Queueing Systems

, 68:339

On Markov–Krein characterization of the mean waiting time in M/G/K and other queueing systems

Article

Abstract

We propose a new research direction to reinvigorate research into better understanding of the M/G/K and other queueing systems—via obtaining tight bounds on the mean waiting time as functions of the moments of the service distribution. Analogous to the classical Markov–Krein theorem, we conjecture that the bounds on the mean waiting time are achieved by service distributions corresponding to the upper/lower principal representations of the moment sequence. We present analytical, numerical, and simulation evidence in support of our conjectures.

Keywords

M/G/K Moments Bounds Multi-server systems Markov-Krein theorem Tchebycheff system 

Mathematics Subject Classification (2000)

60K25 

References

  1. 1.
    Bertsimas, D., Natarajan, K.: A semidefinite optimization approach to the steady-state analysis of queueing systems. Queueing Syst. 56(1), 27–39 (2007) CrossRefGoogle Scholar
  2. 2.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: A convex optimization approach. SIAM J. Optim. 15, 780–804 (2005) CrossRefGoogle Scholar
  3. 3.
    Breuer, L.: Transient and stationary distributions for the GI/G/k queue with Lebesgue-dominated inter-arrival time distribution. Queueing Syst. 45(1), 47–57 (2003) CrossRefGoogle Scholar
  4. 4.
    Burman, D., Smith, D.: A light-traffic theorem for multi-server queues. Math. Oper. Res. 8, 15–25 (1983) CrossRefGoogle Scholar
  5. 5.
    Daley, D., Rolski, T.: Some comparability results for waiting times in single- and many-server queues. J. Appl. Probab. 21, 887–900 (1984) CrossRefGoogle Scholar
  6. 6.
    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, Boca Raton, FL, USA, pp. 35–59 (1997) Google Scholar
  7. 7.
    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) CrossRefGoogle Scholar
  8. 8.
    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) CrossRefGoogle Scholar
  9. 9.
    de Smit, J.H.A.: The queue GI/H m/s in continuous time. J. Appl. Probab. 22(1), 214–222 (1985) CrossRefGoogle Scholar
  10. 10.
    Eckberg, A. Jr.: Sharp bounds on Laplace–Stieltjes transforms, with applications to various queueing problems. Math. Oper. Res. 2(2), 132–142 (1977) CrossRefGoogle Scholar
  11. 11.
    Foss, S., Korshunov, D.: Heavy tails in multi-server queue. Queueing Syst. 52(1), 31–48 (2006) CrossRefGoogle Scholar
  12. 12.
    Gamarnik, D., Momčilović, P.: Steady-state analysis of a multiserver queue in the Halfin–Whitt regime. Adv. Appl. Probab. 40(2), 548–577 (2008) CrossRefGoogle Scholar
  13. 13.
    Gans, N., Koole, G., Mandelbaum, A.: Telephone call centers: tutorial, review, and research prospects. Manuf. Serv. Oper. Manag. 5, 79–141 (2003) CrossRefGoogle Scholar
  14. 14.
    Gupta, V., Dai, J., Harchol-Balter, M., Zwart, B.: On the inapproximability of M/G/K: why two moments of job size distribution are not enough. Queueing Syst. 64(1), 5–48 (2010) CrossRefGoogle Scholar
  15. 15.
    Gupta, V., Osogami, T.: On Markov-Krein characterization of mean sojourn time in queueing systems. Technical Report CMU-CS-11-109, School of Computer Science, Carnegie Mellon University (2011) Google Scholar
  16. 16.
    Hokstad, P.: Approximations for the M/G/m queue. Oper. Res. 26(3), 510–523 (1978) CrossRefGoogle Scholar
  17. 17.
    Hokstad, P.: The steady state solution of the M/K 2/m queue. Adv. Appl. Probab. 12(3), 799–823 (1980) CrossRefGoogle Scholar
  18. 18.
    Karlin, S., Studden, W.J.: Tchebycheff Systems: With Applications in Analysis and Statistics. Wiley, New York (1966) Google Scholar
  19. 19.
    Kimura, T.: Diffusion approximation for an M/G/m queue. Oper. Res. 31, 304–321 (1983) CrossRefGoogle Scholar
  20. 20.
    Kingman, J.: Inequalities in the theory of queues. J. R. Stat. Soc. 32(1), 102–110 (1970) Google Scholar
  21. 21.
    Köllerström, J.: Heavy traffic theory for queues with several servers. I. J. Appl. Probab. 11, 544–552 (1974) CrossRefGoogle Scholar
  22. 22.
    Lee, A., Longton, P.: Queueing process associated with airline passenger check-in. Oper. Res. Q. 10, 56–71 (1959) CrossRefGoogle Scholar
  23. 23.
    Ma, B., Mark, J.: Approximation of the mean queue length of an M/G/c queueing system. Oper. Res. 43(1), 158–165 (1995) CrossRefGoogle Scholar
  24. 24.
    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) CrossRefGoogle Scholar
  25. 25.
    Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley Series in Probability and Statistics. Wiley, Chichester (2002) Google Scholar
  26. 26.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover, New York (1995), revised edn. Google Scholar
  27. 27.
    Nozaki, S., Ross, S.: Approximations in finite-capacity multi-server queues with Poisson arrivals. J. Appl. Probab. 15(4), 826–834 (1978) CrossRefGoogle Scholar
  28. 28.
    Osogami, T., Raymond, R.: Semidefinite optimization for transient analysis of queues. ACM SIGMETRICS Perform. Eval. Rev. 38(1), 363–364 (2010) CrossRefGoogle Scholar
  29. 29.
    Scheller-Wolf, A., Sigman, K.: New bounds for expected delay in FIFO GI/GI/c queues. Queueing Syst. 26(1–2), 169–186 (1997) CrossRefGoogle Scholar
  30. 30.
    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) CrossRefGoogle Scholar
  31. 31.
    Stoyan, D.: Approximations for M/G/s queues. Math. Oper.forsch. Stat., Ser. Optim. 7, 587–594 (1976) Google Scholar
  32. 32.
    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 D.J. Daley Google Scholar
  33. 33.
    Tijms, H.C., Van Hoorn, M.H., Federgruen, A.: Approximations for the steady-state probabilities in the M/G/c queue. Adv. Appl. Probab. 13, 186–206 (1981) CrossRefGoogle Scholar
  34. 34.
    Whitt, W.: The effect of variability in the GI/G/s queue. J. Appl. Probab. 17, 1062–1071 (1980) CrossRefGoogle Scholar
  35. 35.
    Whitt, W.: Comparison conjectures about the M/G/s queue. Oper. Res. Lett. 2(5), 203–209 (1983) CrossRefGoogle Scholar
  36. 36.
    Whitt, W.: On approximations for queues, I: Extremal distributions. AT&T Bell Lab. Tech. J. 63, 115–138 (1984) Google Scholar
  37. 37.
    Whitt, W.: Approximations for the GI/G/m queue. Prod. Oper. Manag. 2(2), 114–161 (1993) CrossRefGoogle Scholar
  38. 38.
    Whitt, W.: A diffusion approximation for the G/GI/n/m queue. Oper. Res. 52, 922–941 (2004) CrossRefGoogle Scholar
  39. 39.
    Wolff, R.W.: Stochastic Modeling and the Theory of Queues. Prentice Hall, New York (1989) Google Scholar
  40. 40.
    Yao, D.: Refining the diffusion approximation for the M/G/m queue. Oper. Res. 33, 1266–1277 (1985) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.IBM Research—TokyoKanagawaJapan

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