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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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References

  1. Adan, I.J.B.F., Resing, J.: Queueing Theory. Eindhoven University of Technology (2002)

  2. Barford, P., Crovella, M.: Generating representative web workloads for network and server performance evaluation. In: Proceedings of ACM SIGMETRICS/Performance’98, pp. 151–160 (1998)

  3. Bertsimas, D., Natarajan, K.: A semidefinite optimization approach to the steady-state analysis of queueing systems. Queueing Syst. 56(1), 27–39 (2007)

    Article  Google Scholar 

  4. Borovkov, A.A.: Stochastic Processes in Queueing Theory. Nauka, Moscow (1972)

    Google Scholar 

  5. Burman, D.Y., Smith, D.R.: A light-traffic theorem for multi-server queues. Math. Oper. Res. 8, 15–25 (1983)

    Article  Google Scholar 

  6. Cosmetatos, G.P.: Some approximate equilibrium results for the multiserver queue (M/G/r). Oper. Res. Q. 27, 615–620 (1976)

    Article  Google Scholar 

  7. Daley, D.J.: Some results for the mean waiting-time and workload in GI/GI/k queues. In: Dshalalow, J.H. (ed.) Frontiers in Queueing: Models and Applications in Science and Engineering, pp. 35–59. CRC Press, Boca Raton (1997)

    Google Scholar 

  8. Daley, D.J., Rolski, T.: Some comparability results for waiting times in single- and many-server queues. J. Appl. Probab. 21, 887–900 (1984)

    Article  Google Scholar 

  9. de Smit, J.H.A.: A numerical solution for the multiserver queue with hyper-exponential service times. Oper. Res. Lett. 2(5), 217–224 (1983)

    Article  Google Scholar 

  10. de Smit, J.H.A.: The queue GI/M/s with customers of different types or the queue GI/H m /s. Adv. Appl. Probab. 15(2), 392–419 (1983)

    Article  Google Scholar 

  11. de Smit, J.H.A.: The queue GI/H m /s in continuous time. J. Appl. Probab. 22(1), 214–222 (1985)

    Article  Google Scholar 

  12. Downy, A., Harchol-Balter, M.: Exploiting process lifetime distributions for dynamic load balancing. ACM Trans. Comput. Syst. 15(3), 253–285 (1997)

    Article  Google Scholar 

  13. Foss, S., Korshunov, D.: Heavy tails in multi-server queue. Queueing Syst. 52(1), 31–48 (2006)

    Article  Google Scholar 

  14. Gans, N., Koole, G., Mandelbaum, A.: Telephone call centers: tutorial, review, and research prospects. Manuf. Serv. Oper. Manag. 5, 79–141 (2003)

    Article  Google Scholar 

  15. Gupta, V., Dai, J.G., Harchol-Balter, M., Zwart, B.: The effect of higher moments of job size distribution on the performance of an M/G/K queueing system. Technical Report CMU-CS-08-106, School of Computer Science, Carnegie Mellon University (2008)

  16. Gupta, V., Harchol-Balter, M., Scheller-Wolf, A., Yechiali, U.: Fundamental characteristics of queues with fluctuating load. In: Proceedings of ACM SIGMETRICS, pp. 203–215 (2006)

  17. Harchol-Balter, M., Schroeder, B.: Evaluation of task assignment policies for supercomputing servers. In: Proceedings of 9th IEEE Symposium on High Performance Distributed Computing (HPDC ’00), (2001)

  18. van Hoorn, M.H., Tijms, H.C., Federgruen, A.: Approximations for the steady-state probabilities in the M/G/c queue. Adv. Appl. Probab. 13, 186–206 (1981)

    Article  Google Scholar 

  19. Hokstad, P.: Approximations for the M/G/m queue. Oper. Res. 26(3), 510–523 (1978)

    Article  Google Scholar 

  20. Hokstad, P.: The steady state solution of the M/K 2/m queue. Adv. Appl. Probab. 12(3), 799–823 (1980)

    Article  Google Scholar 

  21. Kimura, T.: Diffusion approximation for an M/G/m queue. Oper. Res. 31, 304–321 (1983)

    Article  Google Scholar 

  22. Kimura, T.: Approximations for multi-server queues: system interpolations. Queueing Syst. 17(3–4), 347–382 (1994)

    Article  Google Scholar 

  23. Kingman, J.F.C.: Inequalities in the theory of queues. J. R. Stat. Soc. 32(1), 102–110 (1970)

    Google Scholar 

  24. Kleinrock, L.: Queueing Systems, Volume I: Theory. Wiley-Interscience, New York (1975)

    Google Scholar 

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

    Article  Google Scholar 

  26. Lee, A.M., Longton, P.A.: Queueing process associated with airline passenger check-in. Oper. Res. Q. 10, 56–71 (1959)

    Article  Google Scholar 

  27. Lee, H.L., Cohen, M.A.: A note on the convexity of performance measures of M/M/c queueing systems. J. Appl. Probab. 20(4), 920–923 (1983)

    Article  Google Scholar 

  28. Ma, B.N.W., Mark, J.W.: Approximation of the mean queue length of an M/G/c queueing system. Oper. Res. 43(1), 158–165 (1995)

    Article  Google Scholar 

  29. Miyazawa, M.: Approximation of the queue-length distribution of an M/GI/s queue by the basic equations. J. Appl. Probab. 23, 443–458 (1986)

    Article  Google Scholar 

  30. Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley Series in Probability and Statistics. Wiley, Chichester (2002)

    Google Scholar 

  31. Nozaki, S.A., Ross, S.M.: Approximations in finite-capacity multi-server queues with Poisson arrivals. J. Appl. Probab. 15(4), 826–834 (1978)

    Article  Google Scholar 

  32. Boxma, J.C.O., Huffels, N.: Approximations in the mean waiting time in an M/G/s queueing system. Oper. Res. 27, 1115–1127 (1979)

    Article  Google Scholar 

  33. Ross, S.M.: Stochastic Processes, 2nd edn. Wiley, New York (1996)

    Google Scholar 

  34. Scheller-Wolf, A., Sigman, K.: New bounds for expected delay in FIFO GI/GI/c queues. Queueing Syst. 26(1–2), 169–186 (1997)

    Article  Google Scholar 

  35. Scheller-Wolf, A., Vesilo, R.: Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues. Queueing Syst. 54(3), 221–232 (2006)

    Article  Google Scholar 

  36. Stoyan, D.: Approximations for M/G/s queues. Math. Operationsforsch. Stat. Ser. Optim. 7, 587–594 (1976)

    Google Scholar 

  37. Stoyan, D.: A continuity theorem for queue size. Bull. Acad. Sci. Polon. 21, 1143–1146 (1973)

    Google Scholar 

  38. Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, Chichester (1983). Translation from the German edited by Daryl J. Daley

    Google Scholar 

  39. Takagi, H.: Queueing Analysis, Vol. 1: Vacation and Priority Systems. North-Holland, Amsterdam (1991)

    Google Scholar 

  40. Takahashi, Y.: An approximation formula for the mean waiting time of an M/G/c queue. J. Oper. Res. Soc. Jpn. 20, 147–157 (1977)

    Google Scholar 

  41. Thorisson, H.: The queue GI/GI/k: finite moments of the cycle variables and uniform rates of convergence. Commun. Stat. Stoch. Models 1(2), 221–238 (1985)

    Article  Google Scholar 

  42. van Doorn, E.A., Regterschot, J.K.: Conditional PASTA. Oper. Res. Lett. 7, 229–232 (1988)

    Article  Google Scholar 

  43. Whitt, W.: A diffusion approximation for the G/GI/n/m queue. Oper. Res. 52, 922–941 (2004)

    Article  Google Scholar 

  44. Whitt, W.: The effect of variability in the GI/G/s queue. J. Appl. Probab. 17, 1062–1071 (1980)

    Article  Google Scholar 

  45. Whitt, W.: Comparison conjectures about the M/G/s queue. OR Lett. 2(5), 203–209 (1983)

    Google Scholar 

  46. Whitt, W.: On approximations for queues, I: Extremal distributions. AT&T Bell Lab. Tech. J. 63, 115–138 (1984)

    Google Scholar 

  47. Whitt, W.: Approximations for the GI/G/m queue. Prod. Oper. Manag. 2(2), 114–161 (1993)

    Google Scholar 

  48. Yao, D.D.: Refining the diffusion approximation for the M/G/m queue. Oper. Res. 33, 1266–1277 (1985)

    Article  Google Scholar 

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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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