pith. sign in

arxiv: 2606.02311 · v1 · pith:WTCNGX36new · submitted 2026-06-01 · 💻 cs.PF · math.PR

γ-CounterBoost: Optimizing response time tails using job type information only

Nils Charlet , Benny Van Houdt This is my paper

Pith reviewed 2026-06-28 11:28 UTC · model grok-4.3

classification 💻 cs.PF math.PR
keywords schedulingresponse time tailM/G/1 queuepartial informationjob typestail minimizationboost policies
0
0 comments X

The pith

The γ-CounterBoost policy minimizes the tail of the response time distribution among Contextual CounterBoost policies using only job type counts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in light-tailed M/G/1 queues, when only job type information is available and arrival times cannot be used, the γ-CounterBoost policy achieves the minimal response time tail within the class of Contextual CounterBoost policies. This holds for any number of job types d at least 2. For two types it coincides with the Nudge-M policy known to be optimal in a similar setting. A sympathetic reader would care because it provides a practical way to reduce the chance of very long response times with limited information.

Core claim

The γ-CounterBoost policy minimizes the tail in an even broader class of scheduling policies called Contextual CounterBoost policies. The policy is defined for the partial information setting with d job types and does not use arrival times, relying solely on job type counts.

What carries the argument

The γ-CounterBoost policy, a scheduling rule that uses boosted counters based on job types to decide which job to serve next without reference to arrival times.

Load-bearing premise

The queue has light-tailed service times and the scheduler is limited to policies that use only job type counts without arrival times.

What would settle it

Run simulations of an M/G/1 queue with exponential or other light-tailed service distributions and multiple job types, then check if γ-CounterBoost yields a smaller tail probability than alternative Contextual CounterBoost policies for large response time thresholds.

Figures

Figures reproduced from arXiv: 2606.02311 by Benny Van Houdt, Nils Charlet.

Figure 1
Figure 1. Figure 1: The ratio between the tail constant of γ-Boost, γ-CounterBoost, and Nudge-M over FCFS. For Nudge-M, three curves show the three different ways to group two of the three types together, as that policy requires two job types. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 c / cFCFS -CounterBoost CounterBoost +30% CounterBoost -30% CounterBoost +50% CounterBoost -50% (a) Hy… view at source ↗
Figure 2
Figure 2. Figure 2: The ratio between the tail constant of γ-CounterBoost over FCFS, compared to this ratio when over- or underesti￾mating the boosts by 30%. loads, keeping all three types distinct is crucial. The other two ways to merge two types are less effective in this setting. In Figure 1b, γ-CounterBoost has nearly the same tail constant as γ-Boost, the difference is too small to see on the plot. In this setting, mergi… view at source ↗
Figure 3
Figure 3. Figure 3: The ratio between the prefactor of CounterBoost using [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
read the original abstract

In a recent paper the $\gamma$-Boost scheduling policy was shown to minimize the tail of the response time distribution in a light-tailed M/G/1-queue. This policy schedules jobs using a boosted arrival time, defined as the arrival time of a job minus its boost, where the boost of a job depends on its exact job size. The $\gamma$-Boost policy can also be used when only partial job size information is available, such as the type of an incoming job. In such case the boost $b_i$ of a job depends solely on its type $i$ and $\gamma$-Boost was shown to optimize the tail among all boost policies, where a boost policy is fully determined by the $b_i$ values. In the partial information setting $\gamma$-Boost relies on two types of information: job types and arrival times. This paper focuses on the problem of minimizing the tail in a light-tailed M/G/1-queue in the partial job size information setting when the scheduler only makes use of the job types and {\it does not exploit arrival times}. Prior work showed that in case of $2$ job types the so-called Nudge-$M$ policy minimizes the tail in a large class of scheduling policies. In this paper we introduce the $\gamma$-CounterBoost policy in the partial information setting with $d \geq 2$ job types and prove that it minimizes the tail in an even broader class of scheduling policies called Contextual CounterBoost policies. The $\gamma$-CounterBoost policy reduces to the Nudge-$M$ policy in case of $d=2$ job types.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces the γ-CounterBoost policy for a light-tailed M/G/1 queue in the partial-information setting (d ≥ 2 job types known, arrival times unused). It proves that γ-CounterBoost minimizes response-time tail asymptotics inside the explicitly defined class of Contextual CounterBoost policies and shows that the policy coincides exactly with the known Nudge-M policy when d = 2.

Significance. If the proof is correct, the result supplies a clean generalization of the d = 2 Nudge-M optimality result to an arbitrary number of types while remaining inside a well-delimited policy class that never uses arrival epochs. The reduction to Nudge-M supplies an immediate consistency check. The work therefore strengthens the theoretical toolkit for tail-optimal scheduling under type-only information.

minor comments (3)
  1. §2 (model section): the precise definition of the Contextual CounterBoost class should be stated as a numbered definition rather than inline prose so that the optimality statement can cite it directly.
  2. The proof sketch in §4 relies on an inductive argument over the number of types; a short appendix containing the base case (d = 2) written out in full would make the induction step easier to follow.
  3. Notation: the symbol γ is overloaded between the boost parameter and the policy name; a brief clarifying sentence at first use would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the recognition of the generalization from the d=2 Nudge-M result, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; mathematical proof within explicitly defined policy class

full rationale

The paper's central result is a proof that γ-CounterBoost minimizes response-time tail asymptotics inside the explicitly delimited class of Contextual CounterBoost policies (which use only job-type counts). The class is defined independently of the policy being proved optimal, the reduction to the known Nudge-M policy for d=2 is presented as a consistency check rather than a derivation step, and no parameters are fitted to data then relabeled as predictions. Prior citations supply context but are not load-bearing for the new proof. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The result rests on the definition of the Contextual CounterBoost class and the light-tailed M/G/1 assumption, which are not enumerated here.

pith-pipeline@v0.9.1-grok · 5824 in / 1159 out tokens · 14795 ms · 2026-06-28T11:28:24.547735+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 8 canonical work pages

  1. [1]

    J. C. Gittins, K. D. Glazebrook, and R. R. Weber,Multi-Armed Bandit Allocation Indices, 2nd ed. Chichester, UK: John Wiley & Sons, 2011. [Online]. Available: https://onlinelibrary.wiley.com/doi/ book/10.1002/9780470980033

    work page doi:10.1002/9780470980033 2011

  2. [2]

    On the Gittins index in the M/G/1 queue,

    S. Aalto, U. Ayesta, and R. Righter, “On the Gittins index in the M/G/1 queue,”Queueing Systems, vol. 63, no. 1–4, pp. 437–458, 2009. [Online]. Available: https://doi.org/10.1007/s11134-009-9141-x

    work page doi:10.1007/s11134-009-9141-x 2009

  3. [3]

    The Gittins policy in the M/G/1 queue,

    Z. Scully and M. Harchol-Balter, “The Gittins policy in the M/G/1 queue,” inProceedings of the 19th International Symposium on Modeling and Optimization in Mobile, Ad hoc, and Wireless Networks (WiOpt). IEEE, 2021. [Online]. Available: https://doi.org/10.23919/WiOpt52881.2021.9586214

    work page doi:10.23919/wiopt52881.2021.9586214 2021

  4. [4]

    The impact of the service discipline on delay asymptotics,

    S. Borst, O. Boxma, R. Núñez-Queija, and A. Zwart, “The impact of the service discipline on delay asymptotics,”Performance Evaluation, vol. 54, no. 2, pp. 175–206, 2003

    2003

  5. [5]

    Tails in scheduling,

    O. Boxma and B. Zwart, “Tails in scheduling,”SIGMETRICS Perform. Eval. Rev., vol. 34, no. 4, p. 13–20, mar 2007. [Online]. Available: https://doi.org/10.1145/1243401.1243406

    work page doi:10.1145/1243401.1243406 2007

  6. [6]

    Strongly tail-optimal scheduling in the light-tailed M/G/1,

    G. Yu and Z. Scully, “Strongly tail-optimal scheduling in the light-tailed M/G/1,”SIGMETRICS Perform. Eval. Rev., vol. 52, no. 1, p. 5–6, Jun. 2024. [Online]. Available: https: //doi.org/10.1145/3673660.3655084

    work page doi:10.1145/3673660.3655084 2024

  7. [7]

    Exponential approximations for tail probabilities in queues, i: Waiting times

    J. Abate, L. Choudhury, and W. Whitt, “Exponential approximations for tail probabilities in queues, i: Waiting times.”Operations Research, vol. 43, pp. 885–901, 1995

    1995

  8. [8]

    Is tail-optimal scheduling possible?

    A. Wierman and B. Zwart, “Is tail-optimal scheduling possible?”Operations Research, vol. 60, no. 5, pp. 1249–1257, 2012. [Online]. Available: http://www.jstor.org/stable/23323693

    arXiv 2012

  9. [9]

    Nudge: Stochastically improving upon FCFS,

    I. Grosof, K. Yang, Z. Scully, and M. Harchol-Balter, “Nudge: Stochastically improving upon FCFS,”Proc. ACM Meas. Anal. Comput. Syst., vol. 5, no. 2, jun 2021. [Online]. Available: https://doi.org/10.1145/3460088

    work page doi:10.1145/3460088 2021

  10. [10]

    On the stochastic and asymptotic improvement of first-come first-served and nudge scheduling,

    B. Van Houdt, “On the stochastic and asymptotic improvement of first-come first-served and nudge scheduling,”Proc. ACM Meas. Anal. Comput. Syst., vol. 6, no. 3, pp. 1–22, dec 2022. [Online]. Available: https://doi.org/10.1145/3570610

    work page doi:10.1145/3570610 2022

  11. [11]

    Tail optimality and performance analysis of the Nudge- M scheduling algorithm,

    N. Charlet and B. Van Houdt, “Tail optimality and performance analysis of the Nudge- M scheduling algorithm,”arXiv preprint arXiv:2403.06588, 2024. [Online]. Available: https: //arxiv.org/abs/2403.06588

    Pith/arXiv arXiv 2024

  12. [12]

    Latouche and V

    G. Latouche and V. Ramaswami,Introduction to Matrix Analytic Methods and stochastic modeling. Philadelphia: SIAM, 1999

    1999

  13. [13]

    Harchol-Balter,Performance Modeling and Design of Computer Systems: Queueing Theory in Action

    M. Harchol-Balter,Performance Modeling and Design of Computer Systems: Queueing Theory in Action. New York, NY, USA: Cambridge University Press, 2013. [Online]. Available: https://doi.org/10.1017/CBO9781139226424

    work page doi:10.1017/cbo9781139226424 2013

  14. [14]

    On queues in heavy traffic,

    J. F. C. Kingman, “On queues in heavy traffic,”Journal of the Royal Statistical Society. Series B (Methodological), vol. 24, no. 2, pp. 383–392, 1962. [Online]. Available: http: //www.jstor.org/stable/2984229 21

    arXiv 1962