Queueing Systems

, Volume 56, Issue 1, pp 27–39 | Cite as

A semidefinite optimization approach to the steady-state analysis of queueing systems



Computing the steady-state distribution in Markov chains for general distributions and general state space is a computationally challenging problem. In this paper, we consider the steady-state stochastic model \(\boldsymbol {W}\stackrel{d}{=}g(\boldsymbol {W},\boldsymbol {X})\) where the equality is in distribution. Given partial distributional information on the random variables X, we want to estimate information on the distribution of the steady-state vector W. Such models naturally occur in queueing systems, where the goal is to find bounds on moments of the waiting time under moment information on the service and interarrival times. In this paper, we propose an approach based on semidefinite optimization to find such bounds. We show that the classical Kingman’s and Daley’s bounds for the expected waiting time in a GI/GI/1 queue are special cases of the proposed approach. We also report computational results in the queueing context that indicate the method is promising.


Steady-state distribution Waiting time Semidefinite optimization 

Mathematics Subject Classification (2000)

60K25 90C22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003) Google Scholar
  2. 2.
    Bertsimas, D., Natarajan, K., Teo, C.P.: Probabilistic combinatorial optimization: moments, semidefinite programming and asymptotic bounds. SIAM J. Optim. 15(1), 185–209 (2004) CrossRefGoogle Scholar
  3. 3.
    Bertsimas, D., Natarajan, K., Teo, C.P.: Persistence in discrete optimization under data uncertainty. Math. Program. Ser. B 108(2), 251–274 (2006) CrossRefGoogle Scholar
  4. 4.
    Bertsimas, D., Popescu, I.: On the relation between option and stock prices: a convex optimization approach. Oper. Res. 50, 358–374 (2002) CrossRefGoogle Scholar
  5. 5.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005) CrossRefGoogle Scholar
  6. 6.
    Borovkov, A.A., Foss, S.G.: Stochastically recursive sequences and their generalizations. Sib. Adv. Math. 2, 16–81 (1992) Google Scholar
  7. 7.
    Daley, D.J.: Inequalities for moments of tails of random variables with queueing applications. Z. Wahrscheinlichkeitstheor. Verw. Geb. 41, 139–143 (1977) CrossRefGoogle Scholar
  8. 8.
    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, pp. 35–59. CRC Press, Boca Raton (1997) Google Scholar
  9. 9.
    Kiefer, J., Wolfowitz, J.: On the theory of queues with many servers. Trans. Am. Math. Soc. 78, 1–18 (1955) CrossRefGoogle Scholar
  10. 10.
    Kingman, J.F.C.: Some inequalities for the GI/G/1 queue. Biometrika 49, 315–324 (1962) Google Scholar
  11. 11.
    Kingman, J.F.C.: Inequalities in the theory of queues. J. Roy. Stat. Soc. Ser. B 32, 102–110 (1970) Google Scholar
  12. 12.
    Lasserre, J.B.: Bounds on measures satisfying moment conditions. Ann. Appl. Probab. 12, 1114–1137 (2002) CrossRefGoogle Scholar
  13. 13.
    Lasserre, J.B., Hernández-Lerma, O.: Markov Chains and Invariant Probabilities. Progress in Mathematics, vol. 211. Birkhäuser, Basel (2003) Google Scholar
  14. 14.
    Lindley, D.V.: On the theory of queues with a single server. Proc. Camb. Philos. Soc. 48, 277–289 (1952) CrossRefGoogle Scholar
  15. 15.
    Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002) Google Scholar
  16. 16.
    SeDuMi version 1.1. Available from http://sedumi.mcmaster.ca/
  17. 17.
    Wolff, R.W., Wang, C.L.: Idle period approximations and bounds for the GI/GI/1 queue. Adv. Appl. Probab. 35, 773–792 (2003) CrossRefGoogle Scholar
  18. 18.
    Zuluaga, L., Pena, J.F.: A conic programming approach to generalized Tchebycheff inequalities. Math. Oper. Res. 30(2), 369–388 (2005) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Sloan School of Management and Operations Research CenterMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore

Personalised recommendations