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On the inapproximability of M/G/K: why two moments of job size distribution are not enough

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Abstract

The M/G/K queueing system is one of the oldest models for multiserver systems and has been the topic of performance papers for almost half a century. However, even now, only coarse approximations exist for its mean waiting time. All the closed-form (nonnumerical) approximations in the literature are based on (at most) the first two moments of the job size distribution. In this paper we prove that no approximation based on only the first two moments can be accurate for all job size distributions, and we provide a lower bound on the inapproximability ratio, which we refer to as “the gap.” This is the first such result in the literature to address “the gap.” The proof technique behind this result is novel as well and combines mean value analysis, sample path techniques, scheduling, regenerative arguments, and asymptotic estimates. Finally, our work provides insight into the effect of higher moments of the job size distribution on the mean waiting time.

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Authors and Affiliations

  1. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

    Varun Gupta & Mor Harchol-Balter

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    J. G. Dai

  3. CWI, Amsterdam, The Netherlands

    Bert Zwart

Authors
  1. Varun Gupta
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  2. Mor Harchol-Balter
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  3. J. G. Dai
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  4. Bert Zwart
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Corresponding author

Correspondence to Varun Gupta.

Additional information

Bert Zwart is also affiliated with VU University (Amsterdam, The Netherlands), Eurandom (Eindhoven, The Netherlands) and Georgia Institute of Technology (Atlanta, GA, USA).

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Gupta, V., Harchol-Balter, M., Dai, J.G. et al. On the inapproximability of M/G/K: why two moments of job size distribution are not enough. Queueing Syst 64, 5–48 (2010). https://doi.org/10.1007/s11134-009-9133-x

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  • DOI: https://doi.org/10.1007/s11134-009-9133-x

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