pith. sign in

arxiv: 2510.11015 · v2 · submitted 2025-10-13 · 🧮 math.PR · cs.PF

A new 1/(1-rho)-scaling bound for multiserver queues via a leave-one-out technique

Yige Hong This is my paper

Pith reviewed 2026-05-18 08:11 UTC · model grok-4.3

classification 🧮 math.PR cs.PF
keywords G/G/n queuemultiserver queuequeue length bound1/(1-rho) scalingleave-one-out couplinglight-tailed distributionsheavy trafficheterogeneous servers
0
0 comments X

The pith

A leave-one-out coupling produces a universal O(1/(1-ρ)) bound on G/G/n queue length with a tighter explicit constant.

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

This paper establishes a new upper bound on the steady-state queue length for the G/G/n multiserver queue that scales as O(1/(1-ρ)) for any utilization ρ less than one. The bound is derived under light-tailed interarrival and service times and improves prior results by providing a more interpretable and often smaller leading constant. The proof introduces a modified queue, uses the stationarity of a quadratic test function, and applies a novel leave-one-out coupling to keep the argument simple. A reader would care because the O(1/(1-ρ)) order is tight in classical heavy-traffic, Halfin-Whitt, and nondegenerate-slowdown regimes, so sharper constants help practical performance analysis. The same method extends directly to the case of fully heterogeneous service times across servers.

Core claim

We present a new universal bound of order O(1/(1-ρ)) for the G/G/n queue. Our bound, while restricted to the light-tailed case and the first moment of the queue length, has a more interpretable and often tighter leading constant. Our proof is relatively simple, utilizing a modified G/G/n queue, the stationarity of a quadratic test function, and a novel leave-one-out coupling technique. We also extend our method to G/G/n queues with fully heterogeneous service-time distributions.

What carries the argument

A leave-one-out coupling technique applied to a modified G/G/n queue in which a quadratic test function is stationary.

Load-bearing premise

The quadratic test function remains stationary in the modified multiserver queue and the leave-one-out coupling holds when interarrival and service times are light-tailed.

What would settle it

An exact calculation or high-precision simulation of the steady-state expected queue length for a concrete light-tailed G/G/n instance that exceeds the proposed bound by more than the claimed constant factor.

Figures

Figures reproduced from arXiv: 2510.11015 by Yige Hong.

Figure 1
Figure 1. Figure 1: An illustration of 𝐺 𝐼/𝐺 𝐼/𝑛 queue, where each cycle denotes a server, and each rectangle denotes a job. In contrast to the single-server case, finding a similarly simple and accurate upper bound for the 𝐺 𝐼/𝐺 𝐼/𝑛 queue is a much greater challenge. This complexity arises from a rich set of combinations of two basic scalings: the heavy-traffic scaling, where the load 𝜌 ↑ 1, and the many-server scaling, wher… view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of the 𝑗-th leave-one-out system, where the 𝑗-th server does not take jobs from the queue. An attempt to answer this question inspires a simple proof of the asymptotic indepen￾dence. We consider the counterfactual queue length 𝑄e− 𝑗 if 𝑗-th server had not existed in the modified 𝐺 𝐼/𝐺 𝐼/𝑛 queue (see Section 6.1 for the precise definition). Crucially, the in￾dicator 𝟙  𝑄e− 𝑗 = 0 [PITH_FULL… view at source ↗
read the original abstract

Bounding the queue length in a multiserver queue is a central challenge in queueing theory. Even for the classical $G/G/n$ queue with homogeneous servers, it is highly non-trivial to derive a simple and accurate bound for the steady-state queue length that holds for all problem parameters. A recent breakthrough by Li and Goldberg (2025) establishes a universal bound of order $O(1/(1-\rho))$ that holds for any load $\rho < 1$ and any number of servers $n$. This order is tight in many well-known scaling regimes, including classical heavy-traffic, Halfin-Whitt and Nondegenerate-Slowdown. However, their bounds entail large constant factors and a highly intricate proof, suggesting room for further improvement. In this paper, we present a new universal bound of order $O(1/(1-\rho))$ for the $G/G/n$ queue. Our bound, while restricted to the light-tailed case and the first moment of the queue length, has a more interpretable and often tighter leading constant. Our proof is relatively simple, utilizing a modified $G/G/n$ queue, the stationarity of a quadratic test function, and a novel leave-one-out coupling technique. Finally, we also extend our method to $G/G/n$ queues with fully heterogeneous service-time distributions.

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

1 major / 2 minor

Summary. The manuscript claims a new universal bound of order O(1/(1-ρ)) on the steady-state expected queue length for the G/G/n queue (homogeneous servers) under light-tailed assumptions on interarrival and service times. The bound is asserted to have a more interpretable and often tighter leading constant than the Li-Goldberg (2025) result. The proof constructs a modified G/G/n process, invokes stationarity of a quadratic test function (E[generator applied to test function] = 0), and employs a leave-one-out coupling to relate the original and modified systems. The approach is extended to G/G/n queues with fully heterogeneous service-time distributions.

Significance. If the central derivation holds, the result is significant for queueing theory: it supplies a simpler proof for a tight scaling bound that is known to be sharp in heavy-traffic, Halfin-Whitt, and Nondegenerate-Slowdown regimes. The leave-one-out coupling technique is a novel methodological contribution that may apply more broadly, and the heterogeneous-server extension increases practical relevance. The work improves interpretability of the leading constant relative to the recent breakthrough result while preserving universality in the light-tailed regime.

major comments (1)
  1. [§3] §3 (construction of modified queue and quadratic test function): the stationarity argument E[𝒜V(Q)] = 0 is applied to the modified process whose service distribution for one server is altered while keeping overall load ρ fixed. The generator calculation must explicitly cancel the modified drift terms plus the leave-one-out coupling error; if the light-tailed tail parameter only controls moments up to order 2+ε, an uncontrolled remainder that grows with n or with the tail index could appear in the final constant. This step is load-bearing for the claimed improvement over Li-Goldberg.
minor comments (2)
  1. [Abstract] Abstract and §1: the phrase 'often tighter leading constant' is stated without a concrete numerical comparison or reference to a table/figure; adding one explicit example (e.g., M/M/n with ρ=0.9) would make the claim easier to verify.
  2. [§2] Notation in §2: the precise definition of the modified service process (which server is altered, how the distribution is changed) should be stated in a displayed equation rather than inline text to avoid ambiguity when the leave-one-out coupling is introduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful review and the recommendation for minor revision. We are pleased that the significance of the leave-one-out technique and the heterogeneous extension is recognized. We address the major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (construction of modified queue and quadratic test function): the stationarity argument E[𝒜V(Q)] = 0 is applied to the modified process whose service distribution for one server is altered while keeping overall load ρ fixed. The generator calculation must explicitly cancel the modified drift terms plus the leave-one-out coupling error; if the light-tailed tail parameter only controls moments up to order 2+ε, an uncontrolled remainder that grows with n or with the tail index could appear in the final constant. This step is load-bearing for the claimed improvement over Li-Goldberg.

    Authors: We appreciate the referee highlighting the importance of the generator calculation. In Section 3 the modified process alters the service distribution of one server while preserving overall load ρ. The quadratic test function V is chosen so that the modified drift terms cancel exactly under the generator 𝒜. The leave-one-out coupling error is bounded using the light-tailed assumption, which guarantees moment generating functions finite in a neighborhood of zero and thus all moments (stronger than order 2+ε). This controls all remainder terms uniformly in n without growth in the tail index, so they are absorbed into the leading constant. The cancellations and bounds are carried out explicitly in the proof, confirming the improvement over Li-Goldberg. No revision is required. revision: no

Circularity Check

0 steps flagged

Derivation is self-contained via coupling and test-function stationarity

full rationale

The paper establishes the O(1/(1-ρ)) bound for the G/G/n queue through a modified queue construction, application of the generator to a quadratic test function under stationarity, and a leave-one-out coupling argument. These steps are presented as direct consequences of the light-tailed moment assumptions and the coupling construction itself; no parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is imported from prior work by the same author. The derivation therefore does not reduce to its own inputs by construction and remains independent of the specific numerical values being bounded.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard queueing assumptions plus the new coupling construction; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Light-tailed interarrival and service time distributions
    Explicitly required for the bound and proof technique to apply (abstract).
  • domain assumption Stationarity of quadratic test function in the modified queue
    Invoked as the key step that converts the coupling into the bound (abstract, proof description).

pith-pipeline@v0.9.0 · 5773 in / 1422 out tokens · 41390 ms · 2026-05-18T08:11:20.401183+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

82 extracted references · 82 canonical work pages · 2 internal anchors

  1. [1]

    and RAMANAN, K

    AGHAJANI, R. and RAMANAN, K. (2020). The limit of stationary distributions of many-server queues in the Halfin–Whitt regime.Math. Oper. Res.451016-1055

    work page 2020

  2. [2]

    (2003).Applied Probability and Queues

    ASMUSSEN, S. (2003).Applied Probability and Queues. Springer-Verlag New York

    work page 2003

  3. [3]

    ATAR, R. (2012). A diffusion regime with nondegenerate slowdown.Oper. Res.60490-500

    work page 2012

  4. [4]

    and SOLOMON, N

    ATAR, R. and SOLOMON, N. (2011). Asymptotically optimal interruptible service policies for scheduling jobs in a diffusion regime with nondegenerate slowdown.Queueing Syst.69217–235

    work page 2011

  5. [5]

    BOROVKOV, A. A. (1965). Some limit theorems in the theory of mass service, II multiple channels systems. Theory Probab. Appl.10375-400

    work page 1965

  6. [6]

    and REIMAN, M

    BORST, S., MANDELBAUM, A. and REIMAN, M. I. (2004). Dimensioning large call centers.Oper. Res.52 17-34

    work page 2004

  7. [7]

    BRAVERMAN, A. (2017). Stein’s method for steady-state diffusion approximations.arXiv:1704.08398 [math.PR]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  8. [8]

    High order steady-state diffusion approximation of the Erlang-C system

    BRAVERMAN, A. and DAI, J. G. (2016). High order steady-state diffusion approximation of the Erlang-C system.arxiv:1602.02866 [math.PR]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  9. [9]

    and DAI, J

    BRAVERMAN, A. and DAI, J. (2017). Stein’s method for steady-state diffusion approximations of 𝑀/Ph/𝑛+ 𝑀systems.Ann. Appl. Probab.27550-581

    work page 2017

  10. [10]

    BRAVERMAN, A., DAI, J. G. and FENG, J. (2017). Stein’s method for steady-state diffusion approximations: an introduction through the Erlang-A and Erlang-C models.Stoch. Syst.6301–366

    work page 2017

  11. [11]

    BRAVERMAN, A., DAI, J. G. and MIYAZAWA, M. (2017). Heavy traffic approximation for the stationary distribution of a generalized Jackson network: The BAR approach.Stoch. Syst.7143–196

    work page 2017

  12. [12]

    and MIYAZAWA, M

    BRAVERMAN, A., DAI, J. and MIYAZAWA, M. (2023). The BAR-approach for multiclass queueing networks with SBP service policies.arXiv preprint arXiv:2302.05791

    work page arXiv 2023

  13. [13]

    BRAVERMAN, A., DAI, J. G. and FANG, X. (2024). High-order steady-state diffusion approximations.Oper. Res.72604-616

    work page 2024

  14. [14]

    BRAVERMAN, A., DAI, J. G. and MIYAZAWA, M. (2025). The BAR approach for multiclass queueing networks with SBP service policies.Stoch. Syst.151-49

    work page 2025

  15. [15]

    and SCULLY, Z

    BRAVERMAN, A. and SCULLY, Z. (2024). Stein’s method and general clocks: diffusion approximation of the 𝐺/𝐺/1workload.arXiv:2407.12716 [math.PR]

    work page arXiv 2024

  16. [16]

    CHANG, C.-S., THOMAS, J. A. and KIANG, S.-H. (1994). On the stability of open networks: A unified approach by stochastic dominance.Queueing Syst.15239–260

    work page 1994

  17. [17]

    DAI, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models.Ann. Appl. Probab.549 – 77

    work page 1995

  18. [18]

    G., DIEKER, A

    DAI, J. G., DIEKER, A. and GAO, X. (2014). Validity of heavy-traffic steady-state approximations in many-server queues with abandonment.Queueing Syst.781–29

    work page 2014

  19. [19]

    G., GLYNN, P

    DAI, J. G., GLYNN, P. and XU, Y. (2025). Asymptotic product-form steady-state for generalized Jackson networks in multi-scale heavy traffic.arXiv:2304.01499 [math.PR]

    work page arXiv 2025

  20. [20]

    G., GUANG, J

    DAI, J. G., GUANG, J. and XU, Y. (2024). Steady-state convergence of the continuous-time JSQ system with general distributions in heavy traffic.ACM SIGMETRICS Perform. Evaluation Rev.5239–41

    work page 2024

  21. [21]

    G., HE, S

    DAI, J. G., HE, S. and TEZCAN, T. (2010). Many-server diffusion limits for G/Ph/n+GI queues.Ann. Appl. Probab.201854 – 1890

    work page 2010

  22. [22]

    DAI, J. G. and HUO, D. (2024). Asymptotic product-form steady-state for multiclass queueing networks with SBP service policies in multi-scale heavy traffic.arXiv:2403.04090 [math.PR]

    work page arXiv 2024

  23. [23]

    DAI, J. G. and MEYN, S. P. (1995). Stability and convergence of moments for multiclass queueing networks via fluid limit models.IEEE Trans. Autom. Control401889-1904

    work page 1995

  24. [24]

    and CHEN, Y

    DING, L. and CHEN, Y. (2020). Leave-one-out approach for matrix completion: Primal and dual analysis. IEEE Trans. Inf. Theory667274-7301

    work page 2020

  25. [25]

    DOWNEY, P. J. (1991). Bounding synchronization overhead for parallel iteration.ORSA J. Comput.3 288-298

    work page 1991

  26. [26]

    and SRIKANT, R

    ERYILMAZ, A. and SRIKANT, R. (2012). Asymptotically tight steady-state queue length bounds implied by drift conditions.Queueing Syst.72311–359

    work page 2012

  27. [27]

    and GOLDBERG, D

    GAMARNIK, D. and GOLDBERG, D. A. (2013). Steady-state GI/GI/n queue in the Halfin–Whitt regime. Ann. Appl. Probab.232382–2419

    work page 2013

  28. [28]

    and MOM ˇCILOVI ´C, P

    GAMARNIK, D. and MOM ˇCILOVI ´C, P. (2008). Steady-state analysis of a multiserver queue in the Halfin- Whitt regime.Adv. Appl. Probab.40548–577

    work page 2008

  29. [29]

    and STOLYAR, A

    GAMARNIK, D. and STOLYAR, A. L. (2012). Multiclass multiserver queueing system in the Halfin–Whitt heavy traffic regime: asymptotics of the stationary distribution.Queueing Syst.7125–51

    work page 2012

  30. [30]

    and REIMAN, M

    GARNETT, O., MANDELBAUM, A. and REIMAN, M. (2002). Designing a call center with impatient customers.Manuf. Serv. Oper. Manag.4208-227. 30

    work page 2002

  31. [31]

    and SCHELLER-WOLF, A

    GROSOF, I., HARCHOL-BALTER, M. and SCHELLER-WOLF, A. (2022). WCFS: a new framework for analyzing multiserver systems.Queueing Syst.102143–174

    work page 2022

  32. [32]

    and SCHELLER-WOLF, A

    GROSOF, I., HONG, Y., HARCHOL-BALTER, M. and SCHELLER-WOLF, A. (2024). The RESET and MARC techniques, with application to multiserver-job analysis.ACM SIGMETRICS Perform. Evaluation Rev. 516–7

    work page 2024

  33. [33]

    and DAI, J

    GUANG, J., CHEN, X. and DAI, J. G. (2024). Uniform moment bounds for generalized Jackson net- works in multiscale heavy traffic.Math. Oper. Res.0null. Published online ahead of print, DOI: 10.1287/moor.2024.0408

    work page doi:10.1287/moor.2024.0408 2024

  34. [34]

    and MANDELBAUM, A

    GURVICH, I., HUANG, J. and MANDELBAUM, A. (2014). Excursion-based universal approximations for the Erlang-A queue in steady-state.Math. Oper. Res.39325-373

    work page 2014

  35. [35]

    and WHITT, W

    HALFIN, S. and WHITT, W. (1981). Heavy-traffic limits for queues with many exponential servers.Oper. Res.29567–588

    work page 1981

  36. [36]

    HOKSTAD, P. (1985). Relations for the workload of the GI/G/s queue.Advances in Applied Probability17 887–904

    work page 1985

  37. [37]

    and SCULLY, Z

    HONG, Y. and SCULLY, Z. (2024). Performance of the Gittins policy in the G/G/1 and G/G/k, with and without setup times.Perform. Eval.163102377

    work page 2024

  38. [38]

    and WANG, W

    HONG, Y. and WANG, W. (2024). Sharp waiting-time bounds for multiserver jobs.Stoch. Syst.14455-478

    work page 2024

  39. [39]

    IGLEHART, D. L. and WHITT, W. (1970a). Multiple channel queues in heavy traffic. II: sequences, networks, and batches.Adv. Appl. Probab.2355–369

    work page

  40. [40]

    IGLEHART, D. L. and WHITT, W. (1970b). Multiple channel queues in heavy traffic. I.Adv. Appl. Probab.2 150–177

    work page

  41. [41]

    JANSSEN, A. J. E. M.,VANLEEUWAARDEN, J. S. H. and ZWART, B. (2008). Corrected asymptotics for a multi-server queue in the Halfin-Whitt regime.Queueing Syst.58261–301

    work page 2008

  42. [42]

    JANSSEN, A. J. E. M.,VANLEEUWAARDEN, J. S. H. and ZWART, B. (2011). Refining square-root safety staffing by expanding Erlang C.Oper. Res.591512-1522

    work page 2011

  43. [43]

    and MOM ˇCILOVI ´C, P

    JELENKOVI ´C, P., MANDELBAUM, A. and MOM ˇCILOVI ´C, P. (2004). Heavy traffic limits for queues with many deterministic servers.Queueing Syst.4753–69. 2004,

    work page 2004

  44. [44]

    and XU, X

    JIN, X., PANG, G., XU, L. and XU, X. (2025). An approximation to the invariant measure of the limiting diffusion of G/Ph/n + GI queues in the Halfin–Whitt regime and related asymptotics.Math. Oper. Res. 50783-812

    work page 2025

  45. [45]

    and STADJE, W

    KELLA, O. and STADJE, W. (2006). Superposition of renewal processes and an application to multi-server queues.Statistics & Probability Letters761914-1924

    work page 2006

  46. [46]

    KENNEDY, D. P. (1972). Rates of convergence for queues in heavy traffic. II: Sequences of queueing systems. Adv. Appl. Probab.4382–391

    work page 1972

  47. [47]

    KINGMAN, J. F. C. (1962a). Some inequalities for the queue GI/G/1.Biometrika49315–324

    work page

  48. [48]

    KINGMAN, J. F. C. (1962b). On queues in heavy traffic.J. Roy. Stat. Soc. B Met.24383–392

    work page

  49. [49]

    KINGMAN, J. F. C. (1970). Inequalities in the theory of queues.J. Roy. Stat. Soc. B Met.32102-110

    work page 1970

  50. [50]

    KÖLLERSTRÖM, J. (1974). Heavy traffic theory for queues with several servers. I.J. Appl. Probab.11 544–552

    work page 1974

  51. [51]

    KÖLLERSTRÖM, J. (1979). Heavy traffic theory for queues with several servers. II.J. Appl. Probab.16 393–401

    work page 1979

  52. [52]

    LACHENBRUCH, P. A. and MICKEY, M. R. (1968). Estimation of error rates in discriminant analysis. Technometrics101–11

    work page 1968

  53. [53]

    and GOLDBERG, D

    LI, Y. and GOLDBERG, D. A. (2025). Simple and explicit bounds for multiserver queues with 1/(1−𝜌) scaling.Math. Oper. Res.50813-837

    work page 2025

  54. [54]

    LORDEN, G. (1970). On excess over the boundary.Ann. Math. Statist.41520–527

    work page 1970

  55. [55]

    LOULOU, R. (1973). Multi-channel queues in heavy traffic.J. Appl. Probab.10769–777

    work page 1973

  56. [56]

    MANDELBAUM, A., MASSEY, W. A. and REIMAN, M. I. (1998). Strong approximations for Markovian service networks.Queueing Syst.30149–201

    work page 1998

  57. [57]

    and MOM ˇCILOVI ´C, P

    MANDELBAUM, A. and MOM ˇCILOVI ´C, P. (2008a). Queues with many servers: The virtual waiting-time process in the QED regime.Math. Oper. Res.33561-586

    work page

  58. [58]

    and MOM ˇCILOVI ´C, P

    MANDELBAUM, A. and MOM ˇCILOVI ´C, P. (2008b). Queues with many servers: The virtual waiting-time process in the qed regime.Math. Oper. Res.33561-586

    work page

  59. [59]

    MEYN, S. P. and TWEEDIE, R. L. (1993). Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes.Adv. Appl. Probab.25518–548

    work page 1993

  60. [60]

    MIYAZAWA, M. (1994a). Decomposition formulas for single server queues with vacations : A unified approach by the rate conservation law.Commun. Stat. Stoch. Models10389–413

    work page

  61. [61]

    MIYAZAWA, M. (1994b). Rate conservation laws: A survey.Queueing Syst.151–58

    work page

  62. [62]

    MIYAZAWA, M. (2015). Diffusion approximation for stationary analysis of queues and their networks: A review.J. Operat. Res. Soc. Japan58104–148. A NEW1/(1−𝜌)-SCALING BOUND FOR MULTISERVER QUEUES31

    work page 2015

  63. [63]

    MORI, M. (1975). Some bounds for queues.J. Operat. Res. Soc. Japan18152–181

    work page 1975

  64. [64]

    NAGAEV, S. V. (1970a). On the speed of convergence in a boundary problem. I.Theory Probab. Appl.15 163–186

    work page

  65. [65]

    NAGAEV, S. V. (1970b). On the speed of convergence in a boundary problem. II.Theory Probab. Appl.15 403–429

    work page

  66. [66]

    PUHALSKII, A. A. and REIMAN, M. I. (2000). The multiclass GI/PH/N queue in the Halfin-Whitt regime. Adv. Appl. Probab.32564–595

    work page 2000

  67. [67]

    and THEJAMAGULURI, S

    RAJJHUNJHUNWALA, P., HURTADO-LANGE, D. and THEJAMAGULURI, S. (2024). Exponential tail bounds on queues: A confluence of non-asymptotic heavy traffic and large deviations.ACM SIGMETRICS Perform. Evaluation Rev.5118–19

    work page 2024

  68. [68]

    REED, J. (2009). The G/GI/N queue in the Halfin–Whitt regime.The Annals of Applied Probability192211 – 2269

    work page 2009

  69. [69]

    SCHELLER-WOLF, A. (2003). Necessary and sufficient conditions for delay moments in FIFO multiserver queues with an application comparing s slow servers with one fast one.Oper. Res.51748-758

    work page 2003

  70. [70]

    and SIGMAN, K

    SCHELLER-WOLF, A. and SIGMAN, K. (1997). Delay moments for FIFO GI/GI/s queues.Queueing Syst. 2577–95

    work page 1997

  71. [71]

    and VESILO, R

    SCHELLER-WOLF, A. and VESILO, R. (2006). Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues.Queueing Syst.54221–232

    work page 2006

  72. [72]

    and HARCHOL-BALTER, M

    SCULLY, Z., GROSOF, I. and HARCHOL-BALTER, M. (2020). The Gittins policy is nearly optimal in the M/G/𝑘under extremely general conditions.Proc. ACM Meas. Anal. Comput. Syst.4

    work page 2020

  73. [73]

    and SRIDHARAN, K

    SHALEV-SHWARTZ, S., SHAMIR, O., SREBRO, N. and SRIDHARAN, K. (2010). Learnability, stability and uniform convergence.J. Mach. Learn. Res.112635–2670

    work page 2010

  74. [74]

    and SCHELLER-WOLF, A

    VESILO, R. and SCHELLER-WOLF, A. (2013). Delay moment bounds for multiserver queues with infinite variance service times.INFOR51161–174

    work page 2013

  75. [75]

    and HARCHOL-BALTER, M

    WANG, W., XIE, Q. and HARCHOL-BALTER, M. (2021). Zero queueing for multi-server jobs.Proc. ACM Meas. Anal. Comput. Syst.5

    work page 2021

  76. [76]

    and SRIKANT, R

    WANG, W., HARCHOL-BALTER, M., JIANG, H., SCHELLER-WOLF, A. and SRIKANT, R. (2019). Delay asymptotics and bounds for multitask parallel jobs.Queueing Syst.91207–239

    work page 2019

  77. [77]

    WHITT, W. (1992). Understanding the efficiency of multi-server service systems.Manage. Sci.38708-723

    work page 1992

  78. [78]

    (2002).Stochastic-Process Limits, 2002 ed.Springer Series in Operations Research and Financial Engineering

    WHITT, W. (2002).Stochastic-Process Limits, 2002 ed.Springer Series in Operations Research and Financial Engineering. Springer, New York, NY

    work page 2002

  79. [79]

    WOLFF, R. W. (1987). Upper bounds on work in system for multichannel queues.J. Appl. Probab.24 547–551

    work page 1987

  80. [80]

    modified𝐺 𝐼/𝐺 𝐼/𝑛queue

    WORTHINGTON, D. (2009). Reflections on queue modelling from the last 50 years.J. Oper. Res. Soc.60 S83–S92. APPENDIX A: BOUNDED MEAN RESIDUAL TIME IMPLIES EXPONENTIAL TAIL In this section, we show that if a distribution has a bounded mean residual time given any age, it must have an exponentially decaying tail, so all of its moments must be finite. This f...

    work page 2009

Showing first 80 references.