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Outperforming Multiserver SRPT at All Loads

Izzy Grosof, Daniela Hurtado-Lange

TL;DR

The paper addresses the open problem of beating the multiserver SRPT-k policy in M/G/k queues with known job sizes. It introduces SEK-SMOD, a non-index scheduling policy that deviates from SRPT-k in carefully defined 2-mode divergences and uses a coupled SRPT-k system to bound downside via SMOD, all analyzed through a novel hybrid of worst-case, stochastic, and relative analyses. The main result proves E[T^{SEK-SMOD}] < E[T^{SRPT-k}] for all arrival rates, job-size distributions, and k≥2, supported by rigorous stochastic lemmas and worst-case bounds, plus a Practical SEK variant validated by simulation. The work demonstrates meaningful mean-response-time improvements across loads and distributions, including higher-variance jobs and limited-size information, and outlines SEK-n variants for scenarios with more servers. This framework advances the design of global-knowledge, non-index policies for multiserver queues, with practical implications for resource-constrained systems and potential extensions to unknown-size settings and learning-based adaptations.

Abstract

A well-designed scheduling policy can unlock significant performance improvements with no additional resources. Multiserver SRPT (SRPT-$k$) is known to achieve asymptotically optimal mean response time in the heavy traffic limit, as load approaches capacity. No better policy is known for the M/G/$k$ queue in any regime. We introduce a new policy, SRPT-Except-$k+1$ & Modified SRPT (SEK-SMOD), which is the first policy to provably achieve lower mean response time than SRPT-$k$. SEK-SMOD outperforms SRPT-$k$ across all loads and all job size distributions. The key idea behind SEK-SMOD is to prioritize large jobs over small jobs in specific scenarios to improve server utilization, and thereby improve the response time of subsequent jobs in expectation. Our proof is a novel application of hybrid worst-case and stochastic techniques to relative analysis, where we analyze the deviations of our proposed SEK-SMOD policy away from the SRPT-$k$ baseline policy. Furthermore, we design Practical-SEK (a simplified variant of SEK-SMOD) and empirically verify the improvement over SRPT-$k$ via simulation.

Outperforming Multiserver SRPT at All Loads

TL;DR

The paper addresses the open problem of beating the multiserver SRPT-k policy in M/G/k queues with known job sizes. It introduces SEK-SMOD, a non-index scheduling policy that deviates from SRPT-k in carefully defined 2-mode divergences and uses a coupled SRPT-k system to bound downside via SMOD, all analyzed through a novel hybrid of worst-case, stochastic, and relative analyses. The main result proves E[T^{SEK-SMOD}] < E[T^{SRPT-k}] for all arrival rates, job-size distributions, and k≥2, supported by rigorous stochastic lemmas and worst-case bounds, plus a Practical SEK variant validated by simulation. The work demonstrates meaningful mean-response-time improvements across loads and distributions, including higher-variance jobs and limited-size information, and outlines SEK-n variants for scenarios with more servers. This framework advances the design of global-knowledge, non-index policies for multiserver queues, with practical implications for resource-constrained systems and potential extensions to unknown-size settings and learning-based adaptations.

Abstract

A well-designed scheduling policy can unlock significant performance improvements with no additional resources. Multiserver SRPT (SRPT-) is known to achieve asymptotically optimal mean response time in the heavy traffic limit, as load approaches capacity. No better policy is known for the M/G/ queue in any regime. We introduce a new policy, SRPT-Except- & Modified SRPT (SEK-SMOD), which is the first policy to provably achieve lower mean response time than SRPT-. SEK-SMOD outperforms SRPT- across all loads and all job size distributions. The key idea behind SEK-SMOD is to prioritize large jobs over small jobs in specific scenarios to improve server utilization, and thereby improve the response time of subsequent jobs in expectation. Our proof is a novel application of hybrid worst-case and stochastic techniques to relative analysis, where we analyze the deviations of our proposed SEK-SMOD policy away from the SRPT- baseline policy. Furthermore, we design Practical-SEK (a simplified variant of SEK-SMOD) and empirically verify the improvement over SRPT- via simulation.

Paper Structure

This paper contains 38 sections, 13 theorems, 15 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Challenges in beating SRPT-k
  3. Overcoming challenges
  4. Prior Work
  5. Shortest Remaining Processing Time
  6. Stochastic & Worst-case Hybrid Analysis
  7. Relative Analysis
  8. Multiserver Scheduling Analysis: Index Policies
  9. Model and Definitions
  10. Defining SEK-SMOD
  11. Defining performance measures
  12. Main result
  13. Proof outline
  14. Stochastic proofs
  15. Proof of Theorem \ref{['thm:main']}
  16. ...and 23 more sections

Key Result

theorem 1

For all arrival rates $\lambda$, job sizes $S$, and numbers of servers $k\geq 2$, with the SEK-SMOD parameterization given in def:parameter, if M/G/$k$/SRPT is positive recurrent, the mean response time of SEK-SMOD is smaller than that of SRPT-$k$:

Figures (7)

  • Figure 1: Example of two M/G/2 systems operating under scheduling policies $A$ and $B$, respectively. Each gray box represents a job, and the number in each box represents the corresponding remaining size.
  • Figure 2: Diagram of proof of Theorem \ref{['thm:main']}.
  • Figure 3: Diagram of worst-case scenario arguments. The positive part (\ref{['def:positive-part']}), zig-zag matching (\ref{['def:zig-zag']}), and PLN (\ref{['def:pln']}) are defined later in this section; and "work inequality" means that $W^A(t)\leq W^{SRPT- k}$, where policy $A$ can be SMOD or SEK-SMOD depending on the context.
  • Figure 4: Improvement of SEK over SRPT-$k$, PSJF-$k$, and RS-$k$, under varying distributions $S$ and threshold parameters $\epsilon \in \{0.5, 1, 1.5\}$. Improvement ratio compares each policy to SRPT-$k$. Loads simulated: $\rho \in [0.75, 0.996]$, $k=2$ servers. $10^7$ arrivals per data point.
  • Figure 5: Improvement of SEK over SRPT-$k$, PSJF-$k$, and RS-$k$, under varying distributions $S$ and threshold parameters $\epsilon \in \{0.5, 1, 1.5\}$, with an expanded vertical axis. Improvement ratio compares each policy to SRPT-$k$. Loads simulated: $\rho \in [0.75, 0.996]$, $k=2$ servers. $10^7$ arrivals per data point.
  • ...and 2 more figures

Theorems & Definitions (29)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • theorem 1
  • definition 7
  • lemma 1
  • lemma 2: Bad scenario
  • ...and 19 more