arbitrary scheduling policies, particularly in the intermediate-load regime. It introduces the novel Increasing Speed Queue (ISQ-k) and the DiffeDrift method to derive tight bounds on relevant work, which are then translated to mean response time via the Work Integral Number Equality (WINE). The core contributions are three lower bounds—MixEx, ISQ, and ISQ-Recycling—that progressively tighten the gap from naive bounds, with ISQ-Recycling offering substantial improvements in moderate to high load and high-variability job sizes. Empirical results demonstrate significant reductions in the uncertainty region compared to prior bounds and existing SEK upper-bound policies. The framework also lays groundwork for extensions to unknown or estimated job sizes and scalable computation for larger k.

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.

Figures (7)

• Figure 1: $M/G/2$ mean response time under SRPT-2, compared against naive lower bounds ($M/G/1$/SRPT and $M/G/\infty$, and against our lower bounds. See \ref{['sec:wine']} for the definition of MixEX, ISQ and ISQ-Recycling bounds.
• Figure 2: UIR relative to MixEx bound of the ISQ and ISQ-Recycling bound in the $M/G/2$ setting with $\text{Exp}(1)$ job sizes. See \ref{['sec:wine']} for definitions of ISQ and ISQ-Recycling bounds.
• Figure 3: UIR of MixEx bound relative to naive bounds in the $M/G/k$ setting with exponential job sizes. Load ranges from 0.3 to 0.95 with five million simulated arrivals per load.
• Figure 4: UIR of novel bounds relative to MixEx bound in the $M/G/k$ setting with exponential job sizes. Load ranges from 0.3 to 0.95 with five million simulated arrivals per load.
• Figure 5: UIR relative to the MixEx bound for the ISQ and ISQ-Recycling bounds in the $M/G/2$ setting with Uniform job size distributions of various $C^2$. Load ranges from 0.1 to 0.95 with five million simulated arrivals per load.

Theorems & Definitions (59)

• Theorem 4.1
• proof : Proof deferred to \ref{['sec:thm_main_2']}
• Theorem 4.2
• proof : Proof deferred to \ref{['sec:thm_main_k']}
• Theorem 4.3
• Proposition 5.1: WINE Identity
• Proposition 6.1
• proof
• Theorem 6.1
• proof