Queueing Systems

, Volume 64, Issue 1, pp 5–48 | Cite as

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

  • Varun Gupta
  • Mor Harchol-Balter
  • J. G. Dai
  • Bert Zwart
Article

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.

Keywords

Multi-server systems M/G/K Two-moment approximation Inapproximability Higher moments 

Mathematics Subject Classification (2000)

60K25 68M20 90B22 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Varun Gupta
    • 1
  • Mor Harchol-Balter
    • 1
  • J. G. Dai
    • 2
  • Bert Zwart
    • 3
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  2. 2.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.CWIAmsterdamThe Netherlands

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